Concentration Compactness for the Critical Maxwell-Klein-Gordon Equation
aa r X i v : . [ m a t h . A P ] N ov CONCENTRATION COMPACTNESS FOR THE CRITICALMAXWELL-KLEIN-GORDON EQUATION
JOACHIM KRIEGER AND JONAS L ¨UHRMANNA bstract . We prove global regularity, scattering and a priori bounds for the energy critical Maxwell-Klein-Gordon equation relative to the Coulomb gauge on (1 + / rigidity argument by Kenig-Merle [10], following the method developed by the firstauthor and Schlag [20] in the context of critical wave maps. C ontents
1. Introduction 12. Function spaces and technical preliminaries 133. Microlocalized magnetic wave equation 164. Breakdown criterion 355. A concept of weak evolution 366. How to arrive at the minimal energy blowup solution 537. Concentration compactness step 548. Rigidity argument 114References 1581. I ntroduction
The Maxwell-Klein-Gordon system on Minkowski space-time R + n , n ≥
1, is a classical fieldtheory for a complex scalar field φ : R + n → C and a connection 1-form A α : R + n → R for α = , , . . . , n . Defining the covariant derivative D α = ∂ α + iA α and the curvature 2-form F αβ = ∂ α A β − ∂ β A α , the formal Lagrangian action functional of the Maxwell-Klein-Gordon system is given by Z R + n (cid:16) F αβ F αβ + D α φ D α φ (cid:17) dx dt , The first author was supported in part by the Swiss National Science Foundation under Consolidator GrantBSCGI0 157694. The second author was supported in part by the Swiss National Science Foundation under grantSNF 200020-159925. where the Einstein summation convention is in force and Minkowski space R + n is endowed withthe standard metric diag( − , + , . . . , + ∂ β F αβ = Im (cid:0) φ D α φ (cid:1) , (cid:3) A φ = , where (cid:3) A = D α D α is the covariant d’Alembertian. The system has two important features. First, itenjoys the gauge invariance A α A α − ∂ α γ, φ e i γ φ for any suitably regular function γ : R + n → R . Second, it is Lorentz invariant . Moreover, thesystem admits a conserved energy(1.2) E ( A , φ ) : = Z R n (cid:16) X α,β F αβ + X α (cid:12)(cid:12)(cid:12) D α φ (cid:12)(cid:12)(cid:12) (cid:17) dx . Given that the system of equations (1.1) is invariant under the scaling transformation A α ( t , x ) → λ A α ( λ t , λ x ) , φ ( t , x ) → λφ ( λ t , λ x ) for λ > , one distinguishes between the energy sub-critical case corresponding to n ≤
3, the energy criticalcase for n =
4, and the energy super-critical case for n ≥
5. To the best of the authors’ knowledge,at this point no methods are available to prove global regularity for large data for super-criticalnonlinear dispersive equations. The most advanced results for large data can be achieved for criticalequations.Imposing the
Coulomb condition P nj = ∂ j A j = A , the Maxwell-Klein-Gordon system decouples into a system of wave equations for the dynamical variables ( A j , φ ), j = , . . . , n , coupled to an elliptic equation for the temporal compo-nent A ,(MKG-CG) (cid:3) A j = −P j Im (cid:0) φ D x φ (cid:1) , (cid:3) A φ = , ∆ A = − Im (cid:0) φ D φ (cid:1) , where P is the standard projection onto divergence free vector fields.We observe that in the formulation (MKG-CG), the components ( A j , φ ), j = , . . . , n , implicitlycompletely describe ( A α , φ ), since the missing component A is uniquely determined by the ellipticequation(1.3) ∆ A = − Im (cid:0) φ∂ t φ (cid:1) + | φ | A . For this reason, we will mostly work in terms of the dynamical variables ( A x , φ ), it being understoodthat required bounds on A can be directly inferred from (1.3). In particular, to describe initial datafor (MKG-CG), we will use the notation A j [0] : = ( A j , ∂ t A j )(0 , · ) and φ [0] : = ( φ, ∂ t φ )(0 , · ). Often,we will simply denote these by ( A x , φ )[0].The present work will give a complete analysis of the energy critical case n =
4. More precisely,we implement an analysis closely analogous to the one by the first author and Schlag [20] in thecontext of critical wave maps in order to prove existence, scattering and a priori bounds for largeglobal solutions to (MKG-CG). Moreover, we establish a concentration compactness phenomenon,which describes a kind of “atomic decomposition” of sequences of solutions of bounded energy.
ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 3
To formulate our main result, we introduce the following notion of admissible data for the evo-lution problem (MKG-CG) on R + . Definition 1.1.
We call C ∞ -smooth data ( A x , φ )[0] admissible, provided A x [0] satisfy the Coulombcondition and φ [0] as well as all spatial curvature components F jk [0] are Schwartz class. Moreover,we require that for j = , . . . , , (cid:12)(cid:12)(cid:12) A j [0]( x ) (cid:12)(cid:12)(cid:12) . h x i − as | x | → ∞ . In particular, admissible data are of class H sx ( R ) × H s − x ( R ) for any s ≥ Theorem 1.2.
Consider the evolution problem (MKG-CG) on R + . There exists a functionK : (0 , ∞ ) −→ (0 , ∞ ) with the following property. Let ( A x , φ )[0] be an admissible Coulomb class data set such that thecorresponding full set of components ( A α , φ ) has energy E. Then there exists a unique global in timeadmissible solution ( A , φ ) to (MKG-CG) with initial data ( A x , φ )[0] that satisfies for any q + r ≤ with ≤ q ≤ ∞ , ≤ r < ∞ , γ = − q − r the following a priori bound (1.4) (cid:13)(cid:13)(cid:13)(cid:0) ( − ∆ ) − γ ∇ t , x A x , ( − ∆ ) − γ ∇ t , x φ (cid:1)(cid:13)(cid:13)(cid:13) L qt L rx ( R × R ) ≤ C r K ( E ) . The solution scatters in the sense that there exist finite energy free waves f j and g, (cid:3) f j = (cid:3) g = ,such that for j = , . . . , , lim t → + ∞ (cid:13)(cid:13)(cid:13) ∇ t , x A j ( t , · ) − ∇ t , x f j ( t , · ) (cid:13)(cid:13)(cid:13) L x ( R ) = , lim t → + ∞ (cid:13)(cid:13)(cid:13) ∇ t , x φ ( t , · ) − ∇ t , x g ( t , · ) (cid:13)(cid:13)(cid:13) L x ( R ) = , and analogously with di ff erent free waves for t → −∞ . In fact, we will prove the significantly stronger a priori bound (cid:13)(cid:13)(cid:13) ( A x , φ ) (cid:13)(cid:13)(cid:13) S ( R × R ) ≤ K ( E ) , where the precise definition of the S norm will be introduced in Section 2. The purpose of thisnorm is to control the regularity of the solutions.Recently, a proof of the global regularity and scattering a ffi rmations in the preceding theoremwas obtained by Oh-Tataru [32–34], following the method developed by Sterbenz-Tataru [40, 41]in the context of critical wave maps. Our conclusions were reached before the appearance of theirwork and our methods are completely independent.1.1. A history of the problem.
In this subsection we first consider this work in the broader contextof the study of the local and global in time behavior of nonlinear wave equations and highlightsome of the important developments over the last decades that crucially enter the proof of ourmain theorem. Afterwards we give an overview of previous results on the Maxwell-Klein-Gordonequation.
Null structure.
In many geometric wave equations like the wave map equation, the Maxwell-Klein-Gordon equation, and the Yang-Mills equation, the nonlinearities exhibit so-called null structures.Heuristically speaking, such null structures are amenable to better estimates, because they damp the
CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION interactions of parallel waves. The key role that these special nonlinear structures play in the globalregularity theory of nonlinear waves was first highlighted by Klainerman [11]. At that point thetheory of nonlinear wave equations relied for the most part on vector field methods and parametricesin physical space. However, in more recent times the key role that null structures play also within themore technical harmonic analysis approach cannot be overstated. In fact, in [12] a whole programwith precise conjectures pertaining to the sharp well-posedness of a number of nonlinear waveequations with null structure was outlined. The present work may be seen as a further vindicationof the program outlined by Klainerman. Null structures also play a pivotal role in the much morecomplex system of Einstein equations, as evidenced for example in the recent deep sequence ofworks by Klainerman-Rodnianski-Szeftel [17] and Szeftel [44–48] on the bounded L curvatureconjecture. Function spaces.
The development of X s , b spaces by Klainerman-Machedon in the seminal works[13–16] in the low regularity study of nonlinear wave equations provided a powerful tool to takeadvantage of the null structures in geometric wave equations. The fact that the Maxwell-Klein-Gordon and Yang-Mills equations actually display a null structure in the Coulomb gauge was a keyobservation in these works. Moreover, the observation by Klainerman-Machedon that these nullstructures are beautifully compatible with the X s , b functional framework has been highly influentialever since. Di ff erent variants of the X s , b spaces were independently introduced by Bourgain [2] inthe context of the nonlinear Schr¨odinger equation and the Korteweg-de Vries equation.In the quest to prove global regularity for critical wave maps for small initial data it turned out thatnot even the strongest versions of the critical X s , b type spaces yielded good algebra type estimates.This problem was resolved through the development of the null frame spaces in the breakthroughwork of Tataru [52]. We will introduce these spaces later on, see also [51], [53], and [20] for morediscussion. Renormalization.
The key di ffi culty for the (MKG-CG) equation is the equation (cid:3) A φ =
0, which inexpanded form is given by (cid:3) φ = − iA α ∂ α φ + i ( ∂ t A ) φ + A α A α φ. The contribution of the low-high frequency interactions in the magnetic interaction term (1.5) − iA j ∂ j φ turns out to be non-perturbative in the case when the spatial components of the connection formare just free waves. This problem already occurs for small data and is not only a large data issue.One encounters a similar situation in the wave map equation. In the breakthrough works [50, 51]Tao exploited the intrinsic gauge freedom for the wave maps problem to recast the nonlinearity intoa perturbative form. However, for the (MKG-CG) equation the gauge invariance is already spent.Rodnianski and Tao [35] found a way out of this impasse by incorporating the non-perturbativeterm into the linear operator and by deriving Strichartz estimates for the resulting wave operator viaa parametrix construction. This enabled them to prove global regularity for (MKG-CG) for smallcritical Sobolev data in n ≥ n = X s , b type and null frame spaces. The functional calculus from [22] for the paradi ff erentialmagnetic wave operator (cid:3) pA = (cid:3) + i X k ∈ Z P < k − C A f reej P k ∂ j , ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 5 where P k denotes the standard Littlewood-Paley projections and A f reej , j = , . . . ,
4, are free waves,plays an important role in this work and has to be adapted to the large data setting.
Bahouri-G´erard concentration compactness decomposition.
Bahouri and G´erard [1] proved thefollowing description of sequences of solutions to the free wave equation with uniformly boundedenergy.
Let { ( ϕ n , ψ n ) } n ≥ ⊂ ˙ H x ( R ) × L x ( R ) be a bounded sequence and let v n be the solution to thefree wave equation (cid:3) v n = on R × R with initial data ( v n , ∂ t v n ) | t = = ( ϕ n , ψ n ) . Then thereexists a subsequence { v ′ n } n ≥ of { v n } n ≥ and finite energy free waves V ( j ) as well as sequences (cid:8) ( λ ( j ) n , t ( j ) n , x ( j ) n ) (cid:9) n ≥ ⊂ R + × R × R , j ≥ , such that for every l ≥ , (1.6) v ′ n ( t , x ) = l X j = q λ ( j ) n V ( j ) (cid:18) t − t ( j ) n λ ( j ) n , x − x ( j ) n λ ( j ) n (cid:19) + w ( l ) n ( t , x ) and lim l →∞ lim sup n →∞ (cid:13)(cid:13)(cid:13) w ( l ) n (cid:13)(cid:13)(cid:13) L t L x ( R × R ) = . Moreover, there is asymptotic decoupling of the free energy (cid:13)(cid:13)(cid:13) ∇ t , x v ′ n (cid:13)(cid:13)(cid:13) L x = l X j = (cid:13)(cid:13)(cid:13) ∇ t , x V ( j ) (cid:13)(cid:13)(cid:13) L x + (cid:13)(cid:13)(cid:13) ∇ t , x w ( l ) n (cid:13)(cid:13)(cid:13) L x + o (1) as n → ∞ , and for each j , k, we have the asymptotic orthogonality property (1.7) lim n →∞ λ ( j ) n λ ( k ) n + λ ( k ) n λ ( j ) n + | t ( j ) n − t ( k ) n | λ ( j ) n + | x ( j ) n − x ( k ) n | λ ( j ) n = ∞ . The free waves V ( j ) are referred to as concentration profiles and the importance of the linear profiledecomposition (1.6) is that it captures the failure of compactness of the sequence of bounded solu-tions { v n } n ≥ to the free wave equation in terms of the non-compact symmetries of the equation andthe superposition of profiles. Simultaneously to Bahouri and G´erard, Merle and Vega [28] obtainedsimilar concentration compactness decompositions in the context of the mass-critical nonlinearSchr¨odinger equation. This is very analogous to the concentration compactness method originallydeveloped in the context of elliptic equations, see e.g. [23–26] and [43] for a discussion of theoriginal works.Bahouri and G´erard also established an analogous nonlinear profile decomposition for Shatah-Struwe solutions { u n } n ≥ to the energy critical defocusing nonlinear wave equation (cid:3) u n = u n on R × R , see [38], with the same initial data ( u n , ∂ t u n ) | t = = ( ϕ n , ψ n ). Their main application of thisnonlinear profile decomposition was to prove the existence of a function A : (0 , ∞ ) → (0 , ∞ ) withthe property that for any Shatah-Struwe solution u to (cid:3) u = u it holds that k u k L t L x ( R × R ) ≤ A (cid:0) E ( u ) (cid:1) , where E ( u ) denotes the energy functional associated with the quintic nonlinear wave equation.The Bahouri-G´erard profile decomposition is of fundamental importance for the Kenig-Merlemethod that we will describe in the next paragraph. In the proof of our main theorem we will haveto study sequences of solutions to the (MKG-CG) equation with uniformly bounded energy. Akey step will be to obtain an analogous Bahouri-G´erard profile decomposition for such sequences.This poses significant problems, which can be heuristically understood as follows. Very roughlyspeaking, the reason why the Bahouri-G´erard profile decomposition “works” for the energy critical CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION nonlinear wave equation (cid:3) u = u is that in the quintic nonlinearity the interaction of two nonlinearconcentration profiles living at asymptotically orthogonal frequency scales vanishes. This reducesto consider diagonal frequency interactions of the concentration profiles in the nonlinearity, but thenthese profiles must essentially be supported in di ff erent regions of space-time due to the asymptoticorthogonality property (1.7) and therefore do not interact strongly. In contrast, for the (MKG-CG) equation frequency diagonalization appears to fail in the di ffi cult magnetic interaction term(1.5) for low-high interactions. A similar situation occurs for critical wave maps. In the lattercontext the first author and Schlag [20] devised a novel profile decomposition to take into accountthe corresponding low-high frequency interactions. Our approach is strongly influenced by [20],but we will have to use a slightly di ff erent “covariant” wave operator to extract the concentrationprofiles, see the discussion in the next subsection. The Kenig-Merle method.
Kenig and Merle [9, 10] introduced a very general method to proveglobal well-posedness and scattering for critical nonlinear dispersive and wave equations in bothdefocusing and focusing cases, in the latter case only for energies strictly less than the ground stateenergy. Their approach has found a vast amount of applications over the last years. We illustratethe method in the context of the energy critical defocusing nonlinear wave equation (cid:3) u = u on R × R . One can use the L t L x ( R × R ) norm to control the regularity of solutions to this equationand easily prove local well-posedness and small data global well-posedness based on this norm.In the first step of the Kenig-Merle method one assumes that global well-posedness and scatteringfails for some finite energy level. Then let E crit be the critical energy below which all solutions existglobally in time with finite L t L x ( R × R ) bounds, in particular it must hold that E crit >
0. Thus, wefind a sequence of solutions { u n } n ≥ such that E ( u n ) → E crit and k u n k L t L x → ∞ as n → ∞ . Applyingthe Bahouri-G´erard profile decomposition to { u n (0) } n ≥ , we may conclude by the minimality of E crit that there exists exactly one profile in the decomposition. This enables us to extract a minimalblowup solution u C of lifespan I with E ( u C ) = E crit and k u C k L t L x ( I × R ) = ∞ . Moreover, we can infera crucial compactness property of u C , namely that there exist continuous functions x : I → R and λ : I → R + such that the family of functions ((cid:18) λ ( t ) u C (cid:18) t , · − x ( t ) λ ( t ) (cid:19) , λ ( t ) ∂ t u C (cid:18) t , · − x ( t ) λ ( t ) (cid:19)(cid:19) : t ∈ I ) is pre-compact in ˙ H x ( R ) × L x ( R ). The second step of the Kenig-Merle method is a rigidity argu-ment to rule out the existence of such a minimal blowup solution u C by combining the compactnessproperty with conservation laws and other identities of virial or Morawetz type for the energy criti-cal nonlinear wave equation. We will adapt the Kenig-Merle method to the Maxwell-Klein-Gordonequation.We now review previous results on the Maxwell-Klein-Gordon equation. The existence of globalsmooth solutions to the Maxwell-Klein-Gordon equation in n = n = ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 7 also Selberg-Tesfahun [37] for a finite energy global well-posedness result for the Maxwell-Klein-Gordon equation in n = n = n = n ≥ ffi cult magnetic interaction term in (MKG-CG) into the linearwave operator and to derive Strichartz estimates for the resulting wave operator via a parametrixconstruction.Small energy global well-posedness of the energy critical Maxwell-Klein-Gordon equation in n = n = ff erent proof was later obtained by Oh [30, 31], using the Yang-Mills heat flow. Global regular-ity for the Yang-Mills system for small critical Sobolev data for n ≥ (cid:3) u = u on R + and for radial critical wave mapson R + . At this point, with the exception of some special problems, it appears that a general largedata result cannot be inferred by using the small data result as a black box, but instead requires amore or less complete re-working of the small data theory. See, for instance, the works on criticallarge data wave maps [40, 41], [20], [49]. Our approach here is to implement a similar strategy asthe one by the first author and Schlag [20] for critical wave maps, which consists of essentially twosteps. First, a novel “covariant” Bahouri-G´erard procedure to take into account the non-negligibleinfluence of low on high frequencies in the magnetic interaction term. Second, an implementation ofa variant of a concentration compactness / rigidity argument by Kenig-Merle, following more or lessthe sequence of steps in [10]. As the latter was introduced in the context of a scalar wave equation,and we are considering a complex nonlinearly linked system, we believe that the implementation ofthis step for the energy critical Maxwell-Klein-Gordon equation is also of interest in its own right.We expect that our methods extend to prove global regularity, scattering and a priori bounds forthe energy critical Yang-Mills equations in n = Overview of the proof.
In this subsection we give a detailed overview of the proof of Theo-rem 1.2. In fact, the purpose of this paper is to prove a significantly stronger version of Theorem 1.2,
CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION namely the existence of a function K : (0 , ∞ ) → (0 , ∞ ) with (cid:13)(cid:13)(cid:13) ( A x , φ ) (cid:13)(cid:13)(cid:13) S ( R × R ) ≤ K ( E ) , E = E ( A , φ )for any admissible solution ( A , φ ) to (MKG-CG). Once this a priori bound is known, one also obtainsthe scattering assertion in Theorem 1.2. The fact that the dynamical variables of a global admissiblesolution scatter to finite energy free waves, and not to solutions to a suitable linear magnetic waveequation, crucially relies on our strong spatial decay assumptions about the data, see the proof ofscattering at the end of Section 8. The precise definition of the S space and its time localizedversion will be given in Section 2 and Definition 4.1.Beginning the argument at this point, we assume that the existence of such a function K fails forsome finite energy level. Thus, the set of energies E : = n E > ( A ,φ ) admissible E ( A ,φ ) ≤ E (cid:13)(cid:13)(cid:13) ( A x , φ ) (cid:13)(cid:13)(cid:13) S = + ∞ o is non-empty. In view of the small energy global well-posedness result [22], it therefore has apositive infimum, which we denote by E crit , E crit : = inf E . By definition we can then find a sequence of admissible solutions { ( A n , φ n ) } n ≥ to (MKG-CG) suchthat E ( A n , φ n ) → E crit , lim n →∞ (cid:13)(cid:13)(cid:13) ( A nx , φ n ) (cid:13)(cid:13)(cid:13) S = + ∞ . As in [20], we call such a sequence an essentially singular sequence . The goal of this paper is torule out the existence of such an object. This will be accomplished in broad strokes by the followingtwo steps.(1) Extracting an energy class minimal blowup solution (cid:0) A ∞ , Φ ∞ (cid:1) to (MKG-CG) with thecompactness property via a modified Bahouri-G´erard procedure, which consists of an in-ductive sequence of low-frequency approximations and a profile extraction process takinginto account the e ff ect of the magnetic potential interaction. Here we closely follow theprocedure introduced by the first author and Schlag [20], but we have to subtly divergefrom the profile extraction process there to correctly capture the asymptotic evolution ofthe atomic components. We note that the heart of the modified Bahouri-G´erard procedureresides in Section 7.(2) Ruling out the existence of the minimal blowup solution (cid:0) A ∞ , Φ ∞ (cid:1) by essentially follow-ing the method of Kenig and Merle [10]. This step is carried out in Section 8.We now describe these steps in more detail. A concept of weak evolution for energy class data.
In order to extract a minimal blowup solu-tion at the end of the modified Bahouri-G´erard procedure, we first need to introduce the notionof a solution to (MKG-CG) that is merely of energy class. A natural idea here is to approximatea given Coulomb energy class datum by a sequence of admissible data and to define the energyclass solution to (MKG-CG) as a suitable limit of the admissible solutions. One then needs a goodperturbation theory to show that this limit is well-defined and independent of the approximatingsequence. Unfortunately, there is not such a strong perturbation theory for (MKG-CG) as for in-stance for critical wave maps in [20] due to a low frequency divergence. However, the problem withevolving irregular data is really a “high frequency issue” and in Proposition 5.1 we show that thereis a good perturbation theory for perturbing frequency localized data by adding high frequency per-turbations. We can then define the evolution of a Coulomb energy class datum as a suitable limit
ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 9 of the evolutions of low frequency approximations of the energy class datum, provided these lowfrequency approximations exist on some joint time interval and satisfy uniform S norm boundsthere. This is achieved in Proposition 5.2 via a suitable method of localizing the data and exploitinga version of Huygens’ principle together with the gauge invariance of the Maxwell-Klein-Gordonsystem. This step is additionally complicated by the fact that the (MKG-CG) equation does nothave the finite speed of propagation property due to non-local terms in the equation for the spatialcomponents of the connection form A . Bahouri-G´erard I: Filtering out frequency atoms and evolving the “non-atomic” lowest frequencyapproximation.
The extraction of the energy class minimal blowup solution (cid:0) A ∞ , Φ ∞ (cid:1) consists ofa two step Bahouri-G´erard type procedure. This is carried out in Section 7 and forms the core ofour argument. In the first step we consider the initial data ( A nx , φ n )[0] at time t = A n , φ n ) and use a procedure due to M´etivier-Schochet [29] to extract frequencyscales. In what follows we will slightly abuse notation and write ( A n , φ n )[0] instead of ( A nx , φ n )[0].This yields the decompositions A n [0] = Λ X a = A na [0] + A n Λ [0] ,φ n [0] = Λ X a = φ na [0] + φ n Λ [0] , where the “frequency atoms” ( A na , φ na )[0] are essentially frequency localized to scales ( λ na ) − thattend apart as n → ∞ , more precisely lim n →∞ λ na λ na ′ + λ na ′ λ na = ∞ for a , a ′ , while the error ( A n Λ , φ n Λ )[0] satisfieslim sup n →∞ (cid:13)(cid:13)(cid:13) A n Λ [0] (cid:13)(cid:13)(cid:13) ˙ B , ∞ × ˙ B , ∞ + (cid:13)(cid:13)(cid:13) φ n Λ [0] (cid:13)(cid:13)(cid:13) ˙ B , ∞ × ˙ B , ∞ < δ for given δ > Λ = Λ ( δ ) su ffi ciently large. Moreover, we prepare these frequency atoms suchthat their frequency supports are sharply separated as n → ∞ and so that the errors (cid:8) ( A n Λ , φ n Λ )[0] (cid:9) n ≥ are supported away from the frequency scales ( λ na ) − in frequency space. Then we select a numberof “large” frequency atoms ( A na , φ na )[0], a = , . . . , Λ , whose energy E ( A na , φ na ) is above a certainsmall threshold ε depending only on E crit . We order these frequency atoms by the scale aroundwhich they are essentially supported starting with the lowest one.Eventually, we want to conclude that the essentially singular sequence of data (cid:8) ( A n , φ n )[0] (cid:9) n ≥ consists of exactly one non-trivial frequency atom that is composed of exactly one non-trivial phys-ical concentration profile of asymptotic energy E crit . We argue by contradiction and assume thatthis is not the case. Then the idea is to approximate the essentially singular sequence of initialdata ( A n , φ n )[0] by low frequency truncations, obtained by removing all or some of the atoms( A na , φ na )[0], a = , . . . , Λ , and to inductively derive bounds on the S norms of the (MKG-CG)evolutions of the truncated data. As this induction stops after Λ many steps, we obtain an a prioribound on the evolutions(1.8) lim inf n →∞ k ( A n , φ n ) k S < ∞ , contradicting the assumption that { ( A n , φ n ) } n ≥ is an essentially singular sequence of solutions to(MKG-CG). We observe that by construction the “non-atomic” errors ( A n Λ , φ n Λ )[0] are split into Λ + A na , φ na )[0], i.e. we can write A n Λ [0] = Λ + X j = A n j Λ [0] , φ n Λ [0] = Λ + X j = φ n j Λ [0] , where the first pieces ( A n Λ , φ n Λ )[0] have Fourier support in the region closest to the origin. InSubsection 7.3 we then derive a priori S norm bounds on the evolutions of the lowest frequencyapproximations ( A n Λ , φ n Λ )[0]. The problem here is that the pieces ( A n Λ , φ n Λ )[0] might still havelarge energy, which forces us to use a finite number of further delicately chosen low frequencyapproximations (cid:0) P J L A n Λ , P J L φ n Λ (cid:1) [0] of these pieces. Importantly, this number only depends on thesize of E crit . We then inductively obtain bounds on the S norms of the (MKG-CG) evolutions ofthe low frequency approximations (cid:0) P J L A n Λ , P J L φ n Λ (cid:1) [0] by bootstrap. This step is tied together inProposition 7.4. In particular, Step 3 of the proof of Proposition 7.4 is the core perturbative resultof this paper and is used in variations at other instances later on. Bahouri-G´erard II: Selecting concentration profiles and adding the first large frequency atom.
Hav-ing established control over the evolution of the lowest frequency “non-atomic” part ( A n Λ , φ n Λ )[0],we then add the first frequency atom ( A n , φ n )[0] and consider the evolution of the data (cid:0) A n Λ + A n , φ n Λ + φ n (cid:1) [0] . Here we first have to understand the lack of compactness of the functions ( A n , φ n )[0]. It is atthis point that we deviate most significantly from the standard Bahouri-G´erard profile extractionprocedure [1] and also the modified profile extraction procedure developed by the first author andSchlag [20] in the context of critical wave maps. We still extract the concentration profiles for thedata A n [0] using the standard Bahouri-G´erard extraction procedure. However, we evolve the data φ n [0] with respect to the following “covariant” wave operator e (cid:3) A n : = (cid:3) + i (cid:0) A n Λ ,ν + A n , f ree ν (cid:1) ∂ ν and extract the profiles as weak limits of these evolutions to take into account the strong low-highinteractions for (MKG-CG). Here, A n Λ ( t , x ) is the (MKG-CG) evolution of the low frequency data( A n Λ , φ n Λ )[0], while A n , f reej is the free wave evolution of the data A n j [0] for j = , . . . ,
4, and wesimply put A n , f ree =
0. In comparison with [1] and [20], a key di ffi culty in this step is that solutionsto the covariant linear wave equation e (cid:3) A n u = t abn , x abn ) for extracting the concentration profiles both for A n [0] and for φ n [0]. Oncethe profiles have been picked, we use them to construct approximate, but highly accurate, nonlinearprofiles in Theorem 7.14. To this end we solve the (MKG-CG) system in very large but finite space-time boxes centered around ( t abn , x abn ), using the concentration profiles as data, while outside of theseboxes, we use the free wave propagation for A and the “full” covariant wave operator (involving theinfluence of all other profiles) for φ . This is the same strategy as the one pursued for wave mapsin [20]. Provided that all concentration profiles have energy strictly less than E crit with respect tothe Maxwell-Klein-Gordon energy functional, we can then use our perturbation theory to constructthe global (MKG-CG) evolution of the data (cid:0) A n Λ + A n , φ n Λ + φ n (cid:1) [0] and to obtain a priori S normbounds. ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 11
Conclusion of the induction on frequency process.
We may then repeat the preceding steps and “addin” all remaining frequency atoms to conclude a priori global S norm bounds on the evolution ofthe full data ( A n , φ n )[0]. The conclusion of this induction on frequency process is that we arrive ata contradiction, unless the essentially singular sequence of data ( A n , φ n )[0] consists of exactly onefrequency atom that is composed of precisely one concentration profile of asymptotic energy E crit .Due to our relatively poor perturbation theory for (MKG-CG), it then still requires a fair amountof work to extract an energy class minimal blowup solution from this essentially singular sequence( A n , φ n ), see Section 6 and Subsection 7.6. Finally, in Theorem 7.23 we obtain an energy classminimal blowup solution ( A ∞ , Φ ∞ ) to (MKG-CG) with lifespan I and with the crucial compactnessproperty that there exist continuous functions x : I → R and λ : I → (0 , ∞ ) so that each of thefamily of functions ((cid:18) λ ( t ) A ∞ j (cid:18) t , · − x ( t ) λ ( t ) (cid:19) , λ ( t ) ∂ t A ∞ j (cid:18) t , · − x ( t ) λ ( t ) (cid:19)(cid:19) : t ∈ I ) for j = , . . . , ((cid:18) λ ( t ) Φ ∞ (cid:18) t , · − x ( t ) λ ( t ) (cid:19) , λ ( t ) ∂ t Φ ∞ (cid:18) t , · − x ( t ) λ ( t ) (cid:19)(cid:19) : t ∈ I ) is pre-compact in ˙ H x ( R ) × L x ( R ). The Kenig-Merle rigidity argument.
In the final Section 8, we rule out the existence of such aminimal blowup solution ( A ∞ , Φ ∞ ) with the compactness property by following the scheme of theKenig-Merle rigidity argument [10]. The idea is to infer from the compactness property and theminimal energy property of ( A ∞ , Φ ∞ ) the existence of either a static solution to (MKG-CG) or elsethe existence of a self-similar blowup solution to (MKG-CG) and to then exclude the existence ofboth of these objects.A crucial step in the Kenig-Merle rigidity argument is to conclude that the momentum of ( A ∞ , Φ ∞ )must vanish. The proof of this hinges on the relativistic invariance of the Maxwell-Klein-Gordonequation and the transformation behavior of the Maxwell-Klein-Gordon energy functional underLorentz transformations. This step is technically di ffi cult for the Maxwell-Klein-Gordon equation,because the S norm is much more complicated than the Strichartz norms used in [10].We then distinguish between the lifespan I of ( A ∞ , Φ ∞ ) being finite in at least one time directionor not. If I is infinite, we face the possibility of a static solution, which we rule out using virialtype identities for the Maxwell-Klein-Gordon equation, the vanishing momentum condition for( A ∞ , Φ ∞ ) and a Vitali covering argument from [20]. If instead I is finite at one end, we reduceto a self-similar blowup scenario. We then uncover a Lyapunov functional for solutions to theMaxwell-Klein-Gordon equation in self-similar variables. This is the key ingredient, which allowsus to also rule out this scenario. The derivation of this Lyapunov functional appears significantlymore complicated than in [10] or [20] and we use the trick of working in a Cronstrom-type gaugeto simplify the computations.1.3. Overview of the paper.
We now give an overview of the structure of this paper. The twomain steps of the proof of Theorem 1.2 are the modified Bahouri-G´erard procedure in Section 7and the rigidity argument in Section 8. The necessary technical preparations are carried out in thesections leading up to Section 7. • In Section 2 we lay out the functional framework following [22]. • In Section 3 we prove key estimates for the linear magnetic wave equation (cid:3) pA u = f . • In Section 4 we state the property of the S norm as a regularity controlling device. • In Section 5 we show how to unambiguously locally evolve Coulomb energy class data( A x , φ )[0] via approximation by smoothed data and truncation in physical space to reduceto the admissible setup. Here one needs to pay close attention to the fact that solutions to(MKG-CG) do not obey as good a perturbation theory with respect to the S spaces as,say, critical wave maps in a suitable gauge, due to a low frequency divergence. Hence,one needs to be very careful about the correct choice of smoothing, using low frequencytruncations of the data. Moreover, to ensure the existence of an energy class local evolutionof Coulomb energy class data ( A x , φ )[0] on a non-trivial time slice around t =
0, we needto prove uniform S norm bounds for the approximations, which we accomplish similarlyto the procedure in [20] via localization in physical space, see Proposition 5.2. We alsointroduce the concept of the “lifespan” of such an energy class solution and the definitionof its S norm. • In Section 6 we then state that energy class data ( A x , φ )[0] obtained as the limit of the dataof an essentially singular sequence, which will be the outcome of the modified Bahouri-G´erard procedure, lead to a singular solution ( A , φ ) in the sense thatsup J ⊂ I (cid:13)(cid:13)(cid:13) ( A x , φ ) (cid:13)(cid:13)(cid:13) S ( J × R ) = + ∞ , where I denotes its lifespan. The proof of this as well as a number of further technicalassertions will be relegated to Subsection 7.4. • In Section 7 we carry out the modified Bahouri-G´erard procedure. In Subsection 7.1 andSubsection 7.2 we extract the “frequency atoms” mimicking closely the procedure in [20].Then we show in Subsection 7.3 how the lowest frequency “non-atomic” part of the lowfrequency approximation induction can be globally evolved with good S norm bounds.In Subsection 7.4 we prove several technical assertions that all use the core perturbativeresult from Step 3 of the proof of Proposition 7.4. In Subsection 7.5, we add the first“large” frequency atom by extracting concentration profiles and invoking the inductionon energy hypothesis that all profiles have energy strictly less than E crit . The end resultof the modified Bahouri-G´erard procedure is obtained in Subsection 7.6, see in particularTheorem 7.23. We then have a minimal blowup solution ( A ∞ , Φ ∞ ) with the requiredcompactness property. • In Section 8 we rule out the existence of a minimal blowup solution ( A ∞ , Φ ∞ ) with thecompactness property. To this end we largely follow the scheme of the rigidity argumentby Kenig-Merle [10]. In Subsection 8.1 we derive several energy and virial identities forenergy class solutions to (MKG-CG). Then we prove some preliminary properties of theminimal blowup solution ( A ∞ , Φ ∞ ), in particular that its momentum must vanish. Denot-ing by I the lifespan of ( A ∞ , Φ ∞ ), we distinguish between I + : = I ∩ [0 , ∞ ) being a finite oran infinite time interval. In the next Subsection 8.2, we exclude the existence of a minimalblowup solution ( A ∞ , Φ ∞ ) with infinite time interval I + using the virial identities, the factthat the momentum of ( A ∞ , Φ ∞ ) must vanish and an additional Vitali covering argumentintroduced in [20]. Moreover, we reduce the case of finite lifespan I + to a self-similarblowup scenario. In the last Subsection 8.3, we then derive a suitable Lyapunov func-tional for the Maxwell-Klein-Gordon system in self-similar variables, which will enableus to also rule out the self-similar case and thus finishes the rigidity argument. Finally, weaddress the proof of the scattering assertion in Theorem 1.2. ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 13
We remark that we will often abuse notation and denote the spatial components A x of the con-nection form simply by A . Acknowledgments : The authors are grateful to the referee for valuable corrections and suggestions.2. F unction spaces and technical preliminaries
We will be working with the same function spaces that were used for the small data energycritical global well-posedness result for the MKG-CG system [22] together with their time-localizedversions. In this section we briefly recall their definitions. For a more detailed discussion of thesespaces we refer to Section 3 in [22] and [51], [40], [20].In this work we only rely on the precise fine structure of these spaces in that we frequently usethe multilinear estimates from [22] to reduce to “su ffi ciently generic” situations where a divisibilityargument works, i.e. when all inputs are approximately at the same frequency and have angularseparation between their frequency supports.In order to introduce various Littlewood-Paley projection operators, we pick a non-negative evenbump function ϕ ∈ C ∞ ( R ) satisfying ϕ ( y ) = | y | ≤ ϕ ( y ) = | y | > ϕ ( y ) = ϕ ( y ) − ϕ (2 y ). Then we define the standard Littlewood-Paley projection operators for k ∈ Z by d P k f ( ξ ) = ϕ (cid:0) − k | ξ | (cid:1) b f ( ξ ) . We use the concept of modulation to measure proximity of the space-time Fourier support to thelight cone and define for j ∈ Z the projection operators F (cid:0) Q j f (cid:1) ( τ, ξ ) = ϕ (cid:0) − j || τ | − | ξ || (cid:1) F ( f )( τ, ξ ) , F (cid:0) Q ± j f (cid:1) ( τ, ξ ) = ϕ (cid:0) − j || τ | − | ξ || (cid:1) χ {± τ> } F ( f )( τ, ξ ) , where F denotes the space-time Fourier transform. Occasionally, we also need multipliers S l torestrict the space-time frequency and correspondingly set for l ∈ Z , F (cid:0) S l f (cid:1) ( τ, ξ ) = ϕ (cid:0) − l | ( τ, ξ ) | (cid:1) F ( f )( τ, ξ ) . We also use projection operators P ω l to localize the homogeneous variable ξ | ξ | to caps ω ⊂ S ofdiameter ∼ l for integers l < ff s. We assume that for each such l < ff s form a smooth partition of unity subordinate to a uniformly finitely overlapping covering of S by caps ω of diameter ∼ l .With these projection operators in hand we introduce the convention that for any norm k · k S andany p ∈ [1 , ∞ ), k F k ℓ p S = (cid:18)X k ∈ Z k P k F k pS (cid:19) p . Next we define the X s , b type norms applied to functions at spatial frequency ∼ k , k F k X s , bp = sk (cid:18)X j ∈ Z (cid:16) b j k Q j P k F k L t L x (cid:17) p (cid:19) p for s , b ∈ R and p ∈ [1 , ∞ ) with the obvious analogue for p = ∞ , k F k X s , b ∞ = sk sup j ∈ Z b j k Q j P k F k L t L x . We will mainly use three function spaces N , N ∗ , and S . Their dyadic subspaces N k , N ∗ k and S k satisfy N k = L t L x + X , − , N ∗ k = L ∞ t L x ∩ X , ∞ , X , ⊆ S k ⊆ N ∗ k . Then we have k F k N = X k ∈ Z k P k F k N k , k F k N ∗ = X k ∈ Z k P k F k N ∗ k . The space S k is defined by k φ k S k = k φ k S Strk + k φ k S angk + k φ k X , ∞ , where S S trk = \ q + / r ≤ ( q + r − k L qt L rx , k φ k S angk = sup l < X ω k P ω l Q < k + l φ k S ω k ( l ) and the angular sector norms S ω k ( l ) are defined below.To introduce the angular sector norms S ω k ( l ) we first define the plane wave space k φ k PW ± ω ( l ) = inf φ = R φ ω ′ Z | ω − ω ′ |≤ l k φ ω ′ k L ± ω ′ L ∞ ( ± ω ′ ) ⊥ d ω ′ and the null energy space k φ k NE = sup ω k / ∇ ω φ k L ∞ ω L ω ⊥ , where the norms are with respect to ℓ ± ω = t ± ω · x and the transverse variable, while / ∇ ω denotesspatial di ff erentiation in the ( ℓ + ω ) ⊥ plane. We now set k φ k S ω k ( l ) = k φ k S Strk + − k k φ k NE + − k X ± k Q ± φ k PW ∓ ω ( l ) + sup k ′ ≤ k , l ′ ≤ , k + l ≤ k ′ + l ′ ≤ k + l X C k ′ ( l ′ ) (cid:18) k P C k ′ ( l ′ ) φ k S Strk + − k k P C k ′ ( l ′ ) φ k NE + − k ′ − k k P C k ′ ( l ′ ) φ k L t L ∞ x + − k ′ + l ′ ) X ± k Q ± P C k ′ ( l ′ ) φ k PW ∓ ω ( l ) (cid:19) , where P C k ′ ( l ′ ) is a projection operator to a radially directed block C k ′ ( l ′ ) of dimensions 2 k ′ × (2 k ′ + l ′ ) .Then we define k φ k S = X k ∈ Z k∇ t , x P k φ k S k + k (cid:3) φ k ℓ L t ˙ H − x and the higher derivative norms k φ k S N : = k∇ N − t , x φ k S , N ≥ . Moreover, we introduce k u k S ♯ k = k∇ t , x u k L ∞ t L x + k (cid:3) u k N k . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 15
On occasion we need to separate the two characteristic cones { τ = ±| ξ |} . To this end we define N k , ± , N k = N k , + ∩ N k , − S ♯ k , ± , S ♯ k = S ♯ k , + + S ♯ k , − N ∗ k , ± , N ∗ k = N ∗ k , + + N ∗ k , − . We will also use an auxiliary space of L t L ∞ x type, k φ k Z = X k ∈ Z k P k φ k Z k , k φ k Z k = sup l < C X ω l k P ω l Q k + l φ k L t L ∞ x . Finally, to control the component A , we define k A k Y = k∇ t , x A k L ∞ t L x + k A k L t ˙ H / x + k ∂ t A k L t ˙ H / x and the higher derivative norms k A k Y N = k∇ N − t , x A k Y , N ≥ . The link between the S and N spaces is given by the following energy estimate from [22], k∇ t , x φ k S . k∇ t , x φ (0) k L x + k (cid:3) φ k N . We will need to work with time-localized versions of the S k and N k spaces. For any compactinterval I ⊂ R and k ∈ Z , we define k ψ k S k ( I × R ) : = inf ˜ ψ | I = ψ | I k P k ˜ ψ k S k ( R × R ) with ψ and ˜ ψ Schwartz functions. Analogously, we define N k ( I × R ).The following lemma shows that the S k and Z k spaces are compatible with time cuto ff s. We willfrequently use this fact without further mentioning. Lemma 2.1.
Let χ I be a smooth cuto ff to a time interval I ⊂ R . Then it holds for all k ∈ Z that (cid:13)(cid:13)(cid:13) P k ( χ I φ ) (cid:13)(cid:13)(cid:13) S k ( R × R ) . (cid:13)(cid:13)(cid:13) P k φ (cid:13)(cid:13)(cid:13) S k ( R × R ) and (cid:13)(cid:13)(cid:13) P k ( χ I φ ) (cid:13)(cid:13)(cid:13) Z k ( R × R ) . (cid:13)(cid:13)(cid:13) P k φ (cid:13)(cid:13)(cid:13) Z k ( R × R ) . Proof.
This is obvious for the Strichartz type norms. It remains to show it for the X , ∞ and S angk components. We start with the former. For fixed j ∈ Z , we have Q j (cid:0) χ I φ (cid:1) = Q j (cid:0) Q j + O (1) ( χ I ) Q ≤ j − C φ (cid:1) + Q j (cid:0) χ I Q > j − C ( φ ) (cid:1) . Using the bound (cid:13)(cid:13)(cid:13) Q j + O (1) ( χ I ) (cid:13)(cid:13)(cid:13) L t . − j , we obtain2 j (cid:13)(cid:13)(cid:13) Q j (cid:0) Q j + O (1) ( χ I ) P k Q ≤ j − C φ (cid:1)(cid:13)(cid:13)(cid:13) L t L x . j (cid:13)(cid:13)(cid:13) Q j + O (1) ( χ I ) (cid:13)(cid:13)(cid:13) L t (cid:13)(cid:13)(cid:13) P k Q ≤ j − C φ (cid:13)(cid:13)(cid:13) L ∞ t L x . (cid:13)(cid:13)(cid:13) P k φ (cid:13)(cid:13)(cid:13) L ∞ t L x . Moreover, we find2 j (cid:13)(cid:13)(cid:13) Q j (cid:0) χ I P k Q > j − C ( φ ) (cid:1)(cid:13)(cid:13)(cid:13) L t L x . j (cid:13)(cid:13)(cid:13) χ I (cid:13)(cid:13)(cid:13) L ∞ t L ∞ x (cid:13)(cid:13)(cid:13) P k Q > j − C ( φ ) (cid:13)(cid:13)(cid:13) L t , x . (cid:13)(cid:13)(cid:13) P k φ (cid:13)(cid:13)(cid:13) X , ∞ . Thus, we have (cid:13)(cid:13)(cid:13) P k ( χ I φ ) (cid:13)(cid:13)(cid:13) X , ∞ . (cid:13)(cid:13)(cid:13) P k φ (cid:13)(cid:13)(cid:13) S k . Next, we consider the S angk component, which is given by k φ k S angk = sup l < X ω (cid:13)(cid:13)(cid:13) P ω l Q < k + l φ (cid:13)(cid:13)(cid:13) S ω k ( l ) . We write P ω l Q < k + l ( χ I φ ) = P ω l Q < k + l ( χ I Q < k + l + C φ ) + P ω l Q < k + l ( χ I Q ≥ k + l + C φ )Then the first term on the right hand side is bounded by (cid:13)(cid:13)(cid:13) P ω l Q < k + l ( χ I Q < k + l + C φ ) (cid:13)(cid:13)(cid:13) S ω k ( l ) . (cid:13)(cid:13)(cid:13) P ω l Q < k + l + C φ (cid:13)(cid:13)(cid:13) S ω k ( l ) , where we have used the fact that the operator P ω l Q < k + l is disposable. For the second term above,we use that X ω (cid:13)(cid:13)(cid:13) P ω l Q < k + l ( χ I Q ≥ k + l + C φ ) (cid:13)(cid:13)(cid:13) S ω k ( l ) . (cid:13)(cid:13)(cid:13) P k Q < k + l ( χ I Q ≥ k + l + C φ ) (cid:13)(cid:13)(cid:13) X , . (cid:13)(cid:13)(cid:13) φ (cid:13)(cid:13)(cid:13) X , ∞ . For the Z k space, fix a scale l < X ω l (cid:13)(cid:13)(cid:13) P ω l Q k + l (cid:0) χ I φ (cid:1)(cid:13)(cid:13)(cid:13) L t L ∞ x . Write P ω l Q k + l (cid:0) χ I φ (cid:1) = P ω l Q k + l (cid:0) Q < k + l − C ( χ I ) φ (cid:1) + P ω l Q k + l (cid:0) Q ≥ k + l − C ( χ I ) φ (cid:1) . For the first term on the right hand side, we have (cid:13)(cid:13)(cid:13) P ω l Q k + l (cid:0) Q < k + l − C ( χ I ) φ (cid:1)(cid:13)(cid:13)(cid:13) L t L ∞ x . (cid:13)(cid:13)(cid:13) P ω l Q k + l + O (1) φ (cid:13)(cid:13)(cid:13) L t L ∞ x , which leads to an acceptable contribution. For the second term on the right hand side, we use (cid:13)(cid:13)(cid:13) P ω l Q k + l (cid:0) Q ≥ k + l − C ( χ I ) φ (cid:1)(cid:13)(cid:13)(cid:13) L t L ∞ x . (cid:13)(cid:13)(cid:13) Q ≥ k + l − C ( χ I ) (cid:13)(cid:13)(cid:13) L t l + k (cid:0) − l − k (cid:13)(cid:13)(cid:13) P ω l φ k (cid:13)(cid:13)(cid:13) L t L ∞ x (cid:1) It follows that 2 l (cid:13)(cid:13)(cid:13) P ω l Q k + l (cid:0) Q ≥ k + l − C ( χ I ) φ (cid:1)(cid:13)(cid:13)(cid:13) L t L ∞ x . (cid:0) − l − k (cid:13)(cid:13)(cid:13) P ω l φ k (cid:13)(cid:13)(cid:13) L t L ∞ x (cid:1) , which can be square-summed over ω , see (9) in [22]. (cid:3)
3. M icrolocalized magnetic wave equation
In this section we assume that the spatial components of the connection form A are solutions tothe linear wave equation (cid:3) A j = R t × R x for j = , . . . , A is in Coulomb gauge. Wedefine the magnetic wave operator(3.1) (cid:3) pA = (cid:3) + i X k ∈ Z P ≤ k − C A j P k ∂ j . The goal of this section is to derive the following linear estimate for the magnetic wave operator (cid:3) pA . Theorem 3.1.
Suppose that (cid:3) A j = on R t × R x for j = , . . . , and that A is in Coulomb gauge.For all f ∈ N ( R × R ) and ( g , h ) ∈ ˙ H x ( R ) × L x ( R ) , the solution to the magnetic wave equation (3.2) (cid:3) pA φ = f on R × R , ( φ, φ t ) | t = = ( g , h ) ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 17 exists globally and satisfies (3.3) k φ k S ( R × R ) ≤ C (cid:16) k g k ˙ H x + k h k L x + k f k ( N ∩ ℓ L t ˙ H − x )( R × R ) (cid:17) , where the constant C > depends only on k∇ t , x A k L x and grows at most polynomially in k∇ t , x A k L x .Proof. By time reversibility it su ffi ces to prove the existence of the solution φ on the time interval[0 , ∞ ). Let ε > ffi ciently small constant to be fixed later. We may cover the time interval[0 , ∞ ) by finitely many consecutive closed intervals I , . . . , I J with the following properties. Thenumber of intervals J depends only on k∇ t , x A k L x and ε , the intervals I j overlap at most two at atime, consecutive intervals have intersection with non-empty interior and [0 , ∞ ) = ∪ ∞ j = I j . Mostimportantly, the intervals I j are chosen such that a finite number of suitable space-time norms ofthe magnetic potential A that will be specified later are less than ε uniformly on all intervals I j .We first construct suitable local solutions to the magnetic wave equation (3.2) on the intervals I j . The precise statement is summarized in the following theorem. Its proof will be given furtherbelow and is based on a parametrix construction. The accuracy of the parametrix crucially relieson the above mentioned smallness of suitable space-time norms of the magnetic potential A on theintervals I j . We use the notation I j = [ T ( l ) j , T ( r ) j ] for the left and right endpoints of I j . Theorem 3.2.
Let f ∈ N ( R × R ) and (˜ g , ˜ h ) ∈ ˙ H x ( R ) × L x ( R ) . For j = , . . . , J there exists asolution φ ( j ) ∈ S ( R × R ) to (3.4) (cid:3) pA φ ( j ) = f on I j × R , ( φ ( j ) , φ ( j ) t ) | t = T ( l ) j = (˜ g , ˜ h ) in the sense that k χ I j ( (cid:3) pA φ ( j ) − f ) k N ( R × R ) = for a sharp cuto ff χ I j to the time interval I j . Moreover,it holds that (3.5) k φ ( j ) k S ( R × R ) ≤ C (cid:16) k ˜ g k ˙ H x + k ˜ h k L x + k f k ( N ∩ ℓ L t ˙ H − x )( R × R ) (cid:17) , where the constant C > depends only on k∇ t , x A k L x . Finally, we obtain the solution φ to the magnetic wave equation (3.2) on [0 , ∞ ) × R by patchingtogether suitable local solutions on the intervals I j . Given ( g , h ) ∈ ˙ H x ( R ) × L x ( R ) and f ∈ N ( R × R ), Theorem 3.2 yields a solution φ (1) ∈ S ( R × R ) on I = [0 , T ( r )1 ] to (cid:3) pA φ (1) = f on I × R , ( φ (1) , φ (1) t ) | t = = ( g , h ) . Next, we obtain a solution φ (2) ∈ S ( R × R ) on I = [ T ( l )2 , T ( r )2 ] to (cid:3) pA φ (2) = f on I × R , ( φ (2) , φ (2) t ) | t = T ( l )2 = ( φ (1) ( T ( l )2 ) , φ (1) t ( T ( l )2 )) , where we recall that I ∩ I , ∅ with T ( l )2 < T ( r )1 . We proceed analogously for the remainingintervals I , . . . , I J . By uniqueness, we must have φ ( j ) | I j ∩ I j + = φ ( j + | I j ∩ I j + for j = , . . . , J −
1. We choose a smooth partition of unity { χ j } subordinate to the cover { I j } such that supp( χ j ) ⊂ I j andsupp( χ ′ j ) ⊂⊂ ( I j − ∩ I j ) ∪ ( I j ∩ I j + ). We then define φ = J X j = χ j φ ( j ) . Since we have χ ′ j + χ ′ j + = I j ∩ I j + for j = , . . . , J −
1, it follows that J X j = χ ′ j φ ( j ) = R t × R x and hence, ∇ t , x J X j = χ j φ ( j ) = J X j = χ j ∇ t , x φ ( j ) on R t × R x . Similarly, we find that (cid:3) J X j = χ j φ ( j ) = J X j = χ j (cid:3) φ ( j ) . Using Lemma 2.1 and estimate (3.5), we thus conclude that k φ k S ( R × R ) = (cid:13)(cid:13)(cid:13) J X j = χ j φ ( j ) (cid:13)(cid:13)(cid:13) S ( R × R ) . J X j = k φ ( j ) k S ( R × R ) . C ( k∇ t , x A k L x ) (cid:16) J X j = k φ ( j ) ( T ( l ) j ) k ˙ H x + k ∂ t φ ( j ) ( T ( l ) j ) k L x + k f k N ( R × R ) (cid:17) . C ( J ) C ( k∇ t , x A k L x ) (cid:0) k g k ˙ H x + k h k L x + k f k N ( R × R ) (cid:1) . Since J depends only on the size of k∇ t , x A k L x and ε , we obtain the desired estimate (3.3). (cid:3) We proceed with the proof of Theorem 3.2.
Proof of Theorem 3.2.
We begin by considering for every k ∈ Z the frequency localized problem(3.6) (cid:3) pA < k φ ( j ) k = f k on I j × R , ( φ ( j ) k , ∂ t φ ( j ) k ) | t = T ( l ) j = (˜ g k , ˜ h k ) , where (cid:3) pA < k = (cid:3) + iP ≤ k − C A j P k ∂ j . Let χ I j denote a sharp cuto ff to the time interval I j . We first wantto construct an approximate solution φ ( j ) app , k to (3.6) that satisfies(3.7) k φ ( j ) app , k k S k ( R × R ) ≤ C (cid:16) k ˜ g k k ˙ H x + k ˜ h k k L x + k f k k N k ( R × R ) (cid:17) and k φ ( j ) app , k ( T ( l ) j ) − ˜ g k k ˙ H x + k ∂ t φ ( j ) app , k ( T ( l ) j ) − ˜ h k k L x + k χ I j ( (cid:3) pA < k φ app , k − f k ) k ( N k ∩ L t ˙ H − x )( R × R ) . ε (cid:16) k ˜ g k k ˙ H x + k ˜ h k k L x + k f k k ( N k ∩ L t ˙ H − x )( R × R ) (cid:17) . (3.8) ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 19
To this end we split f k = f hypk + f ellk , where f hypk is supported in the region || τ | − | ξ || . k . We note that it holds that(3.9) k (cid:3) − f ellk k S k ( R × R ) . k f ellk k N k ( R × R ) . Theorem 3.3 below then yields an approximate solution ˜ φ ( j ) app , k to(3.10) (cid:3) ˜ φ ( j ) app , k = f hypk on I j × R , ( ˜ φ ( j ) app , k , ∂ t ˜ φ ( j ) app , k ) | t = T ( l ) j = (˜ g k , ˜ h k ) − (( (cid:3) − f ellk )( T ( l ) j ) , ( ∂ t (cid:3) − f ellk )( T ( l ) j ))that satisfies(3.11) (cid:13)(cid:13)(cid:13) ˜ φ ( j ) app , k (cid:13)(cid:13)(cid:13) S k ( R × R ) . k ˜ g k k ˙ H x + k ˜ h k k L x + k f k k N k ( R × R ) and (cid:13)(cid:13)(cid:13) ˜ φ ( j ) app , k ( T ( l ) j ) − (cid:0) ˜ g k − ( (cid:3) − f ellk )( T ( l ) j ) (cid:1)(cid:13)(cid:13)(cid:13) ˙ H x + (cid:13)(cid:13)(cid:13) ∂ t ˜ φ ( j ) app , k ( T ( l ) j ) − (cid:0) ˜ h k − ( ∂ t (cid:3) − f ellk )( T ( l ) j ) (cid:1)(cid:13)(cid:13)(cid:13) L x + (cid:13)(cid:13)(cid:13) χ I j (cid:0) (cid:3) A p < k ˜ φ ( j ) app , k − f hypk (cid:1)(cid:13)(cid:13)(cid:13) N k ( R × R ) . ε (cid:0) k ˜ g k k ˙ H x + k ˜ h k k L x + k f k k N k ( R × R ) (cid:1) . (3.12)We remark that because of scaling invariance Theorem 3.3 below is only formulated for the case k =
0. Next we set φ ( j ) app , k = ˜ φ ( j ) app , k + ( (cid:3) − f ellk )and find that (cid:13)(cid:13)(cid:13) χ I j (cid:0) (cid:3) pA < k φ ( j ) app , k − f k (cid:1)(cid:13)(cid:13)(cid:13) ( N k ∩ L t ˙ H − x )( R × R ) . (cid:13)(cid:13)(cid:13) χ I j (cid:0) (cid:3) pA < k ˜ φ ( j ) app , k − f hypk (cid:1)(cid:13)(cid:13)(cid:13) ( N k ∩ L t ˙ H − x )( R × R ) + (cid:13)(cid:13)(cid:13) χ I j A j < k P k ∂ j ( (cid:3) − f ellk ) (cid:13)(cid:13)(cid:13) ( N k ∩ L t ˙ H − x )( R × R ) . ε k f k k ( N k ∩ L t ˙ H − )( R × R ) . (3.13)Here we used that the intervals I j can be chosen such that uniformly for all j = , . . . , J , k A k L t L x ( I j × R ) ≤ ε and thus, (cid:13)(cid:13)(cid:13) χ I j A j < k P k ∂ j ( (cid:3) − f ellk ) (cid:13)(cid:13)(cid:13) ( N k ∩ L t ˙ H − x )( R × R ) ≤ (cid:13)(cid:13)(cid:13) χ I j A j < k P k ∂ j ( (cid:3) − f ellk ) (cid:13)(cid:13)(cid:13) ( L t L x ∩ L t ˙ H − x )( R × R ) . (cid:13)(cid:13)(cid:13) χ I j A < k (cid:13)(cid:13)(cid:13) ( L t L ∞ x )( R × R ) (cid:13)(cid:13)(cid:13) P k ∇ x ( (cid:3) − f ellk ) (cid:13)(cid:13)(cid:13) ( L t L x ∩ L ∞ t ˙ H − x )( R × R ) . k A k L t L x ( I j × R ) k (cid:13)(cid:13)(cid:13) P k ∇ x ( (cid:3) − f ellk ) (cid:13)(cid:13)(cid:13) ( L t L x ∩ L ∞ t ˙ H − x )( R × R ) . ε (cid:13)(cid:13)(cid:13) (cid:3) − f ellk (cid:13)(cid:13)(cid:13) S k ( R × R ) . ε k f k k N k ( R × R ) . From (3.9), (3.11), (3.12), and (3.13) it now follows immediately that φ ( j ) app , k is an approximatesolution to (3.6) that satisfies the estimates (3.7) and (3.8). Finally, we reassemble the approximate solutions φ ( j ) app , k to the frequency localized problems(3.6) to a full approximate solution φ ( j ) app = P k ∈ Z φ ( j ) app , k to (3.4) satisfying k φ ( j ) app ( T ( l ) j ) − ˜ g k ˙ H x + k ∂ t φ ( j ) app ( T ( l ) j ) − ˜ h k L x + k χ I j ( (cid:3) pA φ ( j ) app − f ) k ( N ∩ ℓ L t ˙ H − x )( R × R ) . ε (cid:0) k ˜ g k ˙ H x + k ˜ h k L x + k f k ( N ∩ ℓ L t ˙ H − x )( R × R ) (cid:1) and (cid:13)(cid:13)(cid:13) φ ( j ) app (cid:13)(cid:13)(cid:13) S ( R × R ) . k ˜ g k ˙ H x + k ˜ h k L x + k f k ( N ∩ ℓ L t ˙ H − x )( R × R ) . Applying this procedure iteratively to the successive errors, we obtain an exact solution φ ( j ) to (3.4)satisfying (3.5). (cid:3) We now turn to the heart of the matter, namely the construction of the approximate solutions tothe frequency localized magnetic wave equations.
Theorem 3.3.
Let (˜ g , ˜ h ) ∈ ˙ H x ( R ) × L x ( R ) and ˜ f ∈ N ( R × R ) . Assume that ˜ f , ˜ g , ˜ h are frequencylocalized at | ξ | ∼ and that ˜ f is localized at modulation || τ | − | ξ || . . For j = , . . . , J there existsan approximate solution ˜ φ ( j ) app to (3.14) (cid:3) pA < φ = ˜ f on I j × R , ( φ, φ t ) | t = T ( l ) j = (˜ g , ˜ h ) in the sense that (3.15) (cid:13)(cid:13)(cid:13) ˜ φ ( j ) app (cid:13)(cid:13)(cid:13) S ( R × R ) . k ˜ g k L x + k ˜ h k L x + k ˜ f k N ( R × R ) and (cid:13)(cid:13)(cid:13) ˜ φ ( j ) app ( T ( l ) j ) − ˜ g (cid:13)(cid:13)(cid:13) L x + (cid:13)(cid:13)(cid:13) ∂ t ˜ φ ( j ) app ( T ( l ) j ) − ˜ h (cid:13)(cid:13)(cid:13) L x + (cid:13)(cid:13)(cid:13) χ I j (cid:0) (cid:3) pA < ˜ φ ( j ) app − ˜ f (cid:1)(cid:13)(cid:13)(cid:13) N ( R × R ) . ε (cid:0) k ˜ g k L x + k ˜ h k L x + k ˜ f k N ( R × R ) (cid:1) , (3.16) where χ I j denotes a sharp cuto ff to the time interval I j .Proof. In order to prove estimates and construct a parametrix for the frequency localized magneticwave equation (3.14) we adapt the scheme in Section 6 of [22] to our time-localized setting. We willuse frequency localized renormalization operators e − i ψ ± < ( t , x , D ) and e + i ψ ± < ( D , y , s ), where P ( x , D )denotes the left quantization and P ( D , y ) the right quantization of a pseudodi ff erential operator P and where the subscript < ≪
1. For the definition of the phase correction ψ ± in the renormalization operator e + i ψ ± < ( D , y , s ) we need to introduce some notation.For any ξ ∈ R \{ } we set ω = ξ | ξ | , L ω ± : = ± ∂ t + ω · ∇ x , ∆ ω ⊥ : = ∆ − ( ω · ∇ x ) . Moreover, for any ω ∈ S and any angle 0 < θ .
1, we define the sector projection Π ω>θ in frequencyspace by the formula [ Π ω>θ f ( ζ ) : = (cid:16) − η (cid:16) ∠ ( ζ, ω ) θ (cid:17)(cid:17)(cid:16) − η (cid:16) ∠ ( − ζ, ω ) θ (cid:17)(cid:17) b f ( ζ ) , ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 21 where η ( y ) is a bump function on R which equals 1 when | y | < and vanishes for | y | >
1, and ∠ ( ζ, ω )is the angle between ζ and ω . Thus, Π ω>θ restricts f smoothly (except at the frequency origin) to thesector of frequencies ζ whose angle with both ω and − ω is & θ . Similarly, we define the Fouriermultipliers Π ωθ , Π ω ≤ θ and Π ωθ > · >θ .Let C , C > ffi ciently large later on depending on the size of k∇ t , x A k L x and let σ > ffi ciently small. We then define the phasecorrection ψ ± by(3.17) ψ ± ( t , x , ξ ) = X − C ≤ k < L ω ± ∆ − ω ⊥ Π ω − C > · > σ k A k · ω + X k < − C L ω ± ∆ − ω ⊥ Π ω> σ k A k · ω. Note that the first sum e ff ectively only starts at k . − C σ . See Section 6 in [22] for a motivation forsuch a choice of phase correction. We emphasize that this phase slightly di ff ers from the one usedin [22], because for intermediate frequencies − C ≤ k < ff .We define the approximate solution ˜ φ ( j ) app to (3.14) by˜ φ ( j ) app = χ I j ( t ) 12 X ± (cid:26) e − i ψ ± < ( t , x , D ) 1 | D | e ± i ( t − T ( l ) j ) | D | e i ψ ± < ( D , y , T ( l ) j )( | D | ˜ g ± ( − i )˜ h ) ± e − i ψ ± < ( t , x , D ) 1 | D | K ± j e i ψ ± < ( D , y , s )( − i ) ˜ f (cid:27) , where K ± j ˜ f ( t ) = Z tT ( l ) j e ± i ( t − s ) | D | ˜ f ( s ) ds . In order to prove the estimates (3.15) and (3.16) we establish the following crucial time-localizedmapping properties of the renormalization operator e ± i ψ ± < ( t , x , D ). Theorem 3.4.
For j = , . . . , J, the frequency localized renormalization operators have the follow-ing mapping properties with Z ∈ { N ( R × R ) , L x ( R ) , N ∗ ( R × R ) } , χ I j e ± i ψ ± < : Z → Z , (3.18) χ I j ∂ t e ± i ψ ± < : Z → ε Z , (3.19) χ I j ( e − i ψ ± < ( t , x , D ) e + i ψ ± < ( D , y , t ) −
1) : Z → ε Z , (3.20) χ I j ( e − i ψ ± < ( t , x , D ) (cid:3) − (cid:3) pA < e − i ψ ± < ( t , x , D )) : N ∗ , ± ( R × R ) → ε N , ± ( R × R ) , (3.21) χ I j e − i ψ ± < ( t , x , D ) : S ♯ ( R × R ) → S ( R × R ) , (3.22) where χ I j denotes a sharp cuto ff to the time interval I j . In the estimates (3.18) and (3.19) , theoperator e ± i ψ ± < , respectively ∂ t e ± i ψ ± < , stands for both left and right quantization. The estimates (3.15) and (3.16) then follow by adapting the manipulations in the proof of Theo-rem 4 in [22] to our time-localized setting. (cid:3)
The remainder of this section is devoted to the proof of Theorem 3.4. To this end we willadapt the general scheme of Sections 7 – 11 in [22] to our large data setting. The accuracy of theapproximate solution ˜ φ ( j ) app relies on the error estimates (3.19), (3.20) and (3.21). While in [22] thesmall energy assumption can be used to achieve smallness in the corresponding error estimates,we have to argue more carefully here, using the high angle cut-o ff in the definition of the phase correction and smallness of suitable space-time norms of A on su ffi ciently small time intervals,namely the intervals I j .3.1. Decomposable function spaces.
We begin by reviewing the notion of decomposable functionspaces and estimates from [35], [21], and [22].Let c ( t , x , D ) be a pseudodi ff erential operator whose symbol c ( t , x , ξ ) is homogeneous of degree0 in ξ . Assume that c has a representation c = X θ ∈ − N c ( θ ) . Let 1 ≤ q , r ≤ ∞ . For every θ ∈ − N , we define k c ( θ ) k D θ ( L qt L rx )( R × R ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) X l = X Γ νθ sup ω ∈ Γ νθ (cid:13)(cid:13)(cid:13) b νθ ( ω )( θ ∇ ξ ) l c ( θ ) (cid:13)(cid:13)(cid:13) L rx (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L qt ( R ) , where { Γ νθ } ν ∈ S is a uniformly finitely overlapping covering of S by caps of diameter ∼ θ and { b νθ } ν ∈ S is a smooth partition of unity subordinate to the covering { Γ νθ } ν ∈ S . Then we define thedecomposable norm k c k D ( L qt L rx )( R × R ) = inf c = P θ c ( θ ) X θ ∈ − N k c ( θ ) k D θ ( L qt L rx )( R × R ) . We will repeatedly use the following decomposable estimates.
Lemma 3.5 ( [22, Lemma 7.1]) . Let P ( t , x , D ) be a pseudodi ff erential operator with symbol p ( t , x , ξ ) .Suppose that P satisfies the fixed-time estimate sup t ∈ R k P ( t , x , D ) k L x → L x . . Let ≤ q , q , q , r , r ≤ ∞ such that q = q + q and r = r + . For any symbol c ( t , x , ξ ) ∈ D ( L q t L r x )( R × R ) that is zero homogeneous in ξ , we have k ( cp )( t , x , D ) φ k L qt L rx ( R × R ) . k c k D ( L q t L r x )( R × R ) k φ k L q t L x ( R × R ) . By duality we obtain decomposable estimates for right quantizations.
Lemma 3.6.
Let P be a pseudodi ff erential operator with symbol p ( t , x , ξ ) . Suppose that P satisfiesthe fixed-time estimate sup t ∈ R k P ( t , x , D ) k L x → L x . . Let ≤ q < ∞ and ≤ q , q ≤ ∞ such that q = q + q . For any symbol c ( t , x , ξ ) ∈ D ( L q t L ∞ x )( R × R ) that is zero homogeneous in ξ , the right-quantized operator ( c p )( D , y , t ) has the followingmapping property (cid:13)(cid:13)(cid:13) ( c p )( D , y , t ) φ (cid:13)(cid:13)(cid:13) L qt L x ( R × R ) . k c k D ( L q t L ∞ x )( R × R ) k φ k L q t L x ( R × R ) . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 23
Proof.
Let 1 < q ′ ≤ ∞ be the the conjugate exponent to q and define q = q + q ′ . By duality,H ¨older’s inequality and Lemma 3.5, we have (cid:13)(cid:13)(cid:13) ( c p )( D , y , t ) φ (cid:13)(cid:13)(cid:13) L qt L x = sup k ψ k Lq ′ t L x ≤ (cid:10) ψ, ( c p )( D , y , t ) φ (cid:11) = sup k ψ k Lq ′ t L x ≤ (cid:10) ( cp )( t , x , D ) ψ, φ (cid:11) ≤ sup k ψ k Lq ′ t L x ≤ (cid:13)(cid:13)(cid:13) ( cp )( t , x , D ) ψ (cid:13)(cid:13)(cid:13) L ˜ qt L x k φ k L q t L x . sup k ψ k Lq ′ t L x ≤ k c k D ( L q t L ∞ x ) k ψ k L q ′ t L x k φ k L q t L x . k c k D ( L q t L ∞ x ) k φ k L q t L x . (cid:3) From [21, Lemma 10.2] we have the following H ¨older-type estimate for decomposable norms(3.23) (cid:13)(cid:13)(cid:13)(cid:13) m Y i = c i (cid:13)(cid:13)(cid:13)(cid:13) D ( L qt L rx ) . m Y i = k c i k D ( L qit L rix ) , where m ∈ N , 1 ≤ q , r , q i , r i ≤ ∞ for i = , . . . , m and ( q , r ) = P mi = ( q i , r i ).3.2. Some symbol bounds for phases.
Recall that the magnetic potential A is assumed to besupported at frequencies .
1. For any integer k < < θ .
1, we usethe notation ψ ( θ ) k ( t , x , ξ ) = L ω ± ∆ − ω ⊥ Π ωθ A k · ω and ψ < k = X l < k P l ψ ± . Lemma 3.7.
For any t , s ∈ R , x , y ∈ R , ξ ∈ R and any integer k < , it holds that (3.24) | ψ ± ( t , x , ξ ) − ψ ± ( s , y , ξ ) | . (2 − C / + − C ) k∇ t , x A (0) k L x ( | t − s | + | x − y | ) . Moreover, we have for any multi-index α ∈ N with ≤ | α | ≤ σ − that (3.25) |∇ αξ ( ψ ( t , x , ξ ) − ψ ( s , y , ξ )) | . h| t − s | + | x − y |i σ ( | α |− ) k∇ t , x A (0) k L x . Proof.
For any t ∈ R , x ∈ R , ξ ∈ R and any integer k <
0, we obtain that | ψ ( θ ) k ( t , x , ξ ) | ≤ k L ω ± ∆ − ω Π ωθ P k A · ω k L ∞ x . ( θ k ) − ∞ k L ω ± ∆ − ω Π ωθ P k A · ω k L x . θ / k θ − k θ − k Π ωθ P j L ω ± A k L x . θ / k∇ t , x A k k L x , where we used Bernstein’s inequality, the Coulomb gauge of A and that | d ∆ − ω ⊥ ( ξ ) | ∼ − k θ − on thefrequency support of Π ωθ P k . Similarly, we find |∇ t , x ψ ( θ ) k ( t , x , ξ ) | . k θ / k∇ t , x A k k L x . Thus, we have | ψ ± ( t , x , ξ ) − ψ ± ( s , y , ξ ) |≤ X − C ≤ k < X σ k <θ< − C | ψ ( θ ) k ( t , x , ξ ) − ψ ( θ ) k ( s , y , ξ ) | + X k < − C X σ k <θ | ψ ( θ ) k ( t , x , ξ ) − ψ ( θ ) k ( s , y , ξ ) |≤ (cid:18) X − C ≤ k < X σ k <θ< − C k θ / + X k < − C X σ k <θ k θ / (cid:19) / k∇ t , x A k L x ( | x − y | + | t − s | ) ≤ (2 − C / + − C ) k∇ t , x A k L x ( | x − y | + | t − s | ) . We now turn to the proof of (3.25). To this end we note that di ff erentiating with respect to ξ yields θ − factors, i.e. for any α ∈ N it holds that |∇ αξ ψ ( θ ) k ( t , x , ξ ) | . θ −| α | k∇ t , x A k k L x and |∇ t , x ∇ αξ ψ ( θ ) k ( t , x , ξ ) | . k θ −| α | k∇ t , x A k k L x . For any 1 ≤ | α | ≤ σ − and l < |∇ αξ ( ψ ± ( t , x , ξ ) − ψ ± ( s , y , ξ )) | . X k < l X σ k <θ k θ −| α | k∇ t , x A k L x ( | x − y | + | t − s | ) + X k ≥ l X σ k <θ θ −| α | k∇ t , x A k L x . l (1 − σ ( | α |− )) k∇ t , x A k L x ( | x − y | + | t − s | ) + − σ l ( | α |− ) k∇ t , x A k L x . Optimizing the choice of l < |∇ αξ ( ψ ± ( t , x , ξ ) − ψ ± ( s , y , ξ )) | . h| t − s | + | x − y |i σ ( | α |− ) k∇ t , x A k L x . (cid:3) We will frequently use the following bounds on decomposable norms of the phase ψ ± . Lemma 3.8 ( [22, Lemma 7.3]) . Let ≤ q , r ≤ ∞ with q + r ≤ . For any integer k < and anydyadic angle θ ∈ − N the component ψ ( θ ) k = L ω ± ∆ − ω ⊥ Π ωθ A k · ω satisfies (3.26) (cid:13)(cid:13)(cid:13) ( ψ ( θ ) k , − k ∇ t , x ψ ( θ ) k ) (cid:13)(cid:13)(cid:13) D θ ( L qt L rx )( R × R ) . − ( q + r ) k θ − q − r k∇ t , x A k L x . Oscillatory integral estimates.
In order to prove the mapping properties in Theorem 3.4, weneed pointwise kernel bounds for operators of the form T a = e − i ψ ± ( t , x , D ) a ( D ) e ± i ( t − s ) | D | e i ψ ± ( D , y , s ) , where a is localized at frequency | ξ | ∼
1. The kernel of T a is given by the oscillatory integral K a ( t , x ; s , y ) = Z R e − i ( ψ ± ( t , x ,ξ ) − ψ ± ( s , y ,ξ )) e i ( t − s ) | ξ | e i ( x − y ) · ξ a ( ξ ) d ξ, where a is a smooth bump function with support on the annulus | ξ | ∼ Lemma 3.9.
For any t , s ∈ R , x , y ∈ R and any integer ≤ N ≤ σ − , we have (3.27) | K a ( t , x ; t , y ) | . k∇ t , x A k L x h| x − y |i N (1 − σ )ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 25 and (3.28) | K a ( t , x ; t , y ) − ˇ a ( x − y ) | . min n (2 − C / + − C ) , | x − y | N (1 − σ ) o k∇ t , x A k L x . Moreover, it holds that (3.29) | K a ( t , x ; s , y ) | . h t − s i − h| t − s | − | x − y |i − N k∇ t , x A k L x . Proof. If | x − y | . | K a ( t , x ; t , y ) | . . If instead | x − y | ≫
1, we use (3.25) and integrate by parts repeatedly to obtain for any 1 ≤ N ≤ σ − that | K a ( t , x ; t , y ) | . k∇ t , x A k L x | x − y | N (1 − σ ) . This proves (3.27). In order to show (3.28), we integrate by parts repeatedly for | x − y | ≫
1, whilefor | x − y | .
1, we use (3.24) to estimate | K a ( t , x ; t , y ) − ˇ a ( x − y ) | ≤ Z R | e i ( ψ ± ( t , x ,ξ ) − ψ ± ( t , y ,ξ )) − | a ( ξ ) d ξ ≤ Z R | ψ ± ( t , x , ξ ) − ψ ± ( t , y , ξ ) | a ( ξ ) d ξ . (2 − C / + − C ) k∇ t , x A k L x | x − y | . Finally, the proof of (3.29) proceeds along the lines of Proposition 6(a) in [22]. We only have toargue a bit more that away from the cone for su ffi ciently large | t − s | the phase is still non-degenerate.But this is because away from the cone (cid:12)(cid:12)(cid:12) − i ∇ ξ ( ψ ± ( t , x , ξ ) − ψ ± ( s , y , ξ )) + i ( t − s ) ξ | ξ | + i ( x − y ) (cid:12)(cid:12)(cid:12) ≥ c h| t − s | + | x − y |i − C k∇ t , x A k L x h| t − s | + | x − y |i σ and we choose 0 < σ ≪ ffi ciently small. (cid:3) To deal with the frequency localized operators e ± i ψ ± < ( t , x , D ) and e i ψ ± < ( D , y , s ), we need to producesimilar estimates for the kernel K a ,< of the operator T a ,< = e − i ψ ± < ( t , x , D ) a ( D ) e ± i ( t − s ) | D | e i ψ ± < ( D , y , s ) . Noting that the frequency localized symbol e ± i ψ ± < can be represented as e ± i ψ ± < = Z R + z m ( z ) e ± iT z ψ ± dz , where m ( z ) is an integrable bump function on the unit scale and T z denotes space-time translationin the direction z ∈ R + , the transition to these frequency localized operators can be made just asin Proposition 7 in [22]. We obtain the following estimates for K a ,< . Lemma 3.10.
For any t , s ∈ R , x , y ∈ R and any integer ≤ N ≤ σ − , we have (3.30) | K a ,< ( t , x ; t , y ) | . k∇ t , x A k L x h| x − y |i N (1 − σ )6 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION and (3.31) | K a ,< ( t , x ; t , y ) − ˇ a ( x − y ) | . min n (2 − C / + − C ) , | x − y | N (1 − σ ) o k∇ t , x A k L x . Moreover, it holds that (3.32) | K a ,< ( t , x ; s , y ) | . h t − s i − h| t − s | − | x − y |i − N k∇ t , x A k L x . Fixed-time L x estimates in Theorem 3.4. In this subsection we prove the fixed-time L x esti-mates in Theorem 3.4 using the above oscillatory integral estimates. To obtain a small factor ε inthe estimates (3.18) and (3.19), we additionally have to fix the constants C , C > ψ ± su ffi ciently large. Lemma 3.11.
For any t ∈ R , we have (3.33) (cid:13)(cid:13)(cid:13) e ± i ψ ± < ( t , x , D ) P φ (cid:13)(cid:13)(cid:13) L x . k∇ t , x A k L x k P φ k L x and (3.34) (cid:13)(cid:13)(cid:13) e ± i ψ ± < ( D , y , t ) P φ (cid:13)(cid:13)(cid:13) L x . k∇ t , x A k L x k P φ k L x Proof.
The claim follows immediately from the kernel bound (3.30) and a
T T ∗ -argument. (cid:3) Lemma 3.12.
For any ε > the constants C , C > in the definition of the phase correction ψ ± can be chosen su ffi ciently large (depending on the size of ε − and k∇ t , x A k L x ) such that we have (3.35) (cid:13)(cid:13)(cid:13) ( ∇ t , x e − i ψ ± < )( t , x , D ) P φ (cid:13)(cid:13)(cid:13) L x . ε k P φ k L x . Proof.
Using Lemma 3.5, Lemma 3.8, and (3.33) we obtain that (cid:13)(cid:13)(cid:13) ( ∇ t , x e − i ψ ± < )( t , x , D ) P φ (cid:13)(cid:13)(cid:13) L x = (cid:13)(cid:13)(cid:13) ( ∇ t , x ψ ± ) e − i ψ ± < ( t , x , D ) P φ (cid:13)(cid:13)(cid:13) L x ≤ X − C ≤ k < X σ k <θ< − C (cid:13)(cid:13)(cid:13) ( ∇ t , x ψ ( θ ) k ) e − i ψ ± < ( t , x , D ) P φ (cid:13)(cid:13)(cid:13) L x + X k < − C X σ k <θ (cid:13)(cid:13)(cid:13) ( ∇ t , x ψ ( θ ) k ) e − i ψ ± < ( t , x , D ) P φ (cid:13)(cid:13)(cid:13) L x . X − C ≤ k < X σ k <θ< − C k∇ t , x ψ ( θ ) k k D θ ( L ∞ t L ∞ x ) k∇ t , x A k L x k P φ k L x + X k < − C X σ k <θ k∇ t , x ψ ( θ ) k k D θ ( L ∞ t L ∞ x ) k∇ t , x A k L x k P φ k L x . (cid:18) X − C ≤ k < X σ k <θ< − C k θ / + X k < − C X σ k <θ k θ / (cid:19) k∇ t , x A k L x k P φ k L x . (2 − C / + − C ) k∇ t , x A k L x k P φ k L x , from which the assertion follows. (cid:3) Lemma 3.13.
For any ε > the constants C , C > in the definition of the phase correction ψ ± can be chosen su ffi ciently large (depending on the size of ε − and k∇ t , x A k L x ) such that we have (3.36) (cid:13)(cid:13)(cid:13) ( e − i ψ ± < ( t , x , D ) e i ψ ± < ( D , y , t ) − P φ (cid:13)(cid:13)(cid:13) L x . ε k P φ k L x . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 27
Proof.
The integral kernel of ( e − i ψ ± < ( t , x , D ) e i ψ ± < ( D , y , t ) − a ( D )is given by K a ,< ( t , x ; t , y ) − ˇ a ( x − y ). Using (3.31) we find thatsup y Z R | K a ,< ( t , x ; t , y ) − ˇ a ( x − y ) | dx . Z R min n (2 − C / + − C ) , | x | N (1 − σ ) o k∇ t , x A k L x dx . inf R > n (2 − C / + − C ) R + R N (1 − σ ) − o k∇ t , x A k L x . Choosing C , C > ffi ciently large depending on the size of ε − and k∇ t , x A k L x , we obtain thatsup y Z R | K a ,< ( t , x ; t , y ) − ˇ a ( x − y ) | dx ≤ ε and similarly for sup x R R | K a ( t , x ; t , y ) − ˇ a ( x − y ) | dy . The assertion then follows from Schur’s lemma. (cid:3) Remark 3.14.
The fixed-time L x bounds from Lemma 3.11, Lemma 3.12 and Lemma 3.13 in facthold for the operators e ± i ψ < l < k ( t , x , D ) , e ± i ψ < l k ( t , x , D ) , and e ± i ψ < l ( t , x , D ) for any k , l < . The proofs inthis and the previous subsection can be easily adapted to obtain this assertion. Modulation localized estimates.
All implicit constants in this subsection may depend on thesize of k∇ t , x A k L x . Proposition 3.15.
For any ε > the intervals I j can be chosen such that uniformly for all j = , . . . , J and all integers k ≤ k ′ ± O (1) < , it holds that (3.37) (cid:13)(cid:13)(cid:13) Q k (cid:0) χ I j e ± i ψ ± k ′ ( t , x , D ) P φ (cid:1)(cid:13)(cid:13)(cid:13) L t L x ( R × R ) . ε − k δ ( k − k ′ ) k P φ k N ∗ ( R × R ) . In the proof of Proposition 3.15 we will use the following result whose proof will be given later.
Lemma 3.16.
Let ≤ q ≤ p ≤ ∞ . For any ε > the intervals I j can be chosen such that uniformlyfor all j = , . . . , J and all integers k + C ≤ l ≤ , the following operator bound holds (cid:13)(cid:13)(cid:13) χ I j ( e ± i ψ < k l )( t , x , D ) (cid:13)(cid:13)(cid:13) L pt L x ( R × R ) → L qt L x ( R × R ) . ε k − l ) ( p − q ) k . Proof of Proposition 3.15.
In the following we denote an interval I j just by I and e ± i ψ ± k ′ stands forthe left quantization e ± i ψ ± k ′ ( t , x , D ).We first reduce to the case k = k ′ ± O (1). To this end we will use that Proposition 9 and Lemma10 in [22] hold without the ε smallness gain also for large energies. We split Q k (cid:0) χ I e ± i ψ ± k ′ P φ (cid:1) = Q k (cid:0) Q < k − C ( χ I ) e ± i ψ ± k ′ P φ (cid:1) + Q k (cid:0) Q [ k − C , k + C ] ( χ I ) e ± i ψ ± k ′ P φ (cid:1) + Q k (cid:0) Q > k + C ( χ I ) e ± i ψ ± k ′ P φ (cid:1) . (3.38)For the first term we obtain2 k (cid:13)(cid:13)(cid:13) Q k (cid:0) Q < k − C ( χ I ) e ± i ψ ± k ′ P φ (cid:1)(cid:13)(cid:13)(cid:13) L t L x = k (cid:13)(cid:13)(cid:13) Q k (cid:0) Q < k − C ( χ I ) Q k + O (1) e ± i ψ ± k ′ P φ (cid:1)(cid:13)(cid:13)(cid:13) L t L x . k (cid:13)(cid:13)(cid:13) Q k + O (1) e ± i ψ ± k ′ P φ (cid:13)(cid:13)(cid:13) L t L x . δ ( k − k ′ ) k P φ k N ∗ , where we used Proposition 9 from [22]. We estimate the second term from (3.38) by2 k (cid:13)(cid:13)(cid:13) Q k (cid:0) Q [ k − C , k + C ] ( χ I ) e ± i ψ ± k ′ P φ (cid:1)(cid:13)(cid:13)(cid:13) L t L x . k (cid:13)(cid:13)(cid:13) Q [ k − C , k + C ] ( χ I ) (cid:13)(cid:13)(cid:13) L t ( R ) (cid:13)(cid:13)(cid:13) e ± i ψ ± k ′ P φ (cid:13)(cid:13)(cid:13) L t L x . k − k (cid:13)(cid:13)(cid:13) e ± i ψ ± k ′ P φ (cid:13)(cid:13)(cid:13) L t L x . Using continuous Littlewood-Paley resolutions to decompose the group element we have e ± i ψ ± k ′ = e ± i ψ < k ′− C k ′ ± i Z l > k ′ − C S k ′ (cid:0) ψ l e ± i ψ < l (cid:1) dl . By Lemma 10 in [22] and the decomposable estimates (3.26) we find (cid:13)(cid:13)(cid:13) e ± i ψ ± k ′ P φ (cid:13)(cid:13)(cid:13) L t L x . (cid:13)(cid:13)(cid:13) e ± i ψ < k ′− C k ′ P φ (cid:13)(cid:13)(cid:13) L t L x + Z l > k ′ − C (cid:13)(cid:13)(cid:13) S k ′ (cid:0) ψ l e ± i ψ < l (cid:1) P φ (cid:13)(cid:13)(cid:13) L t L x dl . − k ′ k P φ k L ∞ t L x + Z l > k ′ − C k ψ l k D ( L t L ∞ x ) k e ± i ψ < l P φ k L ∞ t L x dl . − k ′ k P φ k L ∞ t L x + Z l > k ′ − C − l k∇ t , x A k L x k P φ k L ∞ t L x dl . − k ′ k P φ k N ∗ . Thus, the second term from (3.38) is bounded by2 k (cid:13)(cid:13)(cid:13) Q k (cid:0) Q [ k − C , k + C ] ( χ I ) e ± i ψ ± k ′ P φ (cid:1)(cid:13)(cid:13)(cid:13) L t L x . ( k − k ′ ) k P φ k N ∗ . To estimate the third term from (3.38) we first use Bernstein in time to obtain2 k (cid:13)(cid:13)(cid:13) Q k (cid:0) Q > k + C ( χ I ) e ± i ψ ± k ′ P φ (cid:1)(cid:13)(cid:13)(cid:13) L t L x ≤ k X j ≥ k + C (cid:13)(cid:13)(cid:13) Q k (cid:0) Q j ( χ I ) Q j + O (1) ( e ± i ψ ± k ′ P φ ) (cid:1)(cid:13)(cid:13)(cid:13) L t L x . k X j ≥ k + C k Q j ( χ I ) k L t (cid:13)(cid:13)(cid:13) Q j + O (1) e ± i ψ ± k ′ P φ (cid:13)(cid:13)(cid:13) L t L x . k k ′ + C X j = k + C − j (cid:13)(cid:13)(cid:13) Q j + O (1) e ± i ψ ± k ′ P φ (cid:13)(cid:13)(cid:13) L t L x + k X j > k ′ + C − j (cid:13)(cid:13)(cid:13) Q j + O (1) e ± i ψ ± k ′ P φ (cid:13)(cid:13)(cid:13) L t L x . For the first sum we use Proposition 9 from [22], for the second sum we first note that there is nomodulation interference since k ′ < j − C and then use the fixed-time L x → L x estimate for e ± i ψ ± k ′ .Hence, . k k ′ + C X j = k + C − j − j δ ( j − k ′ ) k∇ t , x A k L x k P φ k N ∗ + k X j > k ′ + C − j k P φ k X , ∞ . (2 δ ( j − k ′ ) + k − k ′ ) k P φ k N ∗ . Putting things together we find that2 k (cid:13)(cid:13)(cid:13) Q k (cid:0) χ I e ± i ψ ± k ′ P φ (cid:1)(cid:13)(cid:13)(cid:13) L t L x . δ ( k − k ′ ) k P φ k N ∗ and for su ffi ciently large | k | ≫ | k ′ | we therefore trivially gain a smallness factor ε from 2 δ ( k − k ′ ) . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 29
We are thus reduced to the case k = k ′ ± O (1) and it remains to show that2 k (cid:13)(cid:13)(cid:13) Q k (cid:0) χ I e ± i ψ ± k P φ (cid:1)(cid:13)(cid:13)(cid:13) L t L x . ε k P φ k N ∗ . As in the proof of Proposition 9 in [22] we expand the untruncated group element e ± i ψ = e ± i ψ < k − C ± i Z l > k − C ψ l e ± i ψ < k − C dl − " l , l ′ > k − C ψ l ψ l ′ e ± i ψ < k − C dl ′ dl ∓ i $ l , l ′ , l ′′ > k − C ψ l ψ l ′ ψ l ′′ e ± i ψ < l ′′ dl ′′ dl ′ dl = Z + L + Q + C and estimate each of these terms seperately. Zero order term Z : From Lemma 3.16 we immediately obtain that (cid:13)(cid:13)(cid:13) Q k (cid:0) χ I e ± i ψ < k − C k P φ (cid:1)(cid:13)(cid:13)(cid:13) L t L x . ε − k k P φ k N ∗ . Linear term L : We have to show that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Q k (cid:16) χ I Z l > k − C S k (cid:0) ψ l e ± i ψ < k − C (cid:1) P φ dl (cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L t L x . ε − k k P φ k N ∗ . To this end we decompose ψ l into a small and a large angular part(3.39) ψ l = X σ l <θ< − C ψ ( θ ) l + X − C ≤ θ . ψ ( θ ) l . In order to bound the small angular part we split χ I = Q ≥ k − C ( χ I ) + Q < k − C ( χ I ) . Using Lemma 3.5, we estimate the first term by (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Q k (cid:18) Q ≥ k − C ( χ I ) Z l > k − C S k (cid:16) X σ l <θ< − C ( ψ ( θ ) l ) e ± i ψ < k − C (cid:17) P φ dl (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L t L x . (cid:13)(cid:13)(cid:13) Q ≥ k − C ( χ I ) (cid:13)(cid:13)(cid:13) L t Z l > k − C X σ l <θ< − C k ψ ( θ ) l k D θ ( L t L ∞ x ) (cid:13)(cid:13)(cid:13) e ± i ψ < k − C P φ (cid:13)(cid:13)(cid:13) L ∞ t L x dl . − k Z l > k − C X σ l <θ< − C θ − l k∇ t , x A k L x k P φ k L ∞ t L x dl . − C − k k P φ k N ∗ . For the second term we have Q k (cid:18) Q < k − C ( χ I ) Z l > k − C S k (cid:16) X σ l <θ< − C ψ ( θ ) l e ± i ψ < k − C (cid:17) P φ dl (cid:19) = Q k (cid:18) Q < k − C ( χ I ) Q k + O (1) Z l > k − C S k (cid:16) X σ l <θ< − C ψ ( θ ) l e ± i ψ < k − C (cid:17) P φ dl (cid:19) . Then since ψ ( θ ) l is a free wave, we can write this as Q k (cid:18) Q < k − C ( χ I ) Q k + O (1) Z l > k − C S k (cid:16) X σ l <θ< − C ψ ( θ ) l e ± i ψ < k − C (cid:17) Q k + O (1) P φ dl (cid:19) and estimate by (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Q k (cid:18) Q < k − C ( χ I ) Q k + O (1) Z l > k − C S k (cid:16) X σ l <θ< − C ψ ( θ ) l e ± i ψ < k − C (cid:17) Q k + O (1) P φ dl (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L t L x . Z l > k − C X σ l <θ< − C (cid:13)(cid:13)(cid:13) ψ ( θ ) l (cid:13)(cid:13)(cid:13) D θ ( L t L ∞ x ) (cid:13)(cid:13)(cid:13) e ± i ψ < k − C Q k + O (1) P φ (cid:13)(cid:13)(cid:13) L t L x dl . Z l > k − C X σ l <θ< − C − l − θ k∇ t , x A k L x (cid:13)(cid:13)(cid:13) Q k + O (1) P φ (cid:13)(cid:13)(cid:13) L t L x dl . − C − k k (cid:13)(cid:13)(cid:13) Q k + O (1) P φ (cid:13)(cid:13)(cid:13) L t L x . − C − k k P φ k N ∗ . Here we used Lemma 3.5, the fixed-time L x → L x estimate for e ± iT z ψ < k − C and then Bernstein intime.The large angular part in (3.39) has to be estimated more carefully. Noting that the symbollocalization S k (cid:0) ψ l e ± i ψ < k − C (cid:1) can be represented as(3.40) S k (cid:0) ψ l e ± i ψ < k − C (cid:1) = Z R + z m k ( z )( T z ψ l ) e ± iT z ψ < k − C dz , where m k is an integrable bump function at scale 2 − k and T z denotes translation in space-timedirection z ∈ R + , we derive the following key estimate for the large angular part (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Q k (cid:18) χ I Z l > k − C Z R + z m k ( z ) (cid:16) X − C ≤ θ . ( T z ψ ( θ ) l ) (cid:17) e ± iT z ψ < k − C P φ dz dl (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L t L x . C / − k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z l > k − C − ( l − k ) Z R + z | m k ( z ) | X − C ≤ θ . X Γ νθ sup ω ∈ Γ νθ (cid:16) − l (cid:13)(cid:13)(cid:13) b νθ ( ω ) Π ωθ ∇ t , x T z A l (cid:13)(cid:13)(cid:13) L x (cid:17) dz dl (cid:19) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L t ( I ) ×× (cid:18)Z l > k − C − ( l − k ) Z R + z | m k ( z ) | (cid:13)(cid:13)(cid:13) e ± iT z ψ < k − C P φ (cid:13)(cid:13)(cid:13) L ∞ t L x dz dl (cid:19) . C / − k ×× (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z l > k − C − ( l − k ) Z R + z | m k ( z ) | X − C ≤ θ . X Γ νθ sup ω ∈ Γ νθ (cid:16) − l (cid:13)(cid:13)(cid:13) b νθ ( ω ) Π ωθ ∇ t , x T z A l (cid:13)(cid:13)(cid:13) L x (cid:17) dz dl (cid:19) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L t ( I ) k P φ k N ∗ . This estimate can be proven by carefully opening up the proof of the decomposable estimates inLemma 3.5. We emphasize that uniformly for all integers k <
0, the quantity (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z l > k − C − ( l − k ) Z R + z | m k ( z ) | X − C ≤ θ . X Γ νθ sup ω ∈ Γ νθ (cid:16) − l (cid:13)(cid:13)(cid:13) b νθ ( ω ) Π ωθ ∇ t , x T z A l (cid:13)(cid:13)(cid:13) L x (cid:17) dz dl (cid:19) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L t ( R ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z R X k < l + C − ( l − k ) Z R + z | m k ( z ) | X − C ≤ θ . X Γ νθ sup ω ∈ Γ νθ (cid:16) − l (cid:13)(cid:13)(cid:13) b νθ ( ω ) Π ωθ ∇ t , x T z A l (cid:13)(cid:13)(cid:13) L x (cid:17) dz dl (cid:19) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L t ( R ) is bounded by k∇ t , x A k L x by Strichartz estimates.By first fixing C > ffi ciently large and then suitably choosing the intervals I j , the estimateof the linear term L follows. ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 31
Quadratic and cubic terms Q and C : Using the above ideas these can be estimated similarly. Weomit the details. (cid:3)
Proof of Lemma 3.16.
As in [22, Lemma 10] we write the symbol as S l e ± i ψ < k = ( ± i ) − l Y r = [ S ( r ) l ∂ t ψ < k ] e ± i ψ < k , where the product denotes a nested multiplication by S l ∂ t ψ < k for a series of frequency cuto ff s S ( r + l S ( r ) l = S ( r ) l ≈ S l with expanding widths. Then we have S ( r ) l ∂ t ψ < k = Z R + zr m ( r ) l ( z r )( T z r ∂ t ψ < k ) dz r , where m ( r ) l is an integrable bump function at scale 2 − l and T z r denotes translation in space-timedirection z r ∈ R + . The claim now reduces to proving that the intervals I j can be chosen such thatuniformly for j = , . . . , J and all integers k ≤ l − C , it holds that(3.41) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) χ I j (cid:18) Y r = Z R + zr m ( r ) l ( z r )( T z r ∂ t ψ < k ) (cid:19) e ± i ψ < k ( t , x , D ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L pt L x → L qt L x . ε k l ( p − q ) k . To this end we show that the intervals I j can be chosen such that uniformly for j = , . . . , J , allintegers k ≤ l − C and all integers k , . . . , k < k , we have the operator bound (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) χ I j (cid:18) Y r = Z R + zr m ( r ) l ( z r )( T z r ∂ t ψ k r ) (cid:19) e ± i ψ < k ( t , x , D ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L pt L x → L qt L x . ε l − q k (1 − q ) k · · · (1 − q ) k , (3.42)where q = q − p . By summing over dyadic frequencies, the estimate (3.41) then follows.In order to prove (3.42), we split ∂ t ψ k into a small and a large angular part ∂ t ψ k = X σ k <θ < − C ∂ t ψ ( θ ) k + X − C ≤ θ . ∂ t ψ ( θ ) k for some constant C > ffi ciently large later in the proof. We estimate the smallangular part using H ¨older-type estimates for decomposable function spaces (3.23) and the bounds(3.26) for the phase, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) χ I j (cid:18) X σ k <θ < − C Z R + z m (1) l ( z )( T z ∂ t ψ ( θ ) k ) dz (cid:19)(cid:18) Y r = Z R + zr m ( r ) l ( z r )( T z r ∂ t ψ k r ) dz r (cid:19) e ± i ψ < k ( t , x , D ) φ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L qt L x . (cid:18) X σ k <θ < − C Z R + z | m (1) l ( z ) | (cid:13)(cid:13)(cid:13) T z ∂ t ψ ( θ ) k (cid:13)(cid:13)(cid:13) D θ ( L qt L ∞ x ) dz (cid:19)(cid:18) Y r = Z R + zr | m ( r ) l ( z r ) | (cid:13)(cid:13)(cid:13) T z r ∂ t ψ k r (cid:13)(cid:13)(cid:13) D ( L qt L ∞ x ) dz r (cid:19) k φ k L pt L x . (cid:18) X σ k <θ < − C (1 − q ) k θ − q k∇ t , x A k L x (cid:19)(cid:18) Y r = (1 − q ) k r k∇ t , x A k L x (cid:19) k φ k L pt L x . − C ( − q ) l − q k (1 − q ) k · · · (1 − q ) k k∇ t , x A k L x k φ k L pt L x . Here we dropped the time cuto ff χ I j and used the space-time translation invariance of the decom-posable function spaces. For the large angular part we establish the crucial estimate (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) χ I j (cid:18) X − C ≤ θ . Z R + z m (1) l ( z )( T z ∂ t ψ ( θ ) k ) dz (cid:19)(cid:18) Y r = Z R + zr m ( r ) l ( z r )( T z r ∂ t ψ k r ) dz r (cid:19) e ± i ψ < k ( t , x , D ) φ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L qt L x . l − q k (1 − q ) k · · · (1 − q ) k k∇ t , x A k L x C / ×× (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) k − l Z R + z | m (1) l ( z ) | X − C ≤ θ . X Γ ν θ sup ω ∈ Γ ν θ (cid:16) ( q + r − k k b ν θ ( ω ) Π ωθ ∇ t , x T z A k k L r x (cid:17) dz (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L qt ( I j ) k φ k L pt L x , where r ≥ q , r ) is sharp wave admissible. This estimate can beproven by carefully opening up the proof of Lemma 3.5 and of H ¨older-type estimates for decom-posable function spaces (3.23).Noting that uniformly for all integers k ≤ l − C , the quantity (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) k − l Z R + z | m (1) l ( z ) | X − C ≤ θ . X Γ ν θ sup ω ∈ Γ ν θ (cid:16) ( q + r − k k b ν θ ( ω ) Π ωθ ∇ t , x T z A k k L r x (cid:17) dz (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L qt ( R ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)X l ∈ Z X k ≤ l − C k − l Z R + z | m (1) l ( z ) | X − C ≤ θ . X Γ ν θ sup ω ∈ Γ ν θ (cid:16) ( q + r − k k b ν θ ( ω ) Π ωθ ∇ t , x T z A k k L r x (cid:17) dz (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L qt ( R ) is bounded by k∇ t , x A k L x by Strichartz estimates, the assertion follows by first choosing C > ffi ciently large and then suitably choosing the intervals I j . (cid:3) Proposition 3.17.
For any ε > the intervals I j can be chosen such that uniformly for all j = , . . . , J and all integers k ≤ k ′ ± O (1) < , it holds that (3.43) (cid:13)(cid:13)(cid:13) Q k (cid:0) χ I j e − i ψ ± k ′ ( D , y , t ) P φ (cid:1)(cid:13)(cid:13)(cid:13) L t L x ( R × R ) . ε − k δ ( k − k ′ ) k P φ k N ∗ ( R × R ) . Proof.
The proof proceeds analogously to the one of Proposition 3.15 using Lemma 3.6 in place ofLemma 3.5. (cid:3)
Proof of the N → N and N ∗ → N ∗ bounds (3.18) for χ I j e ± i ψ ± < .Proposition 3.18. For j = , . . . , J it holds that (3.44) (cid:13)(cid:13)(cid:13) χ I j e ± i ψ ± < ( t , x , D ) P φ (cid:13)(cid:13)(cid:13) N ( R × R ) . (cid:13)(cid:13)(cid:13) P φ (cid:13)(cid:13)(cid:13) N ( R × R ) and (3.45) (cid:13)(cid:13)(cid:13) χ I j e ± i ψ ± < ( D , y , t ) P φ (cid:13)(cid:13)(cid:13) N ( R × R ) . (cid:13)(cid:13)(cid:13) P φ (cid:13)(cid:13)(cid:13) N ( R × R ) . Proof.
We begin with the proof of (3.44). To simplify the notation we denote an interval I j justby I in what follows. If φ is an L t L x atom, the claim follows immediately from the fixed-time L x → L x estimate for e ± i ψ ± < ( t , x , D ). The key point is therefore to show that if φ is an X , − atom atmodulation k , then we have (cid:13)(cid:13)(cid:13) χ I e ± i ψ ± < ( t , x , D ) Q k P φ (cid:13)(cid:13)(cid:13) N . − k k P φ k L t L x . By duality, this is equivalent to proving (cid:13)(cid:13)(cid:13) Q k (cid:0) χ I e ± i ψ ± < ( D , y , t ) P φ (cid:1)(cid:13)(cid:13)(cid:13) L t L x . − k k P φ k N ∗ . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 33
As in [22, Proposition 9.1] we now write Q k (cid:0) χ I e ± i ψ ± < ( D , y , t ) P φ (cid:1) = Q k (cid:0) χ I e ± i ψ ± < k − C ( D , y , t ) P φ (cid:1) + Q k (cid:0) χ I ( e ± i ψ ± < − e ± i ψ ± < k − C )( D , y , t ) P φ (cid:1) = Q k (cid:0) Q < k − C ( χ I ) e ± i ψ ± < k − C ( D , y , t ) P φ (cid:1) + Q k (cid:0) Q ≥ k − C ( χ I ) e ± i ψ ± < k − C ( D , y , t ) P φ (cid:1) + Q k (cid:0) χ I ( e ± i ψ ± < − e ± i ψ ± < k − C )( D , y , t ) P φ (cid:1) . For the first term we obtain2 k (cid:13)(cid:13)(cid:13) Q k (cid:0) Q < k − C ( χ I ) e ± i ψ ± < k − C ( D , y , t ) P φ (cid:1)(cid:13)(cid:13)(cid:13) L t L x . k P φ k X , ∞ . k P φ k N ∗ , because the output modulation directly transfers to φ . We estimate the second term by2 k (cid:13)(cid:13)(cid:13) Q k (cid:0) Q ≥ k − C ( χ I ) e ± i ψ ± < k − C ( D , y , t ) P φ (cid:1)(cid:13)(cid:13)(cid:13) L t L x . k (cid:13)(cid:13)(cid:13) Q ≥ k − C ( χ I ) (cid:13)(cid:13)(cid:13) L t (cid:13)(cid:13)(cid:13) e ± i ψ ± < k − C ( D , y , t ) P φ (cid:13)(cid:13)(cid:13) L ∞ t L x . k P φ k N ∗ , where we used that (cid:13)(cid:13)(cid:13) Q ≥ k − C ( χ I ) (cid:13)(cid:13)(cid:13) L t . − k . To deal with the last term we use Proposition 3.17.The proof of (3.45) works similarly using Proposition 3.15. (cid:3) In a similar vein we obtain the following N ∗ → N ∗ bounds. Proposition 3.19.
For j = , . . . , J it holds that (3.46) (cid:13)(cid:13)(cid:13) χ I j e ± i ψ ± < ( t , x , D ) P φ (cid:13)(cid:13)(cid:13) N ∗ ( R × R ) . (cid:13)(cid:13)(cid:13) P φ (cid:13)(cid:13)(cid:13) N ∗ ( R × R ) and (3.47) (cid:13)(cid:13)(cid:13) χ I j e ± i ψ ± < ( D , y , t ) P φ (cid:13)(cid:13)(cid:13) N ∗ ( R × R ) . (cid:13)(cid:13)(cid:13) P φ (cid:13)(cid:13)(cid:13) N ∗ ( R × R ) . Proof of the N → ε N and N ∗ → ε N ∗ bounds (3.19) for χ I j ∂ t e ± i ψ ± < .Proposition 3.20. For any ε > the intervals I j can be chosen such that uniformly for all j = , . . . , J it holds that (3.48) (cid:13)(cid:13)(cid:13) χ I j ∂ t e ± i ψ ± < ( t , x , D ) P φ (cid:13)(cid:13)(cid:13) N ( R × R ) . ε k P φ k N ( R × R ) and (3.49) (cid:13)(cid:13)(cid:13) χ I j ∂ t e ± i ψ ± < ( D , y , t ) P φ (cid:13)(cid:13)(cid:13) N ( R × R ) . ε k P φ k N ( R × R ) . Proof.
We proceed as in the proof of Proposition 3.18 using the L x → ε L x bound for ∂ t e ± i ψ ± < andthat we have for k ≤ k ′ ± O (1), (cid:13)(cid:13)(cid:13) Q k (cid:0) χ I j ∂ t e − i ψ ± k ′ P φ (cid:1)(cid:13)(cid:13)(cid:13) L t L x ( R × R ) . ε − k δ ( k − k ′ ) k P φ k N ∗ ( R × R ) for both left and right quantization. The latter estimate can be proven similarly to the proof ofProposition 3.15. (cid:3) Proposition 3.21.
For any ε > the intervals I j can be chosen such that uniformly for all j = , . . . , J it holds that (3.50) (cid:13)(cid:13)(cid:13) χ I j ∂ t e ± i ψ ± < ( t , x , D ) P φ (cid:13)(cid:13)(cid:13) N ∗ ( R × R ) . ε k P φ k N ∗ ( R × R ) and (3.51) (cid:13)(cid:13)(cid:13) χ I j ∂ t e ± i ψ ± < ( D , y , t ) P φ (cid:13)(cid:13)(cid:13) N ∗ ( R × R ) . ε k P φ k N ∗ ( R × R ) . Proof of the renormalization error estimate (3.20) .Proposition 3.22.
For any ε > the intervals I j can be chosen such that uniformly for all j = , . . . , J we have (3.52) (cid:13)(cid:13)(cid:13) χ I j (cid:0) e − i ψ ± < ( t , x , D ) e + i ψ ± < ( D , y , t ) − (cid:1) P φ (cid:13)(cid:13)(cid:13) N ( R × R ) . ε k P φ k N ( R × R ) and (3.53) (cid:13)(cid:13)(cid:13) χ I j (cid:0) e − i ψ ± < ( t , x , D ) e + i ψ ± < ( D , y , t ) − (cid:1) P φ (cid:13)(cid:13)(cid:13) N ∗ ( R × R ) . ε k P φ k N ∗ ( R × R ) . Proof.
We prove the N ∗ → ε N ∗ estimate (3.53). The bound (3.52) then follows by duality. The L ∞ t L x part of (3.53) follows immediately from the fixed-time L x → ε L x estimate (3.36). The X , ∞ part reduces to showing that we can choose the intervals I j such that uniformly for j = , . . . , J andall k ∈ Z , 2 k (cid:13)(cid:13)(cid:13) Q k (cid:0) χ I j (cid:0) e − i ψ ± < ( t , x , D ) e i ψ ± < ( D , y , t ) − (cid:1) P φ (cid:1)(cid:13)(cid:13)(cid:13) L t L x ( R × R ) . ε k P φ k N ∗ ( R × R ) . We use the notation R < k = e − i ψ ± < k ( t , x , D ) e i ψ ± < k ( D , y , t )to write Q k (cid:0) χ I j ( R < − P φ (cid:1) = Q k (cid:0) χ I j ( R < − Q > k − C P φ (cid:1) + Q k (cid:0) χ I j ( R < k − C − Q ≤ k − C P φ (cid:1) + Q k (cid:0) χ I j ( R < − R < k − C ) Q ≤ k − C P φ (cid:1) . (3.54)Using the fixed-time L x → ε L x estimate (3.36) for ( R < − k (cid:13)(cid:13)(cid:13) Q k (cid:0) χ I j ( R < − Q > k − C P φ (cid:1)(cid:13)(cid:13)(cid:13) L t L x . k (cid:13)(cid:13)(cid:13) ( R < − Q > k − C P φ (cid:13)(cid:13)(cid:13) L t L x . k ε k Q > k − C P φ k L t L x . ε k P φ k X , ∞ . To estimate the second term in (3.54) we observe that we have Q k (cid:0) χ I j ( R < k − C − Q ≤ k − C P φ (cid:1) = Q k (cid:0) ( Q [ k − C , k + C ] χ I j )( R < k − C − Q ≤ k − C P φ (cid:1) and hence by the fixed-time L x → ε L x estimate for ( R < k − C − k (cid:13)(cid:13)(cid:13) Q k (cid:0) χ I j ( R < k − C − Q ≤ k − C P φ (cid:1)(cid:13)(cid:13)(cid:13) L t L x . k (cid:13)(cid:13)(cid:13) Q [ k − C , k + C ] ( χ I j ) (cid:13)(cid:13)(cid:13) L t (cid:13)(cid:13)(cid:13) ( R < k − C − Q ≤ k − C P φ (cid:13)(cid:13)(cid:13) L ∞ t L x . ε k P φ k L ∞ t L x . Finally, we expand the third term in (3.54) as follows Q k (cid:0) χ I j ( R < − R < k − C ) Q ≤ k − C P (cid:1) = Q k (cid:16) χ I j (cid:0) e − i ψ ± < ( t , x , D ) − e − i ψ ± < k − C ( t , x , D ) (cid:1) e i ψ ± < ( D , y , t ) Q ≤ k − C P φ (cid:17) + Q k (cid:16) χ I j e − i ψ ± < k − C ( t , x , D ) (cid:0) e i ψ ± < ( D , y , t ) − e i ψ ± < k − C ( D , y , t ) (cid:1) Q ≤ k − C P φ (cid:17) . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 35
To handle the first term in the above expansion we use Proposition 3.15 and the N ∗ → N ∗ estimate(3.47) for e i ψ ± < ( D , y , t ) to find that2 k (cid:13)(cid:13)(cid:13) Q k (cid:0) χ I j (cid:0) e − i ψ ± < ( t , x , D ) − e − i ψ ± < k − C ( t , x , D ) (cid:1) e i ψ ± < ( D , y , t ) Q ≤ k − C P φ (cid:13)(cid:13)(cid:13) L t L x . k ′ = X k ′ = k − C k (cid:13)(cid:13)(cid:13) Q k (cid:0) χ I j e − i ψ ± k ′ ( t , x , D ) e i ψ ± < ( D , y , t ) Q ≤ k − C P φ (cid:13)(cid:13)(cid:13) L t L x . k ′ = X k ′ = k − C ε δ ( k − k ′ ) (cid:13)(cid:13)(cid:13) e i ψ ± < ( D , y , t ) Q ≤ k − C P φ (cid:13)(cid:13)(cid:13) N ∗ . ε k P φ k N ∗ . Observing that Q k (cid:16) χ I j e − i ψ ± < k − C ( t , x , D ) (cid:0) e i ψ ± < ( D , y , t ) − e i ψ ± < k − C ( D , y , t ) (cid:1) Q ≤ k − C P φ (cid:17) = Q k (cid:16) e − i ψ ± < k − C ( t , x , D ) Q k + O (1) (cid:16) χ I j (cid:0) e i ψ ± < ( D , y , t ) − e i ψ ± < k − C ( D , y , t ) (cid:1) Q ≤ k − C P φ (cid:17)(cid:17) , we estimate the second term analogously using the fixed-time L x → L x estimate for e − i ψ ± < k − C ( t , x , D )and Proposition 3.17. (cid:3) Proof of the renormalization error estimate (3.21) . This estimate can be proven by adaptingthe proof in [22, Section 10.2] to our large data setting using similar ideas as above. The additionalerrors generated by the high-angle cuto ff for intermediate frequencies in the definition of the phasecorrection ψ ± can be controlled by divisibility of suitable space-time norms of A .3.10. Proof of the dispersive estimate (3.22) . Since the S space is compatible with time local-izations by Lemma 2.1, the dispersive estimate (3.22) follows immediately from the estimate (83)in [22]. 4. B reakdown criterion Definition 4.1.
Let T , T > . For any admissible solution ( A , φ ) to the MKG-CG system on ( − T , T ) × R , we define k ( A , φ ) k S (( − T , T ) × R ) : = sup < T < T , < T ′ < T (cid:18) X j = k A j k S ([ − T , T ′ ] × R ) + k φ k S ([ − T , T ′ ] × R ) (cid:19) . We establish the following blowup criterion for admissible solutions to the MKG-CG system.
Proposition 4.2.
Let ( − T , T ) be the maximal interval of existence of an admissible solution ( A , φ ) to the MKG-CG system. If k ( A , φ ) k S (( − T , T ) × R ) < ∞ , then it must hold that T = T = ∞ . The idea of the proof of Proposition 4.2 is to establish an a priori bound on a subcritical normsup t ∈ ( − T , T ) 4 X j = (cid:13)(cid:13)(cid:13) A j [ t ] (cid:13)(cid:13)(cid:13) H sx × H s − x + (cid:13)(cid:13)(cid:13) φ [ t ] (cid:13)(cid:13)(cid:13) H sx × H s − x < ∞ for some s >
1. By the local well-posedness result [36] for the MKG-CG system it then followsthat the solution can be smoothly extended beyond the time interval ( − T , T ). To this end, we will use Tao’s device of frequency envelopes. For su ffi ciently small σ > k ∈ Z , c k : = (cid:18)X l ∈ Z − σ | k − l | (cid:16) X j = (cid:13)(cid:13)(cid:13) P l A j [0] (cid:13)(cid:13)(cid:13) H x × L x + (cid:13)(cid:13)(cid:13) P l φ [0] (cid:13)(cid:13)(cid:13) H x × L x (cid:17)(cid:19) . Proposition 4.2 is then a consequence of the following result.
Proposition 4.3.
Let ( − T , T ) be the maximal interval of existence of an admissible solution ( A , φ ) to the MKG-CG system. If k ( A , φ ) k S (( − T , T ) × R ) < ∞ , there exists C = C ( k ( A , φ ) k S (( − T , T ) × R ) ) < ∞ such that for all k ∈ Z , (4.1) (cid:13)(cid:13)(cid:13) P k A (cid:13)(cid:13)(cid:13) S k (( − T , T ) × R ) + (cid:13)(cid:13)(cid:13) P k φ (cid:13)(cid:13)(cid:13) S k (( − T , T ) × R ) ≤ Cc k . Proof.
A sketch of the proof is given in Subsection 7.4. (cid:3)
5. A concept of weak evolution
In order to implement the contradiction argument after the concentration compactness step, wehave to define the notion of a solution to the MKG-CG system that is merely of energy class. Inthe context of critical wave maps in [20] this is achieved by first approximating an energy classdatum by Schwartz class data in the energy topology. One then defines the energy class solutionas a suitable limit of the associated Schwartz class solutions. Using perturbation theory, one showsthat this limit is well-defined and independent of the approximating sequence.For the MKG-CG system we have to argue more carefully, because it appears that the strongperturbative step in the context of the critical wave maps in [20] is not available due to a lowfrequency divergence. However, the problem with evolving irregular data is really a “high frequencyissue” and it appears that truncating high frequencies away does not lead to the same problems as ageneral perturbative step. More concretely, consider Coulomb energy class data at time t =
0. Bytruncating in frequency, we can assume that the frequency support of either input is at | ξ | ≤ K forsome K >
0. Then the problem becomes to show that we can add high-frequency perturbations tothe data, i.e. supported in frequency space at | ξ | > K at time t =
0, and to obtain a perturbed globalevolution.
Proposition 5.1.
Let ( A , φ ) be an admissible solution to the MKG-CG system on [ − T , T ] × R for some T , T > . Assume that ( A , φ )[0] have frequency support at | ξ | ≤ K for some K > andthat k ( A , φ ) k S ([ − T , T ] × R ) = L < ∞ . Then there exists δ ( L ) > with the following property: Let ( A + δ A , φ + δφ ) be any other admissible solution to the MKG-CG system defined locally aroundt = such that E ( δ A , δφ )(0) = δ < δ ( L ) and such that ( δ A , δφ )[0] have frequency support at | ξ | > K. Then ( A + δ A , φ + δφ ) extends to anadmissible solution to the MKG-CG system on the whole time interval [ − T , T ] and satisfies k ( A + δ A , φ + δφ ) k S ([ − T , T ] × R ) ≤ ˜ L ( L , δ ) . Moreover, we have k ( δ A , δφ ) k S ([ − T , T ] × R ) → as δ → .Proof. A sketch of the proof is given in Subsection 7.4. (cid:3)
ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 37
The above high-frequency perturbation result suggests that we could define the MKG-CG evo-lution of energy class Coulomb data as a suitable limit of the evolutions of low frequency approxi-mations of the energy class data. More precisely, for Coulomb data ( A , φ )[0] ∈ ˙ H x × L x , we pick asequence of smoothings ( A n , φ n )[0] by truncating the frequency support of ( A , φ )[0] so thatlim n →∞ ( A n , φ n )[0] = ( A , φ )[0]in the sense of ˙ H x × L x . Here the rather technical issue appears whether there exists a smooth (local)solution ( A n , φ n ) to the MKG-CG system with initial data ( A n , φ n )[0]. The hypothesis ( A , φ )[0] ∈ ˙ H x × L x does not guarantee that A (0) and φ (0) are L integrable in the low frequencies. For thisreason we cannot directly invoke the local well-posedness result [36] to obtain a smooth localsolution. The natural way around this is to localize in physical space. This will be explained inmore detail in Subsection 5.2 below.For each smooth local solution ( A n , φ n ) to the MKG-CG system with initial data ( A n , φ n )[0] wethen define I n : = ∪ ˜ I ∈A n ˜ I , where A n : = (cid:8) ˜ I ⊂ R open interval with 0 ∈ ˜ I : sup J ⊂ ˜ I , Jclosed k ( A n , φ n ) k S ( J × R ) < ∞ (cid:9) . We call I n the maximal lifespan of the solution ( A n , φ n ).In order to define a canonical evolution of Coulomb energy class data, we have to show thatthe low frequency approximations ( A n , φ n ) exist on some joint time interval and satisfy uniform S norm bounds there. Proposition 5.2.
Let ( A , φ )[0] be Coulomb energy class data and let (cid:8) ( A n , φ n )[0] (cid:9) n be a sequenceof smooth low frequency truncations of ( A , φ )[0] such that lim n →∞ ( A n , φ n )[0] = ( A , φ )[0] in the sense of ˙ H x × L x . Denote by ( A n , φ n ) the smooth solutions to the MKG-CG system withinitial data ( A n , φ n )[0] and with maximal intervals of existence I n . Then there exists a time T ≡ T ( A , φ ) > such that [ − T , T ] ⊂ I n for all su ffi ciently large n and lim sup n →∞ k ( A n , φ n ) k S ([ − T , T ] × R ) ≤ C ( A , φ ) , where C ( A , φ ) > is a constant that depends only on the energy class data ( A , φ )[0] .Proof. The proof is given in Subsection 5.1 below. (cid:3)
Using Proposition 5.1 and Proposition 5.2, we may introduce the following notion of energyclass solutions to the MKG-CG system that we outlined above.
Definition 5.3.
Let ( A , φ )[0] be Coulomb energy class data and let { ( A n , φ n )[0] } n be a sequence ofsmooth low frequency truncations of ( A , φ )[0] such that lim n →∞ ( A n , φ n )[0] = ( A , φ )[0] in the sense of ˙ H x × L x . We denote by ( A n , φ n ) the smooth solutions to the MKG-CG system withinitial data ( A n , φ n )[0] and define I = ( − T , T ) = ∪ ˜ I to be the union of all open time intervals ˜ Icontaining with the property that sup J ⊂ ˜ I , Jclosed lim inf n →∞ k ( A n , φ n ) k S ( J × R ) < ∞ . Then we define the MKG-CG evolution of ( A , φ )[0] on I × R to be ( A , φ )[ t ] : = lim n →∞ ( A n , φ n )[ t ] , t ∈ I , where the limit is taken in the energy topology. We call I the maximal lifespan of ( A , φ ) . For anyclosed interval J ⊂ I, we set k ( A , φ ) k S ( J × R ) : = lim n →∞ k ( A n , φ n ) k S ( J × R ) . We obtain the following characterization of the maximal lifespan I of an energy class solution. Lemma 5.4.
Let ( A , φ ) , ( A n , φ n ) and I be as in Definition 5.3. Assume in addition that I , ( −∞ , ∞ ) .Then sup J ⊂ I , Jclosed lim inf n →∞ k ( A n , φ n ) k S ( J × R ) = ∞ . We call an energy class solution ( A , φ ) with maximal lifespan I singular , if either I , R , or if I = R and sup J ⊂ I , Jclosed k ( A , φ ) k S ( J × R ) = ∞ . Proof of Proposition 5.2.
A natural idea is to localize the data ( A n , φ n )[0] in physical spaceto ensure smallness of the energy and to then try to “patch together” the local solutions obtainedfrom the small energy global well-posedness result [22]. The problem is that the MKG-CG systemdoes not have the finite speed of propagation property due to non-local terms in the equation forthe magnetic potential A . To overcome this di ffi culty, we exploit that the Maxwell-Klein-Gordonsystem enjoys gauge invariance.We first describe how we suitably localize the data ( A n , φ n )[0] in physical space to obtain admis-sible Coulomb data with small energy that can be globally evolved by [22]. Let χ ∈ C ∞ c ( R ) be asmooth cuto ff function with support in the ball B (0 , ) and such that χ ≡ B (0 , ). For x ∈ R and r >
0, we set χ {| x − x | . r } ( x ) : = χ ( x − x r ). Then we define(5.1) γ n (0 , · ) : = ∆ − ∂ j (cid:0) χ {| x − x | . r } ( · ) A jn (0 , · ) (cid:1) and for j = , . . . , A n , j (0 , · ) : = χ {| x − x | . r } ( · ) A n , j (0 , · ) − ∂ j γ n (0 , · ) . We determine ˜ A n , (0 , · ) as the solution to the elliptic equation(5.3) ∆ ˜ A n , = − Im (cid:0) χ {| x − x | . r } φ n χ {| x − x | . r } ∂ t φ n (cid:1) + | χ {| x − x | . r } φ n | A n , on R , where φ n and A n , are evaluated at time t =
0. We note that ˜ A n is in Coulomb gauge. Then we set(5.4) ∂ t γ n (0 , · ) : = A n , (0 , · ) − ˜ A n , (0 , · )and define ∂ t ˜ A n , j (0 , · ) for j = , . . . ,
4, first just on B ( x , r ), by setting(5.5) ∂ t ˜ A n , j | B ( x , r ) (0 , · ) : = (cid:0) ∂ t A n , j (0 , · ) − ∂ j ∂ t γ n (0 , · ) (cid:1)(cid:12)(cid:12)(cid:12) B ( x , r ) . We observe that ∆ ( A n , (0 , · ) − ˜ A n , (0 , · )) = B ( x , r ) by the definition of ˜ A n , (0 , · ). Thus, thedata ∂ t ˜ A n | B ( x , r ) (0 , · ) satisfy the Coulomb compatibility condition ∂ j ( ∂ t ˜ A jn )(0 , · ) = B ( x , r ).Using [7, Proposition 2.1], we extend ( ∂ t ˜ A n , j )(0 , · ) | B ( x , r ) to the whole of R while maintaining ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 39 the Coulomb compatibility condition and such that k ∂ t ˜ A n , j k L x ( R ) . k ∂ t ˜ A n , j k L x ( B ( x , r )) . Finally, wedefine(5.6) ˜ φ n (0 , · ) : = e i γ n (0 , · ) χ {| x − x | . r } ( · ) φ n (0 , · )and(5.7) ∂ t ˜ φ n (0 , · ) : = i ∂ t γ n (0 , · ) e i γ n (0 , · ) χ {| x − x | . r } ( · ) φ n (0 , · ) + e i γ n (0 , · ) χ {| x − x | . r } ( · ) ∂ t φ n (0 , · ) . In the next lemma we prove that by choosing r > ffi ciently small, we can ensure that theCoulomb data ( ˜ A n , ˜ φ n )[0] have small energy for all su ffi ciently large n . Here we exploit that theconvergence ( A n , φ n )[0] → ( A , φ )[0] in the energy topology as n → ∞ implies a uniform non-concentration property of the energy of the data ( A n , φ n )[0]. We denote by ε > Lemma 5.5.
Let ( ˜ A n , ˜ φ n ) be defined as in (5.1) – (5.7) . Given ε > there exists r > such thatuniformly for all x ∈ R and for all su ffi ciently large n, it holds thatE ( ˜ A n , ˜ φ n ) < ε . Proof.
We start with the components ˜ A n , j . Suppressing that A n is evaluated at time t =
0, we havefor j = , . . . , k∇ x ˜ A n , j k L x ( R ) . k∇ x ( χ {| x − x | . r } A n , j ) k L x ( R ) + k∇ x ∂ j γ n k L x ( R ) . X i = k∇ x ( χ {| x − x | . r } A n , i ) k L x ( R ) . X i = r Z B ( x , r ) | A n , i ( x ) | dx + Z B ( x , r ) |∇ x A n , i ( x ) | dx . X i = (cid:18)Z B ( x , r ) | A n , i ( x ) | dx (cid:19) / + Z B ( x , r ) |∇ x A n , i ( x ) | dx . (5.8)Next we note that we can pick r > A thatsup x ∈ R X i = Z B ( x , r ) |∇ x A i ( x ) | dx + Z B ( x , r ) | A i ( x ) | dx ≪ ε . Since A n → A in ˙ H x ( R ) as n → ∞ , we also obtain for su ffi ciently large n thatsup x ∈ R X i = Z B ( x , r ) |∇ x A n , i ( x ) | dx + Z B ( x , r ) | A n , i ( x ) | dx ≪ ε . From (5.8) we conclude that k∇ x ˜ A n , j k L x ( R ) . ε . In a similar manner we argue that r > ffi ciently large n we also have X i = k ∂ t ˜ A n , i k L x ( R ) + k∇ x ˜ A n , k L x ( R ) + k∇ t , x ˜ φ n k L x ( R ) . ε and hence, E ( ˜ A n , ˜ φ n ) . ε . (cid:3) By Lemma 5.5 we can pick r > A n , ˜ φ n )[0] can be globally evolved forsu ffi ciently large n by the small energy global well-posedness result [22] and we obtain global S norm bounds on their evolutions ( ˜ A n , ˜ φ n ). For t > ∂ t γ n ( t , · ) : = A n , ( t , · ) − ˜ A n , ( t , · ) , which implies that(5.10) γ n ( t , · ) = γ n (0 , · ) + Z t (cid:0) A n , ( s , · ) − ˜ A n , ( s , · ) (cid:1) ds . Our next goal is to relate the evolutions ( ˜ A n , ˜ φ n ) and ( A n , φ n ) on the light cone K x , r = (cid:8) ( t , x ) : 0 ≤ t < r , | x − x | < r − t (cid:9) over the ball B ( x , r ). These identities will be the key ingredient to recover S norm bounds for( A n , φ n ) from those of ( ˜ A n , ˜ φ n ). Lemma 5.6.
Let ( ˜ A n , ˜ φ n ) and γ n be defined as in (5.1) – (5.7) and (5.9) – (5.10) such that E ( ˜ A n , ˜ φ n ) < ε .For all su ffi ciently large n it holds that ˜ A n , j = A n , j − ∂ j γ n on K x , r for j = , . . . , and that ˜ φ n = e i γ n φ n on K x , r . Proof.
To simplify the notation we omit the subscript n . Using the equations that ( A , φ ), ( ˜ A , ˜ φ ), and γ satisfy, we obtain that(5.11) (cid:3) ˜ A j = − Im (cid:0) ˜ φ ˜ D j ˜ φ (cid:1) + ∂ j ∆ − ∂ i Im (cid:0) ˜ φ ˜ D i ˜ φ (cid:1) on R t × R x and(5.12) (cid:3) ( A j − ∂ j γ ) = − Im (cid:0) φ D j φ (cid:1) + ∂ j ∆ − ∂ i Im (cid:0) ˜ φ ˜ D i ˜ φ (cid:1) − ∂ j Z t n Im (cid:0) φ D t φ (cid:1) − Im (cid:0) ˜ φ D t ˜ φ (cid:1)o ds on K x , r , where we use the notation ˜ D α = ∂ α + i ˜ A α . Next we introduce the quantities B j = ˜ A j − ( A j − ∂ j γ )and ψ = ˜ φ − e i γ φ. From (5.11) and (5.12) we infer that (cid:3) B j = Im (cid:0) φ D j φ (cid:1) − Im (cid:0) ˜ φ ˜ D j ˜ φ (cid:1) − ∂ j Z t n Im (cid:0) φ D t φ (cid:1) − Im (cid:0) ˜ φ ˜ D t ˜ φ (cid:1)o ds on K x , r . The first two terms in the above equation can be rewritten asIm (cid:0) φ D j φ (cid:1) − Im (cid:0) ˜ φ ˜ D j ˜ φ (cid:1) = B j | φ | − Im (cid:0) ψ ( ∂ j + i ˜ A j )( ψ + e i γ φ ) (cid:1) − Im (cid:0) e i γ φ ( ∂ j + i ˜ A j ) ψ (cid:1) and similarly we obtain for the last term thatIm (cid:0) φ D t φ (cid:1) − Im (cid:0) ˜ φ ˜ D t ˜ φ (cid:1) = − Im (cid:0) ψ ( ∂ t + i ˜ A )( ψ + e i γ φ ) (cid:1) − Im (cid:0) e i γ ( ∂ t + i ˜ A ) ψ (cid:1) . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 41
We conclude that the wave equation for B j on the light cone K x , r is of the schematic form (cid:3) B j = f B j + f | ψ | + f ψ + f ψ + f ( ∂ j ψ ) + f ( ∂ j ψ ) + ∂ j Z t n f | ψ | + f ψ + f ψ + f ( ∂ t ψ ) + f ( ∂ t ψ ) o ds , (5.13)where f , . . . , f are smooth functions on K x , r . To obtain a wave equation for ψ , we note that B = ˜ A − ( A − ∂ t γ ) = = (cid:3) ˜ A ˜ φ − e i γ (cid:3) A φ = (cid:3) ˜ A ( ψ + e i γ φ ) − e i γ (cid:3) A φ = (cid:3) ˜ A ψ + (cid:3) B + A − ∂γ ( e i γ φ ) − e i γ (cid:3) A φ = (cid:3) ˜ A ψ + (cid:3) B ( e i γ φ ) − (cid:3) ( e i γ φ ) − B j ( A j − ∂ j γ ) e i γ φ + (cid:3) A − ∂γ ( e i γ φ ) − e i γ (cid:3) A φ = (cid:3) ˜ A ψ + i ( ∂ j B j )( e i γ φ ) + iB j ∂ j ( e i γ φ ) − B j B j ( e i γ φ ) − B j ( A j − ∂ j γ ) e i γ φ. Thus, ψ satisfies a wave equation on the light cone K x , r of the schematic form(5.14) (cid:3) ψ = f ψ + f α ∂ α ψ + g j B j + gB j B j + h ( ∂ j B j ) , where f , f α , g , g j , h are smooth functions on K x , r . Since B [0] and ψ [0] vanish on B ( x , r ) byour choice of the initial data ( ˜ A , ˜ φ )[0], we conclude from (5.13) and (5.14) by a standard energyargument that indeed ˜ A j = A j − ∂ j γ on K x , r and ˜ φ = e i γ φ on K x , r . (cid:3) It is clear that given ε >
0, there exists R > ffi ciently large n , it holds that E (cid:0) χ {| x | > R } ( · ) A n (0 , · ) , χ {| x | > R } ( · ) φ n (0 , · ) (cid:1) < ε . For our later purposes we have to localize the initial data ( A n , φ n ) outside the large ball B (0 , R ) ina scaling invariant way. For any x l ∈ R with | x l | ∼ R m for some m ∈ N , we set r l : = R m − .Then we define γ ( l ) n (0 , · ) : = ∆ − ∂ j (cid:0) χ {| x − x l | . r l } ( · ) A jn (0 , · ) (cid:1) and for j = , . . . ,
4, ˜ A ( l ) n , j (0 , · ) : = χ {| x − x l | . r l } ( · ) A n , j (0 , · ) − ∂ j γ ( l ) n (0 , · ) . We define ˜ A ( l ) n , (0 , · ) , ∂ t ˜ A ( l ) n , j (0 , · ) , ˜ φ ( l ) n (0 , · ) , ∂ t ˜ φ ( l ) n (0 , · ) analogously to (5.3) – (5.7) and γ ( l ) n ( t , · ), ∂ t γ ( l ) n ( t , · )for t > Lemma 5.7.
Given ε > there exists R > such that the initial data ( ˜ A ( l ) n , ˜ φ ( l ) n ) defined as abovesatisfy for all su ffi ciently large n that E ( ˜ A ( l ) n , ˜ φ ( l ) n ) < ε . and Lemma 5.8.
For all su ffi ciently large n it holds that ˜ A ( l ) n , j = A n , j − ∂ j γ ( l ) n on K x l , r l for j = , . . . , , and that ˜ φ ( l ) n = e i γ ( l ) n φ n on K x l , r l , where K x l , r l : = (cid:8) ( t , x ) : 0 ≤ t < r l , | x − x l | < r l − t (cid:9) . We now begin with the proof of Proposition 5.2 where we suitably “patch together” the smallenergy global evolutions constructed above.
Proof of Proposition 5.2.
By time reversibility, it su ffi ces to only prove the statement in forwardtime. We pick r > ffi ciently small and R > ffi ciently large according to Lemma 5.5and Lemma 5.7. Then we cover the ball B (0 , R ) ⊂ R by the supports of finitely many cuto ff s χ {| x − x l | . r l } with r l = r for l = , . . . , L for some L ∈ N . We divide the complement B (0 , R ) c of theball B (0 , R ) into dyadic annulli A m : = (cid:8) x ∈ R : 2 R m − < | x | ≤ R m (cid:9) , m ∈ N , and cover each A m by the supports of finitely many suitable cuto ff s χ {| x − x l | . r l } ( · ) with | x l | ∼ R m and r l ∼ R m − .This can be carried out in such a way that (cid:8) supp( χ {| x − x l | . r l } ) (cid:9) ∞ l = is a uniformly finitely overlappingcovering of R . We denote by ( ˜ A ( l ) n , ˜ φ ( l ) n ) the associated global solutions to MKG-CG with smallenergy data given by Lemma 5.5, respectively Lemma 5.7. Fix 0 < T ≪ r such that[0 , T ] × R ⊂ ∞ [ l = K x l , r l . Then Lemma 5.6 and Lemma 5.8 imply that the evolutions ( A n , φ n ) exist on the time interval [0 , T ]uniformly for all su ffi ciently large n . The covering of R by the supports of the cuto ff s χ {| x − x l | . r l } ( · )can be done in such a way that there exists a uniformly finitely overlapping, smooth partition ofunity { χ l } l ∈ N ⊂ C ∞ c ( R × R ),(5.15) 1 = ∞ X l = χ l on [0 , T ] × R , so that each cuto ff function χ l ( t , x ) is non-zero only for t ∈ [ − T , T ] and satisfies K x l , r l ∩{ t }× R ⊂ supp( χ l ( t , · )) ⊂ K x l , r l ∩ { t } × R for t ∈ [0 , T ].In order to obtain uniform S norm bounds on the evolutions of ( A n , φ n ) on [0 , T ] × R , itsu ffi ces to establish uniform bounds on the Strichartz and X , ∞ components of the S norms of( A n , φ n ) on [0 , T ] × R . These bounds then imply uniform bounds on the full S norms of ( A n , φ n )on [0 , T ] × R by a bootstrap argument as in the proof of Proposition 8.7, see the key Observation 1and Observation 2 there. Since the argument in Proposition 8.7 is self-contained, we omit the detailshere. To facilitate the notation in the following, we introduce the ˜ S norm k u k S : = X k ∈ Z k P k ∇ t , x u k S k . Its dyadic subspaces ˜ S k are given by k u k S k : = k u k S Strk + k u k X , ∞ , ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 43 where we recall that S S trk = \ q + / r ≤ ( q + r − k L qt L rx . We begin by deriving uniform ˜ S norm bounds on the evolutions A n on [0 , T ] × R . To this endwe define for i , j = , . . . , l ∈ N , the curvature tensors F n , i j = ∂ i A n , j − ∂ j A n , i and ˜ F ( l ) n , i j = ∂ i ˜ A ( l ) n , j − ∂ j ˜ A ( l ) n , i . From Lemma 5.6 and Lemma 5.8 we conclude that F n , i j = ∞ X l = χ l F n , i j = ∞ X l = χ l ˜ F ( l ) n , i j on [0 , T ] × R . Using the Coulomb gauge, we find for j = , . . . , A n , j = ∆ − ∂ i F n , i j = ∞ X l = ∆ − ∂ i (cid:0) χ l ˜ F ( l ) n , i j (cid:1) on [0 , T ] × R . In order to infer ˜ S norm bounds on A n , j from the finite S norm bounds of the globally definedevolutions ˜ A ( l ) n , we invoke the following almost orthogonality estimate. We defer its proof to theend of this subsection. Lemma 5.9.
There exists a constant C ( A , φ ) > so that uniformly for all n, we have for j = , . . . , that (5.16) (cid:13)(cid:13)(cid:13)(cid:13) ∞ X l = ∆ − ∂ i (cid:0) χ l ˜ F ( l ) n , i j (cid:1)(cid:13)(cid:13)(cid:13)(cid:13) ˜ S ([0 , T ] × R ) ≤ C ( A , φ ) ∞ X l = (cid:13)(cid:13)(cid:13) ∆ − ∂ i (cid:0) χ l ˜ F ( l ) n , i j (cid:1)(cid:13)(cid:13)(cid:13) S ( R × R ) ! / . The constant C ( A , φ ) > depends only on the size of T > , which is determined by the energyclass data ( A , φ )[0] . Hence, by (5.16) we obtain for j = , . . . , k A n , j k ˜ S ([0 , T ] × R ) = (cid:13)(cid:13)(cid:13)(cid:13) ∞ X l = ∆ − ∂ i (cid:0) χ l ˜ F ( l ) n , i j (cid:1)(cid:13)(cid:13)(cid:13)(cid:13) ˜ S ([0 , T ] × R ) ≤ C ( A , φ ) ∞ X l = (cid:13)(cid:13)(cid:13) ∆ − ∂ i (cid:0) χ l ˜ F ( l ) n , i j (cid:1)(cid:13)(cid:13)(cid:13) S ( R × R ) ! / . C ( A , φ ) ∞ X l = (cid:13)(cid:13)(cid:13) ∆ − ∇ x (cid:0) χ l ∇ x ˜ A ( l ) n (cid:1)(cid:13)(cid:13)(cid:13) S ( R × R ) ! / . Next we will invoke the following multiplier bound for the ˜ S norm that will be proven at the endof this subsection. Lemma 5.10.
Let χ ∈ C ∞ ( R × R ) satisfy max k = , ,..., k∇ kt , x χ k L qt L rx ( R × R ) ≤ D for all ≤ q , r ≤ ∞ for some D > . Then there exists a constant C > independent of χ such that for all ψ ∈ ˜ S ( R × R ) ,it holds that (5.18) (cid:13)(cid:13)(cid:13) ∆ − ∇ x (cid:0) χ ∇ x ψ (cid:1)(cid:13)(cid:13)(cid:13) ˜ S ( R × R ) ≤ CD k ψ k ˜ S ( R × R ) . By scaling invariance of the ˜ S norm and the scaling invariant setup of the partition of unity { χ l } l ∈ N , we are in a position to apply Lemma 5.10 uniformly for all multipliers χ l to estimate theright-hand side of (5.17) by C ( A , φ ) ∞ X l = (cid:13)(cid:13)(cid:13) ˜ A ( l ) n (cid:13)(cid:13)(cid:13) S ( R × R ) ! / . By the small energy global well-posedness result [22], this is in turn bounded by(5.19) C ( A , φ ) ∞ X l = k∇ t , x ˜ φ ( l ) n (0) k L x ( R ) + k∇ t , x ˜ A ( l ) n (0) k L x ( R ) ! / . It remains to square sum in l over the ˙ H x × L x norms of the initial data ( ˜ φ ( l ) n , ˜ A ( l ) n )[0], which we deferto the end of the proof of Proposition 5.2.To deduce uniform ˜ S norm bounds on the evolutions φ n on [0 , T ] × R , we use Lemma 5.6 andLemma 5.8 to write(5.20) φ n = ∞ X l = χ l φ n = ∞ X l = χ l e − i γ ( l ) n ˜ φ ( l ) n on [0 , T ] × R . Next, we apply the following almost orthogonality estimate whose proof we defer to the end of thissubsection.
Lemma 5.11.
There exists a constant C ( A , φ ) > so that uniformly for all n, (5.21) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X l = χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ˜ S ([0 , T ] × R ) ≤ C ( A , φ ) ∞ X l = (cid:13)(cid:13)(cid:13) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:13)(cid:13)(cid:13) S ( R × R ) ! / . The constant C ( A , φ ) > depends only on the size of T > , which is determined by the energyclass data ( A , φ )[0] . Thus, by (5.20) and (5.21) we find that(5.22) k φ n k ˜ S ([0 , T ] × R ) ≤ C ( A , φ ) ∞ X l = (cid:13)(cid:13)(cid:13) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:13)(cid:13)(cid:13) S ( R × R ) ! / . Here it is not immediate how to obtain ˜ S norm bounds for χ l e − i γ ( l ) n ˜ φ ( l ) n from the finite S normbounds of the globally defined ˜ φ ( l ) n , because γ ( l ) n implicitly depends on the unknown quantity φ n .Indeed, we defined in (5.10) for t > γ ( l ) n ( t , · ) = γ ( l ) n (0 , · ) + Z t (cid:0) A n , ( s , · ) − ˜ A ( l ) n , ( s , · ) (cid:1) ds and we have ∆ A n , = − Im (cid:0) φ n D t φ n (cid:1) . We will overcome this di ffi culty by exploiting that ∂ t γ ( l ) n isa harmonic function on every fixed-time slice of K ( x l , r l ) in view of Lemma 5.6, respectivelyLemma 5.8, and its definition ∂ t γ ( l ) n ( t , · ) = A n , ( t , · ) − ˜ A n , ( t , · ) for t ≥ . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 45
It therefore enjoys the interior derivative estimates for harmonic functions on every fixed-time sliceof K ( x l , r l ). The partition of unity (5.15) was chosen in such a way that the cuto ff functions χ l satisfy for all 0 ≤ t ≤ T and x ∈ supp( χ l ( t , · )) that B ( x , r l ) ⊂ K x l , r l ∩ { t } × R . Thus, for allintegers k ≥
0, we obtain from the interior derivative estimates for harmonic functions that (cid:12)(cid:12)(cid:12) χ l ∇ kx ∂ t γ ( l ) n ( t , x ) (cid:12)(cid:12)(cid:12) . C ( k ) r + kl (cid:13)(cid:13)(cid:13)(cid:0) A n , − ˜ A ( l ) n , (cid:1) ( t , · ) (cid:13)(cid:13)(cid:13) L x ( B ( x , rl )) . C ( k ) r + kl (cid:13)(cid:13)(cid:13)(cid:0) A n , − ˜ A ( l ) n , (cid:1) ( t , · ) (cid:13)(cid:13)(cid:13) L x ( R ) . C ( k ) r + kl E ( A , φ ) / . We may therefore conclude that(5.23) r + kl (cid:13)(cid:13)(cid:13) χ l ∇ kx ∂ t γ ( l ) n (cid:13)(cid:13)(cid:13) L ∞ t L ∞ x ( R × R ) ≤ C ( k , A , φ ) . Similarly, we observe that by Lemma 5.6, ∂ t γ ( l ) n ( t , · ) = ∂ t A n , ( t , · ) − ∂ t ˜ A n , ( t , · )is harmonic on every fixed-time slice of K ( x l , r ). The interior derivative estimates for harmonicfunctions then yield for all integers k ≥ (cid:12)(cid:12)(cid:12) χ l ∇ kx ∂ t γ ( l ) n ( t , x ) (cid:12)(cid:12)(cid:12) . C ( k ) r + kl (cid:13)(cid:13)(cid:13)(cid:0) ∂ t A n , − ∂ t ˜ A n , (cid:1) ( t , · ) (cid:13)(cid:13)(cid:13) L x ( B ( x , rl )) . C ( k ) r + kl (cid:13)(cid:13)(cid:13)(cid:0) ∂ t A n , − ∂ t ˜ A n , )( t , · ) (cid:13)(cid:13)(cid:13)(cid:13) L x ( R ) . Since we have (cid:13)(cid:13)(cid:13) ∂ t A n ( t , · ) (cid:13)(cid:13)(cid:13) L x ( R ) . (cid:13)(cid:13)(cid:13) ∇ x ∂ t A n ( t , · ) (cid:13)(cid:13)(cid:13) L x . X i = (cid:13)(cid:13)(cid:13) Im (cid:0) φ n D i φ n (cid:1)(cid:13)(cid:13)(cid:13) L x . X i = k φ n k L x k D i φ n k L x . E ( A , φ )and analogously for k ∂ t ˜ A n ( t , · ) k L x ( R ) , it follows that(5.24) r + kl (cid:13)(cid:13)(cid:13) χ l ∇ kx ∂ t γ ( l ) n (cid:13)(cid:13)(cid:13) L ∞ t L ∞ x ( R × R ) . C ( k , A , φ ) . Next, we note that γ ( l ) n (0 , · ) as defined in (5.1) is harmonic on the ball B ( x l , r l ). As before, theinterior derivative estimates for harmonic functions give for all integers k ≥ r kl (cid:13)(cid:13)(cid:13) χ l ∇ kx γ ( l ) n (0 , · ) (cid:13)(cid:13)(cid:13) L ∞ x ( R ) ≤ C ( k , A , φ ) . We then obtain from γ ( l ) n ( t , x ) = γ ( l ) n (0 , x ) + Z t ∂ t γ ( l ) n ( s , x ) ds that(5.26) r kl (cid:13)(cid:13)(cid:13) χ l ∇ kx γ ( l ) n (cid:13)(cid:13)(cid:13) L ∞ t L ∞ x ( R × R ) . C ( k , A , φ ) . From (5.23) – (5.26) we conclude that for all integers k ≥ C ( k , A , φ ) > k and the energy class data ( A , φ )[0], so that for all su ffi ciently large n and all l ∈ N ,(5.27) max m = , , r k + ml (cid:13)(cid:13)(cid:13) ∇ kx ∂ mt (cid:0) χ l e − i γ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13) L ∞ t L ∞ x ( R × R ) ≤ C ( k , A , φ ) . Similarly to Lemma 5.10, we also have the following multiplier bound for the ˜ S norm. Lemma 5.12.
Let χ ∈ C ∞ ( R × R ) satisfy (5.28) max k = ,..., max m = , , (cid:13)(cid:13)(cid:13) ∇ kx ∂ mt χ (cid:13)(cid:13)(cid:13) L qt L rx ( R × R ) ≤ D for all ≤ q , r ≤ ∞ for some D > . Then there exists a constant C > independent of χ such that for all ψ ∈ ˜ S ( R × R ) , (5.29) k χψ k ˜ S ( R × R ) ≤ CD k ψ k ˜ S ( R × R ) . In view of (5.27), the scaling invariance of the ˜ S norm and the scaling invariant setup of thepartition of unity { χ l } l ∈ N , we can apply Lemma 5.12 uniformly for all multipliers χ l to estimate theright hand side of (5.22) by C ( A , φ ) ∞ X l = (cid:13)(cid:13)(cid:13) ˜ φ ( l ) n (cid:13)(cid:13)(cid:13) S ( R × R ) ! / . By the small energy global well-posedness result [22], this is in turn bounded by(5.30) C ( A , φ ) (cid:18) ∞ X l = k∇ t , x ˜ φ ( l ) n (0) k L x + k∇ t , x ˜ A ( l ) n (0) k L x (cid:19) / . It remains to square sum in l over the ˙ H x × L x norms of the initial data ( ˜ φ ( l ) n , ˜ A ( l ) n )[0] in (5.19) and in(5.30). Here we have, for example, from the definition˜ φ ( l ) n (0 , · ) : = e i γ ( l ) n (0 , · ) χ {| x − x l | . r l } ( · ) φ n (0 , · )that ∞ X l = Z R |∇ x ˜ φ ( l ) n (0 , x ) | dx . ∞ X l = Z R (cid:0) |∇ x γ ( l ) n (0 , x ) | | χ {| x − x l | . r l } ( x ) | + |∇ x χ {| x − x l | . r l } ( x ) | (cid:1) | φ n (0 , x ) | dx + ∞ X l = Z R | χ {| x − x l | . r l } ( x ) | |∇ x φ n (0 , x ) | dx . (5.31)By the construction of the partition of unity, we have uniformly for all l ∈ N and x ∈ R that |∇ x χ {| x − x l | . r l } ( x ) | . C ( A , φ ) | x | and, using also (5.25), that |∇ x γ ( l ) n (0 , x ) | | χ {| x − x l | . r l } ( x ) | . C ( A , φ ) | x | . By Hardy’s inequality and the uniformly finite overlap of the supports of the cuto ff s χ {| x − x l | . r l } ( · ),we conclude that (5.31) is bounded by C ( A , φ ) k∇ x φ n (0 , · ) k L x . C ( A , φ ) E ( A , φ )uniformly for all su ffi ciently large n . Proceeding similarly with the other terms in (5.30), we finallyobtain that (5.30) is bounded by C ( A , φ ) E ( A , φ ) uniformly for all su ffi ciently large n . This finishesthe proof of Proposition 5.2. (cid:3) Next, we turn to the proofs of Lemma 5.9 and of Lemma 5.11. We only give the proof ofLemma 5.11, the other one being similar.
ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 47
Proof of Lemma 5.11.
In view of the setup of the partition of unity { χ l } l ∈ N , we may assume in thisproof that the spatial support of χ l is at scale ∼ l for l ∈ N . Moreover, we recall that χ l ( t , · ) isnon-zero only for t ∈ [ − T , T ].We first consider the S S trk component of the ˜ S norm. Here we want to show that X k ∈ Z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P k ∞ X l = ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) S Strk ([0 , T ] × R ) . ∞ X l = X k ∈ Z (cid:13)(cid:13)(cid:13) P k ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13) S Strk ( R × R ) . Let ( q , r ) be a wave-admissible pair. Then we have(5.32) X k ∈ Z q + r − k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P k ∞ X l = ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L qt L rx ([0 , T ] × R ) . X k ≤ q + r − k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P k − k X l = ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L qt L rx ([0 , T ] × R ) + X k ∈ Z q + r − k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P k X l > − k ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L qt L rx ([0 , T ] × R ) . In order to bound the first term on the right-hand side of (5.32), we introduce slightly fattened cuto ff functions ˜ χ l ∈ C ∞ c ( R × R ) such that supp( χ l ) ⊂ supp( ˜ χ l ). Then we obtain from H ¨older’s inequalityand Bernstein’s estimate that X k ≤ q + r − k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P k − k X l = ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L qt L rx ([0 , T ] × R ) . X k ≤ T q ( q + k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − k X l = ˜ χ l ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ t L x ([0 , T ] × R ) ! . X k ≤ − k X l = T q k k ˜ χ l k L ∞ t L x (cid:13)(cid:13)(cid:13) ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13) L ∞ t L x ([0 , T ] × R ) ! . Since the spatial support of ˜ χ l is at scale 2 l , this is bounded by T q X k ≤ − k X l = k + l ) (cid:13)(cid:13)(cid:13) ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13) L ∞ t L x ([0 , T ] × R ) ! . Finally, using Young’s inequality, we arrive at the desired bound T q ∞ X l = (cid:13)(cid:13)(cid:13) ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13) L ∞ t L x ([0 , T ] × R ) . T q ∞ X l = (cid:13)(cid:13)(cid:13) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:13)(cid:13)(cid:13) S ( R × R ) . Regarding the second term on the right-hand side of (5.32),(5.33) X k ∈ Z q + r − k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X l > − k P k ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L qt L rx ([0 , T ] × R ) , we note that the spatial support of the cuto ff χ l is at scale 2 l , while the projection P k lives at spatialscale 2 − k . Thus, for l > − k the projection P k approximately preserves the spatial localizations of the cuto ff s χ l , up to exponential tails that can be treated easily. Since the family of cuto ff s { χ l } l ∈ N isuniformly finitely overlapping, we may therefore bound (5.33) schematically by X k ∈ Z X l > − k q + r − k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P k ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L qt L rx ([0 , T ] × R ) . ∞ X l = (cid:13)(cid:13)(cid:13) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:13)(cid:13)(cid:13) S ( R × R ) , which is of the desired form.It remains to consider the X , ∞ component of the ˜ S norm. Here our goal is to prove that X k ∈ Z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P k ∞ X l = ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X , ∞ ([0 , T ] × R ) . ∞ X l = (cid:13)(cid:13)(cid:13) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:13)(cid:13)(cid:13) S ( R × R ) . To this end we distinguish between small and large modulations. For small modulations j ≤
0, wemay just dispose of the projection Q j and trivially estimate X k ∈ Z sup j ≤ j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P k Q j ∞ X l = ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L t L x ( R × R ) . X k ∈ Z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X l = P k ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L t L x ( R × R ) . By the space-time support properties of the cuto ff s χ l and H ¨older’s inequality in time, this isbounded by . X k ∈ Z T (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P k ∞ X l = ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ t L x ( R × R ) and then we obtain as in the previous considerations on the S S trk component of the ˜ S norm thedesired bound . T ∞ X l = X k ∈ Z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P k ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ t L x ( R × R ) . For large modulations j > k >
0, the space-time supports of the cuto ff s χ l are approximately preserved, up to exponential tails that can be treated easily. Denoting by ˜ χ l slightly fattended versions of the cuto ff s χ l , we may therefore estimate schematically X k > sup j > j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X l = P k Q j ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L t L x ( R × R ) ≃ X k > sup j > j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X l = ˜ χ l P k Q j ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L t L x ( R × R ) . ∞ X l = X k > sup j > j (cid:13)(cid:13)(cid:13) P k Q j ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13) L t L x ( R × R ) . ∞ X l = X k > (cid:13)(cid:13)(cid:13) P k ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13) X , ∞ , which is of the desired form. It therefore remains to consider the case of large modulations j > k ≤
0. Here we distinguish the cases l > − k and 1 ≤ l < − k . For l > − k ,the projection P k Q j approximately preserves the space-time localization of χ l and we immediately ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 49 obtain the schematic estimate X k ≤ sup j > j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X l > − k P k Q j ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L t L x ( R × R ) . X k ≤ X l > − k (cid:13)(cid:13)(cid:13) P k ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13) X , ∞ ( R × R ) , which is of the desired form. Finally, for 1 ≤ l < − k and large modulations j > Q j approximatelypreserves the time localization of χ l . Thus, we obtain for slightly fattened cuto ff s ˜ χ l that X k ≤ sup j > j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − k X l = P k Q j ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L t L x ( R × R ) ≃ X k ≤ sup j > j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − k X l = P k (cid:16) ˜ χ l Q j ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L t L x ( R × R ) . Since χ l and ˜ χ l approximately live at frequency 2 − l , this is basically a high-high interaction termand we may write schematically ≃ X k ≤ sup j > j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − k X l = P k (cid:16) P − l (cid:0) ˜ χ l (cid:1) P − l (cid:0) Q j ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:1)(cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L t L x ( R × R ) . X k ≤ sup j > j − k X l = (cid:13)(cid:13)(cid:13)(cid:13) P k (cid:16) P − l (cid:0) ˜ χ l (cid:1) P − l (cid:0) Q j ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:1)(cid:17)(cid:13)(cid:13)(cid:13)(cid:13) L t L x ( R × R ) ! . By Bernstein’s estimate and H ¨older’s inequality we then find . X k ≤ sup j > j − k X l = k (cid:13)(cid:13)(cid:13) ˜ χ l (cid:13)(cid:13)(cid:13) L ∞ t L x ( R × R ) (cid:13)(cid:13)(cid:13) P − l Q j ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13) L t L x ( R × R ) ! . X k ≤ − k X l = k + l (cid:13)(cid:13)(cid:13) P − l ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13) X , ∞ ( R × R ) ! , where in the last line we used that the spatial support of ˜ χ l is at scale 2 l . Using Young’s inequality,we arrive at the desired bound . ∞ X l = (cid:13)(cid:13)(cid:13) P − l ∇ t , x (cid:0) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:1)(cid:13)(cid:13)(cid:13) X , ∞ ( R × R ) . ∞ X l = (cid:13)(cid:13)(cid:13) χ l e − i γ ( l ) n ˜ φ ( l ) n (cid:13)(cid:13)(cid:13) S ( R × R ) . This finishes the proof of Lemma 5.11. (cid:3)
It remains to prove Lemma 5.10 and Lemma 5.12. We only give the proof of Lemma 5.12, theother one being similar.
Proof of Lemma 5.12.
We have to prove that for any ψ ∈ ˜ S ( R × R ), k χψ k ˜ S ( R × R ) = X k ∈ Z (cid:13)(cid:13)(cid:13) P k ∇ t , x (cid:0) χψ (cid:1)(cid:13)(cid:13)(cid:13) S Strk ( R × R ) + (cid:13)(cid:13)(cid:13) P k ∇ t , x (cid:0) χψ (cid:1)(cid:13)(cid:13)(cid:13) X , ∞ ( R × R ! / ≤ CD k ψ k ˜ S ( R × R ) . To this end we will constantly invoke the assumed space-time bounds (5.28) for the multiplier χ .We first consider the S S trk component of the ˜ S k norm. Here we denote by ( q , r ) any wave-admissible exponent pair, i.e. satisfying 2 ≤ q , r ≤ ∞ and q + r ≤ . For any k ∈ Z we have(5.34) (cid:13)(cid:13)(cid:13) P k ∇ t , x (cid:0) χψ (cid:1)(cid:13)(cid:13)(cid:13) S Strk ≤ (cid:13)(cid:13)(cid:13) P k (cid:0) ( ∇ t , x χ ) ψ (cid:1)(cid:13)(cid:13)(cid:13) S Strk + (cid:13)(cid:13)(cid:13) P k (cid:0) χ ∇ t , x ψ (cid:1)(cid:13)(cid:13)(cid:13) S Strk and begin with estimating the first term on the right hand side of (5.34). If k ≤
0, we obtain by theBernstein and Sobolev inequalities uniformly for all ( q , r ) that2 ( q + r − k (cid:13)(cid:13)(cid:13) P k (cid:0) ( ∇ t , x χ ) ψ (cid:1)(cid:13)(cid:13)(cid:13) L qt L rx . q k k (cid:13)(cid:13)(cid:13) ∇ t , x χ (cid:13)(cid:13)(cid:13) L qt L x k ψ k L ∞ t L x . k (cid:13)(cid:13)(cid:13) ∇ t , x χ (cid:13)(cid:13)(cid:13) L qt L x k∇ x ψ k L ∞ t L x . k (cid:13)(cid:13)(cid:13) ∇ t , x χ (cid:13)(cid:13)(cid:13) L qt L x k ψ k ˜ S . Here we used that k∇ x ψ k L ∞ t L x . X k ∈ Z k P k ∇ x ψ k L ∞ t L x ! / . X k ∈ Z k P k ∇ t , x ψ k S Strk ! / . k ψ k ˜ S . If k >
0, we have2 ( q + r − k (cid:13)(cid:13)(cid:13) P k (cid:0) ( ∇ t , x χ ) ψ (cid:1)(cid:13)(cid:13)(cid:13) L qt L rx . ( q + r − k (cid:13)(cid:13)(cid:13) P k (cid:0)(cid:0) P > k − C ( ∇ t , x χ ) (cid:1) ψ (cid:1)(cid:13)(cid:13)(cid:13) L qt L rx + ( q + r − k (cid:13)(cid:13)(cid:13) P k (cid:0)(cid:0) P ≤ k − C ( ∇ t , x χ ) (cid:1) P k + O (1) ψ (cid:1)(cid:13)(cid:13)(cid:13) L qt L rx . q k (cid:13)(cid:13)(cid:13) P > k − C ∇ t , x χ (cid:13)(cid:13)(cid:13) L qt L x k ψ k L ∞ t L x + ( q + r − k (cid:13)(cid:13)(cid:13) P ≤ k − C ∇ t , x χ (cid:13)(cid:13)(cid:13) L ∞ t L ∞ x (cid:13)(cid:13)(cid:13) P k + O (1) ψ (cid:13)(cid:13)(cid:13) L qt L rx . q k − k (cid:13)(cid:13)(cid:13) ∇ x ∇ t , x χ (cid:13)(cid:13)(cid:13) L qt L x k∇ x ψ k L ∞ t L x + ( q + r − k (cid:13)(cid:13)(cid:13) P ≤ k − C ∇ t , x χ (cid:13)(cid:13)(cid:13) L ∞ t L ∞ x − k (cid:13)(cid:13)(cid:13) P k + O (1) ∇ x ψ (cid:13)(cid:13)(cid:13) L qt L rx . − k (cid:13)(cid:13)(cid:13) ∇ x ∂ t χ (cid:13)(cid:13)(cid:13) L qt L x k ψ k ˜ S + (cid:13)(cid:13)(cid:13) ∇ t , x χ (cid:13)(cid:13)(cid:13) L ∞ t L ∞ x ( q + r − k (cid:13)(cid:13)(cid:13) P k + O (1) ∇ x ψ (cid:13)(cid:13)(cid:13) L qt L rx , where we used the reverse Bernstein inequality (cid:13)(cid:13)(cid:13) P > k − C ∇ t , x χ (cid:13)(cid:13)(cid:13) L qt L x . − k (cid:13)(cid:13)(cid:13) ∇ x ∇ t , x χ (cid:13)(cid:13)(cid:13) L qt L x . Square-summing over k ∈ Z yields the desired bound. We continue with the second term on theright hand side of (5.34). If k ≤
0, we use Bernstein’s inequality to bound2 ( q + r − k (cid:13)(cid:13)(cid:13) P k (cid:0) χ ∇ t , x ψ (cid:1)(cid:13)(cid:13)(cid:13) L qt L rx . q k k (cid:13)(cid:13)(cid:13) χ (cid:13)(cid:13)(cid:13) L qt L x k∇ t , x ψ k L ∞ t L x . k (cid:13)(cid:13)(cid:13) χ (cid:13)(cid:13)(cid:13) L qt L x k ψ k ˜ S . For k > ( q + r − k (cid:13)(cid:13)(cid:13) P k (cid:0) χ ∇ t , x ψ (cid:1)(cid:13)(cid:13)(cid:13) L qt L rx . ( q + r − k (cid:13)(cid:13)(cid:13) P k (cid:0)(cid:0) P > k − C χ (cid:1) ∇ t , x ψ (cid:13)(cid:13)(cid:13) L qt L rx + ( q + r − k (cid:13)(cid:13)(cid:13) P k (cid:0)(cid:0) P ≤ k − C χ (cid:1) P k + O (1) ∇ t , x ψ (cid:13)(cid:13)(cid:13) L qt L rx . q k (cid:13)(cid:13)(cid:13) P > k − C χ (cid:13)(cid:13)(cid:13) L qt L ∞ x k∇ t , x ψ k L ∞ t L x + ( q + r − k (cid:13)(cid:13)(cid:13) P ≤ k − C χ (cid:13)(cid:13)(cid:13) L ∞ t L ∞ x (cid:13)(cid:13)(cid:13) P k + O (1) ∇ t , x ψ (cid:13)(cid:13)(cid:13) L qt L rx . − k (cid:13)(cid:13)(cid:13) ∇ x χ (cid:13)(cid:13)(cid:13) L qt L ∞ x k ψ k ˜ S + (cid:13)(cid:13)(cid:13) χ (cid:13)(cid:13)(cid:13) L ∞ t L ∞ x ( q + r − k (cid:13)(cid:13)(cid:13) P k + O (1) ∇ t , x ψ (cid:13)(cid:13)(cid:13) L qt L rx . The desired bound again follows after square-summing over k ∈ Z .Next we consider the X , ∞ component of the ˜ S norm. For any k ∈ Z we have(5.35) (cid:13)(cid:13)(cid:13) P k ∇ t , x (cid:0) χψ (cid:1)(cid:13)(cid:13)(cid:13) X , ∞ ≤ (cid:13)(cid:13)(cid:13) P k (cid:0) ( ∇ t , x χ ) ψ (cid:1)(cid:13)(cid:13)(cid:13) X , ∞ + (cid:13)(cid:13)(cid:13) P k (cid:0) χ ∇ t , x ψ (cid:1)(cid:13)(cid:13)(cid:13) X , ∞ . We start with the first term on the right hand side of (5.35). If k ≤
0, we split into a small and alarge modulation term(5.36) P k (cid:0) ( ∇ t , x χ ) ψ (cid:1) = P k Q ≤ (cid:0) ( ∇ t , x χ ) ψ (cid:1) + P k Q > (cid:0) ( ∇ t , x χ ) ψ (cid:1) . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 51
We easily estimate the small modulation term using Bernstein’s inequality, (cid:13)(cid:13)(cid:13) P k Q ≤ (cid:0) ( ∇ t , x χ ) ψ (cid:1)(cid:13)(cid:13)(cid:13) X , ∞ . (cid:13)(cid:13)(cid:13) P k (cid:0) ( ∇ t , x χ ) ψ (cid:1)(cid:13)(cid:13)(cid:13) L t L x . k (cid:13)(cid:13)(cid:13) ∇ t , x χ (cid:13)(cid:13)(cid:13) L t L x k ψ k L ∞ t L x . k (cid:13)(cid:13)(cid:13) ∇ t , x χ (cid:13)(cid:13)(cid:13) L t L x k ψ k ˜ S . To estimate the large modulation term we consider for any j > j (cid:13)(cid:13)(cid:13) P k Q j (cid:0) ( ∇ t , x χ ) ψ (cid:1)(cid:13)(cid:13)(cid:13) L t L x . j (cid:13)(cid:13)(cid:13) P k Q j (cid:0)(cid:0) P > j − C ( ∇ t , x χ ) ψ (cid:1)(cid:13)(cid:13)(cid:13) L t L x + j (cid:13)(cid:13)(cid:13) P k Q j (cid:0)(cid:0) P ≤ j − C Q > j − C ( ∇ t , x χ ) ψ (cid:1)(cid:13)(cid:13)(cid:13) L t L x + j (cid:13)(cid:13)(cid:13) P k Q j (cid:0)(cid:0) P ≤ j − C Q ≤ j − C ( ∇ t , x χ ) Q j + O (1) ψ (cid:1)(cid:13)(cid:13)(cid:13) L t L x . (5.37)We bound the first term using the reverse Bernstein inequality,2 j (cid:13)(cid:13)(cid:13) P k Q j (cid:0)(cid:0) P > j − C ( ∇ t , x χ ) (cid:1) ψ (cid:13)(cid:13)(cid:13) L t L x . k j (cid:13)(cid:13)(cid:13)(cid:0) P > j − C ( ∇ t , x χ ) (cid:1) ψ (cid:13)(cid:13)(cid:13) L t L x . k − j (cid:13)(cid:13)(cid:13) ∇ x ∇ t , x χ (cid:13)(cid:13)(cid:13) L t L x k ψ k ˜ S . For the second term on the right hand side of (5.37) we obtain from a reverse Bernstein estimate intime that 2 j (cid:13)(cid:13)(cid:13) P k Q j (cid:0)(cid:0) P ≤ j − C Q > j − C ( ∇ t , x χ ) (cid:1) ψ (cid:13)(cid:13)(cid:13) L t L x . k − j (cid:13)(cid:13)(cid:13) ∇ t , x ∂ t χ (cid:13)(cid:13)(cid:13) L t L x k ψ k ˜ S . The third term on the right hand side of (5.37) can be estimated via a Littlewood-Paley trichotomy2 j (cid:13)(cid:13)(cid:13) P k Q j (cid:0)(cid:0) P ≤ j − C Q ≤ j − C ( ∇ t , x χ ) (cid:1) Q j + O (1) ψ (cid:13)(cid:13)(cid:13) L t L x . j − C X l = k + C j (cid:13)(cid:13)(cid:13)(cid:0) P l Q ≤ j − C ( ∇ t , x χ )) (cid:1) P l + O (1) Q j + O (1) ψ (cid:13)(cid:13)(cid:13) L t L x + j (cid:13)(cid:13)(cid:13)(cid:0) P k + O (1) Q ≤ j − C ( ∇ t , x χ ) (cid:1) P ≤ k + O (1) Q j + O (1) ψ (cid:13)(cid:13)(cid:13) L t L x + j (cid:13)(cid:13)(cid:13)(cid:0) P ≤ k − C Q ≤ j − C ( ∇ t , x χ ) (cid:1) P k + O (1) Q j + O (1) ψ (cid:13)(cid:13)(cid:13) L t L x . (5.38)We bound the high-high case by j − C X l = k + C k j (cid:13)(cid:13)(cid:13) P l Q ≤ j − C ( ∇ t , x χ ) (cid:13)(cid:13)(cid:13) L ∞ t L x (cid:13)(cid:13)(cid:13) P l + O (1) Q j + O (1) ψ (cid:13)(cid:13)(cid:13) L t L x . k j − C X l = k + C (cid:13)(cid:13)(cid:13) ∇ t , x χ (cid:13)(cid:13)(cid:13) L ∞ t L x − l (cid:13)(cid:13)(cid:13) P l + O (1) ∇ x ψ (cid:13)(cid:13)(cid:13) X , ∞ . k (cid:13)(cid:13)(cid:13) ∇ t , x χ (cid:13)(cid:13)(cid:13) L ∞ t L x k ψ k ˜ S . The high-low case is estimated by X l ≤ k + O (1) j (cid:13)(cid:13)(cid:13) P k + O (1) Q ≤ j − C ( ∇ t , x χ ) (cid:13)(cid:13)(cid:13) L ∞ t L x (cid:13)(cid:13)(cid:13) P l Q j + O (1) ψ (cid:13)(cid:13)(cid:13) L t L ∞ x . X l ≤ k + O (1) (cid:13)(cid:13)(cid:13) ∇ t , x χ (cid:13)(cid:13)(cid:13) L ∞ t L x l (cid:13)(cid:13)(cid:13) P l ∇ x ψ (cid:13)(cid:13)(cid:13) X , ∞ . k (cid:13)(cid:13)(cid:13) ∇ t , x χ (cid:13)(cid:13)(cid:13) L ∞ t L x k ψ k ˜ S and the low-high case by2 k j (cid:13)(cid:13)(cid:13) P ≤ k − C Q ≤ j − C ( ∇ t , x χ ) (cid:13)(cid:13)(cid:13) L ∞ t L x (cid:13)(cid:13)(cid:13) P k + O (1) Q j + O (1) ψ (cid:13)(cid:13)(cid:13) L t L x . k (cid:13)(cid:13)(cid:13) ∇ t , x χ (cid:13)(cid:13)(cid:13) L ∞ t L x (cid:13)(cid:13)(cid:13) P k + O (1) ∇ x ψ (cid:13)(cid:13)(cid:13) X , ∞ . Thus, we obtain the following estimate for the large modulation term in (5.36), (cid:13)(cid:13)(cid:13) P k Q > (cid:0) ( ∇ t , x χ ) ψ (cid:1)(cid:13)(cid:13)(cid:13) X , ∞ . k (cid:13)(cid:13)(cid:13) ∇ t , x χ (cid:13)(cid:13)(cid:13) L t L x k ψ k ˜ S + k (cid:13)(cid:13)(cid:13) ∇ t , x χ (cid:13)(cid:13)(cid:13) L ∞ t L x k ψ k ˜ S + (cid:13)(cid:13)(cid:13) ∇ t , x χ (cid:13)(cid:13)(cid:13) L ∞ t L x (cid:13)(cid:13)(cid:13) P k + O (1) ∇ x ψ (cid:13)(cid:13)(cid:13) X , ∞ . If k > P k (cid:0) ( ∇ t , x χ ) ψ (cid:1) = P k Q ≤ k (cid:0) ( ∇ t , x χ ) ψ (cid:1) + P k Q > k (cid:0) ( ∇ t , x χ ) ψ (cid:1) . We can immediately dispose of the small modulation term as follows (cid:13)(cid:13)(cid:13) P k Q ≤ k (cid:0) ( ∇ t , x χ ) ψ (cid:1)(cid:13)(cid:13)(cid:13) X , ∞ . k (cid:13)(cid:13)(cid:13) P k (cid:0) ( ∇ t , x χ ) ψ (cid:1)(cid:13)(cid:13)(cid:13) L t L x . − k (cid:16)(cid:13)(cid:13)(cid:13) ∇ x ∇ t , x χ (cid:13)(cid:13)(cid:13) L t L x k ψ k L ∞ t L x + (cid:13)(cid:13)(cid:13) ∇ t , x χ (cid:13)(cid:13)(cid:13) L t L ∞ x k∇ x ψ k L ∞ t L x (cid:17) . − k (cid:16)(cid:13)(cid:13)(cid:13) ∇ x ∇ t , x χ (cid:13)(cid:13)(cid:13) L t L x + (cid:13)(cid:13)(cid:13) ∇ t , x χ (cid:13)(cid:13)(cid:13) L t L ∞ x (cid:17) k ψ k ˜ S . To treat the large modulation term, we find that for any j > k ,2 j (cid:13)(cid:13)(cid:13) P k Q j (cid:0) ( ∇ t , x χ ) ψ (cid:1)(cid:13)(cid:13)(cid:13) L t L x . j (cid:13)(cid:13)(cid:13) P k Q j (cid:0)(cid:0) P > j − C ( ∇ t , x χ ) (cid:1) ψ (cid:1)(cid:13)(cid:13)(cid:13) L t L x + j (cid:13)(cid:13)(cid:13) P k Q j (cid:0)(cid:0) P ≤ j − C Q > j − C ( ∇ t , x χ ) (cid:1) ψ (cid:1) k L t L x + j (cid:13)(cid:13)(cid:13) P k Q j (cid:0)(cid:0) P ≤ j − C Q ≤ j − C ( ∇ t , x χ ) (cid:1) Q j + O (1) ψ (cid:1)(cid:13)(cid:13)(cid:13) L t L x . (5.40)We estimate the first term by2 j (cid:13)(cid:13)(cid:13) P > j − C ∇ t , x χ (cid:13)(cid:13)(cid:13) L t L x k ψ k L ∞ t L x . − j (cid:13)(cid:13)(cid:13) ∇ x ∇ t , x χ (cid:13)(cid:13)(cid:13) L t L x k∇ x ψ k L ∞ t L x . − k (cid:13)(cid:13)(cid:13) ∇ x ∇ t , x χ (cid:13)(cid:13)(cid:13) L t L x k ψ k ˜ S . The second term on the right hand side of (5.40) is bounded by2 j (cid:13)(cid:13)(cid:13) P ≤ j − C Q > j − C ∇ t , x χ (cid:13)(cid:13)(cid:13) L t L x k ψ k L ∞ t L x . − j (cid:13)(cid:13)(cid:13) ∇ t , x ∂ t χ (cid:13)(cid:13)(cid:13) L t L x k∇ x ψ k L ∞ t L x . − k (cid:13)(cid:13)(cid:13) ∇ t , x ∂ t χ (cid:13)(cid:13)(cid:13) L t L x k ψ k ˜ S . Using a Littlewood-Paley trichotomy we obtain the following estimate of the third term in (5.40)2 − k (cid:16)(cid:13)(cid:13)(cid:13) ∇ t , x χ (cid:13)(cid:13)(cid:13) L ∞ t L ∞ x + (cid:13)(cid:13)(cid:13) ∇ x ∇ t , x χ (cid:13)(cid:13)(cid:13) L ∞ t L x (cid:17) k ψ k ˜ S + (cid:13)(cid:13)(cid:13) ∇ t , x χ (cid:13)(cid:13)(cid:13) L ∞ t L ∞ x (cid:13)(cid:13)(cid:13) P k + O (1) ∇ x ψ (cid:13)(cid:13)(cid:13) X , ∞ . Finally, square-summing over k ∈ Z gives the desired bound for the first term on the right hand sideof (5.35). The second term on the right hand side of (5.35) can be handled similarly. (cid:3) ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 53
Localizing in physical space.
In this subsection we consider Coulomb data ( A , φ )[0] ∈ ˙ H sx × ˙ H s − x for all s ≥
1. We show that there exists T > C ∞ solution of the same regularity class oneach time slice of the space-time slab [ − T , T ] × R satisfying the required S norm bound k ( A , φ ) k S ([ − T , T ] × R ) < ∞ . To this end we fix a large R >
1. For each R ≥ R , we consider a cuto ff χ R ∈ C ∞ c ( R ) thatequals 1 on the ball B R (0) and has support in a dilate of B R (0). In the previous Subsection 5.1 wedemonstrated that upon writing ˜ A R : = χ R A − ∇ x γ R for the spatial components of a new connection form ˜ A R , where γ R = ∆ − ∂ j (cid:0) χ R A j (cid:1) , there is a way to pick the remaining data ∂ t ˜ A R (0) and ˜ φ R [0], so that the corresponding data are allof class H + x × L + x and of Coulomb class. Thus, we obtain local solutions with these data fromthe local well-posedness result [36]. We can also arrange that ˜ φ R [0] is supported within the ball ofradius R centered at the origin. It is then also easy to verify that( ˜ A R , ˜ φ R )[0] → ( A , φ )[0] as R → ∞ with respect to the ˙ H sx × ˙ H s − x topology for any s ≥
1. Moreover, the argument in the previoussubsection together with Proposition 4.3 implies that these solutions extend of class H sx × H s − x to aspace-time slab [ − T , T ] × R , where T > R ≥ R . It then remains to check thatthe corresponding local solutions on [ − T , T ] × R , call them again ( ˜ A R , ˜ φ R ), converge with respectto the S norm. This will essentially follow from the perturbation theory developed later on in thekey Step 3 of the proof of Proposition 7.4. The following proposition can be proved. Proposition 5.13.
The sequence (cid:8) ( ˜ A R , ˜ φ R ) (cid:9) R ≥ R converges in S ([ − T , T ] × R ) as R → ∞ . The limitis also of class ˙ H sx × ˙ H s − x for all s ≥ on each time slice of [ − T , T ] × R , hence of class C ∞ , and asmooth solution to the MKG-CG system on [ − T , T ] × R with initial data ( A , φ )[0] .Proof. A sketch of the proof is given in Subsection 7.4. (cid:3)
6. H ow to arrive at the minimal energy blowup solution
In this section we address another delicate issue arising due to the di ffi culties with the perturba-tion theory for the MKG-CG system. Assume that ( A n , φ n ) is an “essentially singular sequence” ofadmissible solutions to the MKG-CG system that converges at time t = A , φ )[0] with E ( A , φ ) = E crit ,lim n →∞ ( A n , φ n )[0] = ( A , φ )[0] . Using the concept of MKG-CG evolution for energy class data from Definition 5.3, we obtain anenergy class solution ( A , φ ) with maximal lifespan I . We then want to infer that(6.1) sup J ⊂ I , Jclosed k ( A , φ ) k S ( J × R ) = ∞ , while by construction it holds that E ( A , φ ) = E crit . In view of Lemma 5.4, it su ffi ces to consider thecase I = R . The problem here is that while we havelim n →∞ k ( A n , φ n ) k S ( I n × R ) = ∞ , where I n are suitably chosen time intervals, we cannot use an immediate perturbative argument toobtain (6.1) as is possible for wave maps in [20]. The reason comes from the fact that the ( A n , φ n )may have non-negligible low-frequency components. Nevertheless, we obtain the following result. Proposition 6.1.
Let ( A n , φ n ) be an essentially singular sequence of admissible solutions to theMKG-CG system. Assume that lim n →∞ ( A n , φ n )[0] = ( A , φ )[0] in the energy topology for some Coulomb energy class data pair ( A , φ )[0] . Let I be the maximallifespan of the MKG-CG evolution ( A , φ ) of this data pair given by Definition 5.3. Then it holds that sup J ⊂ I , Jclosed k ( A , φ ) k S ( J × R ) = ∞ . Proof.
A sketch of the proof can be found in Subsection 7.4. (cid:3)
We shall later on need certain variations of the preceding proposition.
Corollary 6.2.
Let { ( A n , φ n )[0] } n ∈ N and ( A , φ )[0] be Coulomb energy class data such that lim n →∞ ( A n , φ n )[0] = ( A , φ )[0] in the energy topology and let I be the maximal lifespan of the MKG-CG evolution of ( A , φ )[0] . IfJ ⊂ I is a compact time interval, then it holds that lim sup n →∞ k ( A n , φ n ) k S ( J × R ) < ∞ . This entails the following important corollary.
Corollary 6.3.
Let { ( A n , φ n )[0] } n ∈ N ⊂ ˙ H x × L x be a compact subset of Coulomb energy class data.Then there exists an open interval I ∗ centered at t = with the property thatI ∗ ⊂ I n for all n ∈ N , where I n denotes the maximal lifespan of the MKG-CG evolutions of ( A n , φ n )[0] given by Defini-tion 5.3.Proof. We argue by contradiction. Assume that there exists a subsequence { ( A n k , φ n k )[0] } k ∈ N forwhich at least one of the lifespan endpoints of the associated MKG-CG evolutions converges to t =
0. Passing to a further subsequence, which we again denote by { ( A n k , φ n k )[0] } k ∈ N , we mayassume that lim k →∞ ( A n k , φ n k )[0] = ( A , φ )[0]in the energy topology for some Coulomb energy class data ( A , φ )[0]. The contradiction now fol-lows from Corollary 6.2. (cid:3)
7. C oncentration compactness step
General considerations.
We begin by sorting out the relationship between the conservedenergy and the ˙ H x × L x -norm of solutions ( A , φ ) to the MKG-CG system. Recall that the conservedenergy is given by the expression E ( A , φ ) = X α,β Z R ( ∂ α A β − ∂ β A α ) dx + X α Z R | ∂ α φ + iA α φ | dx . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 55
Using the Coulomb gauge condition, this can be written as E ( A , φ ) = X i < j Z R ( ∂ i A j ) dx + X i Z R ( ∂ t A i ) + ( ∂ i A ) dx + X α Z R | ∂ α φ + iA α φ | dx , which immediately implies E ( A , φ ) . X i k∇ t , x A i k L x + k∇ x A k L x + k∇ t , x φ k L x + X α k∇ x A α k L x k∇ x φ k L x . Conversely, in order to exploit the conserved energy, we also need to show that the expression X α k∇ t , x A α k L x + k∇ t , x φ k L x is bounded in terms of E ( A , φ ). Here the only issue comes from bounding the terms k∇ t , x φ k L x and k ∂ t A k L x . However, the diamagnetic inequality gives the pointwise estimate | ∂ α | φ || ≤ | ( ∂ α + iA α ) φ | and Sobolev’s inequality then yields k φ k L x . k∇ x | φ |k L x . X j k ( ∂ j + iA j ) φ k L x . Thus, we find k ∂ α φ k L x ≤ k ( ∂ α + iA α ) φ k L x + k A α φ k L x . k ( ∂ α + iA α ) φ k L x + k A α k L x + k φ k L x . E ( A , φ ) + E ( A , φ ) . In order to bound the time derivative k ∂ t A k L x , we use the compatibility relation ∆ ∂ t A = − X j ∂ j Im (cid:0) φ D j φ )to obtain k ∂ t A k L x . k∇ x ∂ t A k L x . X j k φ D j φ k L x . X j k φ k L x k D j φ k L x . E ( A , φ ) . We also recall that the notation ( A , φ )[0] for initial data for the MKG-CG system only refers tothe prescribed data A j [0], j = , . . . ,
4, for the evolution of the spatial components of the connectionform A . The component A is determined via the compatibility relations.7.2. Setting up the induction on frequency scales.
Our final goal will be to show the following.
Let ( A , φ )[0] be admissible Coulomb data. Then the corresponding MKG-CG evolution existsglobally in time and denoting its energy byE = X α,β Z R ( ∂ α A β − ∂ β A α ) dx + X α Z R | ∂ α + iA α φ | dx , there exists an increasing function K : R + → R + such that k ( A , φ ) k S ( R × R ) ≤ K ( E ) . To prove this result we proceed by contradiction. By the small data global well-posedness result[22] we know that the assertion holds for su ffi ciently small energies. So assume that it does nothold for all energies E >
0. Then the set of exceptional energies has a positive infimum, which we denote by E crit , and we can find a sequence of admissible data { ( A n , φ n )[0] } n ∈ N with evolutions { ( A n , φ n ) } n ∈ N defined on ( − T n , T n ) × R such thatlim n →∞ E ( A n , φ n ) = E crit , lim n →∞ k ( A n , φ n ) k S (( − T n , T n ) × R ) = + ∞ . We call such a sequence of initial data essentially singular .We now implement a two step Bahouri-G´erard type procedure. The first step consists in selectingfrequency atoms. Here we largely follow the setup of Subsection 9.1 and Subsection 9.2 in [20],which in turn is partially based on Section III.1 of [1]. We recall the following terminology from [1].A scale is a sequence of positive numbers { λ n } n ∈ N . We say that two scales { λ na } n ∈ N and { λ nb } n ∈ N are orthogonal if lim n →∞ λ na λ nb + λ nb λ na = + ∞ . Let { λ n } n ∈ N be a scale and let { ( f n , g n ) } n ∈ N be a bounded sequence of functions in ˙ H x ( R ) × L x ( R ).Then we say that { ( f n , g n ) } n ∈ N is λ n -oscillatory iflim R → + ∞ lim sup n →∞ Z { λ n | ξ |≤ R } | [ ∇ x f n ( ξ ) | + | b g n ( ξ ) | d ξ + Z { λ n | ξ |≥ R } | [ ∇ x f n ( ξ ) | + | b g n ( ξ ) | d ξ ! = { ( f n , g n ) } n ∈ N is λ n -singular if for all b > a > n →∞ Z { a ≤ λ n | ξ |≤ b } | [ ∇ x f n ( ξ ) | + | b g n ( ξ ) | d ξ = . We obtain the following decomposition of the essentially singular sequence of data { ( A n , φ n )[0] } n ∈ N into frequency atoms. Proposition 7.1.
Let { ( A n , φ n )[0] } n ∈ N be a sequence of admissible data with energy bounded by E.Up to passing to a subsequence the following holds. Given δ > , there exists an integer Λ =Λ ( δ, E ) > and for every n ∈ N a decompositionA n [0] = Λ X a = A na [0] + A n Λ [0] ,φ n [0] = Λ X a = φ na [0] + φ n Λ [0] . For a = , . . . , Λ , the frequency atoms ( A na , φ na )[0] are λ na -oscillatory for a family of pairwiseorthogonal frequency scales { λ na } n . The error ( A n Λ , φ n Λ )[0] is λ na -singular for every ≤ a ≤ Λ andsatisfies the smallness condition lim sup n →∞ (cid:13)(cid:13)(cid:13) A n Λ [0] (cid:13)(cid:13)(cid:13) ˙ B , ∞ × ˙ B , ∞ < δ, lim sup n →∞ (cid:13)(cid:13)(cid:13) φ n Λ [0] (cid:13)(cid:13)(cid:13) ˙ B , ∞ × ˙ B , ∞ < δ. Moreover, for a = , . . . , Λ , the frequency atoms ( A na , φ na )[0] have sharp frequency support in thefrequency intervals | ξ | ∈ [( λ na ) − R − n , ( λ na ) − R n ] for a suitable sequence R n → + ∞ . For di ff erentvalues of a, these frequency intervals [( λ na ) − R − n , ( λ na ) − R n ] are mutually disjoint for su ffi cientlylarge n. Finally, we have asymptotic decoupling of the energyE ( A n , φ n ) = Λ X a = E ( A na , φ na ) + E ( A n Λ , φ n Λ ) + o (1) as n → ∞ , ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 57 where the temporal components A na are determined via the compatibility relation (cid:0) ∆ − | φ na | (cid:1) A na = − Im (cid:0) φ na ∂ t φ na (cid:1) , and similarly for A n Λ , .Proof. We suppress the notation [0] in the proof. As in Section III.1 of [1], we obtain a decompo-sition of the data { ( A n , φ n ) } n ∈ N into frequency atoms A n = Λ X a = ˜ A na + ˜ A n Λ , φ n = Λ X a = ˜ φ na + ˜ φ n Λ , where ( ˜ A na , ˜ φ na ) are λ na -oscillatory for a family of pairwise orthogonal scales { λ na } n ∈ N for a = , . . . , Λ . The error ( ˜ A n Λ , ˜ φ n Λ ) is λ na -singular for a = , . . . , Λ and satisfies the smallness conditionlim sup n →∞ (cid:13)(cid:13)(cid:13) ˜ A n Λ (cid:13)(cid:13)(cid:13) ˙ B , ∞ × ˙ B , ∞ < δ, lim sup n →∞ (cid:13)(cid:13)(cid:13) ˜ φ n Λ (cid:13)(cid:13)(cid:13) ˙ B , ∞ × ˙ B , ∞ < δ. In order to get a clean separation of the frequency atoms in frequency space, we have to preparethem a bit more, because their decay from the scale ( λ na ) − might be arbitrarily slow. To this end let R n → ∞ be a sequence growing su ffi ciently slowly such that the intervals [( λ na ) − R − n , ( λ na ) − R n ]are mutually disjoint for n large enough and for di ff erent values of a . Then we replace the error ˜ A n Λ by A n Λ = P ∩ Λ a ′ = [ µ na ′ − log R n ,µ na ′ + log R n ] c ˜ A n Λ + Λ X a = P ∩ Λ a ′ = [ µ na ′ − log R n ,µ na ′ + log R n ] c ˜ A na , where µ na = − log λ na , and the frequency atoms ˜ A na by A na = P [ µ na − log R n ,µ na + log R n ] ˜ A n Λ + P [ µ na − log R n ,µ na + log R n ] Λ X a ′ = ˜ A na ′ for a = , . . . , Λ . In order to remove the dependence on Λ in the new profiles, we may replace Λ by Λ n with Λ n → ∞ su ffi ciently slowly as n → ∞ . Analogously, we define φ na and φ n Λ . This newdecomposition(7.1) A n = Λ X a = A na + A n Λ , φ n = Λ X a = φ na + φ n Λ , has the same properties as the original one, but that we have now arranged for a sharp separation ofthe frequency supports of the frequency atoms.Finally, we turn to the asymptotic decoupling of the energy. Here we recall that the “ellipticcomponents” A na associated with a frequency atom ( A na , φ na ) are determined via the elliptic com-patibility equations. It therefore su ffi ces to show that the decomposition (7.1) (which only refers tothe spatial components of the connection form A n ) implies a similar frequency atom decomposition(7.2) A n = Λ X a = A na + A n Λ , + o ˙ H x (1) as n → ∞ , where A na is λ na -oscillatory and A n Λ , is λ na -singular for each a = , . . . , Λ . Then the decoupling ofthe energy is an immediate consequence of the construction of the frequency atoms. For example,we have the limiting relations lim n →∞ Z R ∂ α φ na A na ′ α φ na ′′ dx = , if not all of a , a ′ , a ′′ are equal, as well aslim n →∞ Z R A na α φ na ′ A na ′′ α φ na ′′′ dx = , if not all of a , a ′ , a ′′ , a ′′′ are equal. It remains to prove the decomposition (7.2). To show this, wefirst observe that (at fixed time t = − Im (cid:0) φ n ∂ t φ n (cid:1) = Λ X a = − Im (cid:0) φ na ∂ t φ na (cid:1) − Im (cid:0) φ n Λ ∂ t φ n Λ (cid:1) + o L x (1) as n → ∞ . It is then enough to show that (cid:0) ∆ − | φ n | (cid:1) A na = − Im (cid:0) φ na ∂ t φ na (cid:1) + o L x (1) as n → ∞ , (cid:0) ∆ − | φ n | (cid:1) A n Λ , = − Im (cid:0) φ n Λ , ∂ t φ n Λ , (cid:1) + o L x (1) as n → ∞ . This in turn will easily follow once we have shown that each A na is λ na -oscillatory, while A n Λ , is λ na -singular for a = , . . . , Λ . We demonstrate this for a =
1, where we may assume by scalinginvariance of these assertions that λ n = ∆ A n − | φ n | A n = − Im (cid:0) φ n ∂ t φ n (cid:1) and distinguish between small and large frequencies.We begin with the small frequencies. For R ≪ − ∆ P ≤ R A n − P ≤ R (cid:0) | φ n | A n (cid:1) = − P ≤ R (cid:0) Im (cid:0) φ n ∂ t φ n (cid:1)(cid:1) , where we have lim R →−∞ lim sup n →∞ (cid:13)(cid:13)(cid:13) P ≤ R (cid:0) Im (cid:0) φ n ∂ t φ n (cid:1)(cid:1)(cid:13)(cid:13)(cid:13) L x = . Next, we split P ≤ R (cid:0) | φ n | A n (cid:1) = P ≤ R (cid:0) P ≤ R (cid:0) | φ n | (cid:1) A n (cid:1) + P ≤ R (cid:0) P > R ( | φ n | ) A n (cid:1) . Then we have lim R →−∞ lim sup n →∞ (cid:13)(cid:13)(cid:13) P ≤ R | φ n | (cid:13)(cid:13)(cid:13) L x = , whence lim R →−∞ lim sup n →∞ (cid:13)(cid:13)(cid:13) P ≤ R (cid:0) P ≤ R (cid:0) | φ n | ) A n (cid:1)(cid:13)(cid:13)(cid:13) L x = , while for the second term above, we obtain from Bernstein’s inequality that (cid:13)(cid:13)(cid:13) P ≤ R (cid:0) P > R (cid:0) | φ n | (cid:1) A n (cid:1)(cid:13)(cid:13)(cid:13) L x ≤ X k = k + O (1) , k > R (cid:13)(cid:13)(cid:13) P ≤ R (cid:0) P k ( | φ n | ) P k A n (cid:1)(cid:13)(cid:13)(cid:13) L x . R X k = k + O (1) , k > R (cid:13)(cid:13)(cid:13) P k | φ n | (cid:13)(cid:13)(cid:13) L x (cid:13)(cid:13)(cid:13) P k A n (cid:13)(cid:13)(cid:13) L x . R (cid:13)(cid:13)(cid:13) φ n (cid:13)(cid:13)(cid:13) L x (cid:13)(cid:13)(cid:13) A n (cid:13)(cid:13)(cid:13) ˙ H x . We immediately conclude thatlim R →−∞ lim sup n →∞ (cid:13)(cid:13)(cid:13) P ≤ R (cid:0) P > R ( | φ n | ) A n (cid:1)(cid:13)(cid:13)(cid:13) L x = . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 59
Using Sobolev’s inequality, it then follows thatlim sup n →∞ (cid:13)(cid:13)(cid:13) P ≤ R A n (cid:13)(cid:13)(cid:13) ˙ H x . lim sup n →∞ (cid:13)(cid:13)(cid:13) P ≤ R (cid:0) Im (cid:0) φ n ∂ t φ n (cid:1)(cid:1)(cid:13)(cid:13)(cid:13) L x + lim sup n →∞ (cid:13)(cid:13)(cid:13) P ≤ R (cid:0) | φ n | A n (cid:1)(cid:13)(cid:13)(cid:13) L x → R → −∞ . Next, we consider the large frequencies. For R ≫
1, we write ∆ P > R A n − P > R (cid:0) | φ n | A n (cid:1) = − P > R (cid:0) Im (cid:0) φ n ∂ t φ n (cid:1)(cid:1) , where we have lim R →∞ lim sup n →∞ (cid:13)(cid:13)(cid:13) P > R (cid:0) Im (cid:0) φ n ∂ t φ n (cid:1)(cid:1)(cid:13)(cid:13)(cid:13) L = . Then we split P > R (cid:0) | φ n | A n (cid:1) = P > R (cid:0) P > R ( | φ n | ) A n (cid:1) + P > R (cid:0) P ≤ R ( | φ n | ) A n (cid:1) . By frequency localization we havelim R →∞ lim sup n →∞ (cid:13)(cid:13)(cid:13) P > R ( | φ n | ) (cid:13)(cid:13)(cid:13) L x = , and thus, lim R →∞ lim sup n →∞ (cid:13)(cid:13)(cid:13) P > R (cid:0) P > R ( | φ n | ) A n (cid:1)(cid:13)(cid:13)(cid:13) L x = . On the other hand, we have (cid:13)(cid:13)(cid:13) P > R (cid:0) P ≤ R ( | φ n | ) A n (cid:1)(cid:13)(cid:13)(cid:13) L x . X k = k + O (1) , k > R (cid:13)(cid:13)(cid:13) P k (cid:0) P ≤ R ( | φ n | ) P k A n (cid:1)(cid:13)(cid:13)(cid:13) L x ≤ X k = k + O (1) , k > R (cid:13)(cid:13)(cid:13) P ≤ R ( | φ n | ) (cid:13)(cid:13)(cid:13) L x (cid:13)(cid:13)(cid:13) P k A n (cid:13)(cid:13)(cid:13) L x . X k > R R − k (cid:13)(cid:13)(cid:13) φ n (cid:13)(cid:13)(cid:13) L x (cid:13)(cid:13)(cid:13) P k A n (cid:13)(cid:13)(cid:13) ˙ H x . − R (cid:13)(cid:13)(cid:13) φ n (cid:13)(cid:13)(cid:13) L x (cid:13)(cid:13)(cid:13) A n (cid:13)(cid:13)(cid:13) ˙ H x and hence, lim R →∞ lim sup n →∞ (cid:13)(cid:13)(cid:13) P > R (cid:0) P ≤ R ( | φ n | ) A n (cid:1)(cid:13)(cid:13)(cid:13) L x = . We then conclude from Sobolev’s inequality thatlim R →∞ lim sup n →∞ (cid:13)(cid:13)(cid:13) P > R A n (cid:13)(cid:13)(cid:13) ˙ H x = . (cid:3) Given an essentially singular sequence of initial data, by Proposition 7.1 for any δ > { ( A n , φ n )[0] } n ∈ N of the form A n [0] = Λ X a = A na [0] + A n Λ [0] ,φ n [0] = Λ X a = φ na [0] + φ n Λ [0](7.3)with lim sup n →∞ (cid:13)(cid:13)(cid:13) A n Λ [0] (cid:13)(cid:13)(cid:13) ˙ B , ∞ × ˙ B , ∞ < δ, lim sup n →∞ (cid:13)(cid:13)(cid:13) φ n Λ [0] (cid:13)(cid:13)(cid:13) ˙ B , ∞ × ˙ B , ∞ < δ. Eventually, we will prove that necessarily only one frequency atom ( A na , φ na )[0] in the decompo-sition (7.3) is non-trivial and has to be asymptotically of energy E crit . In fact, the subsequentconsiderations will show that if there are at least two frequency atoms ( A n , φ n )[0] , ( A n , φ n )[0] that both do not vanish asymptotically, or if there is only one frequency atom ( A n , φ n )[0] with theerror satisfying lim sup n →∞ k ( A n , φ n )[0] k ˙ H x × L x > , then we get an a priori bound on the S norm of the evolutions lim inf n →∞ k ( A n , φ n ) k S (( − T n , T n ) × R ) < ∞ , contradicting the assumption that { ( A n , φ n )[0] } n ∈ N is essentially singular. We now introduce a smallness parameter ε > ffi ciently smalldepending only on E crit . In particular, we assume that ε is less than the small energy threshold ofthe small energy global well-posedness result [22].By passing to a suitable subsequence and by renumbering the frequency atoms, if necessary, wemay assume that for some integer Λ > X a ≥ Λ + lim sup n →∞ E ( A na , φ na ) < ε . Moreover, we may assume that the frequency atoms { ( A na , φ na )[0] } n ∈ N , a = , . . . , Λ , have “in-creasing frequency supports” in the sense that ( λ na ) − is growing in terms of a (for each fixed n ).The key idea now is as follows. We approximate the initial data ( A n , φ n )[0] by low frequency truncations, obtained by removingall or some of the atoms ( A na , φ na )[0] , a = , . . . , Λ , and inductively obtain bounds on the S normof the MKG-CG evolutions of the truncated data. As this induction stops after Λ many steps, wewill have obtained the desired contradiction, forcing eventually that there has to be exactly onefrequency atom ( A n , φ n )[0] that is asymptotically of energy E crit . Evolving the “non-atomic” lowest frequency approximation.
From now on we suppressthe notation [0] for the initial data. The errors ( A n Λ , φ n Λ ) in the decomposition (7.3) are by con-struction supported away in frequency space from the frequency scales ( λ na ) − , a = , , . . . , Λ .It is then clear that the errors { ( A n Λ , φ n Λ ) } n ∈ N can be written as the sum of Λ + Λ + A na , φ na ). Thus, we can write(7.4) A n Λ = Λ + X j = A n j Λ , φ n Λ = Λ + X j = φ n j Λ , where the first pieces ( A n Λ , φ n Λ ) have Fourier support in the region closest to the origin, i.e. in | ξ | ≤ ( λ n ) − ( R n ) − . In other words, one essentially obtains the “lowest frequency approximations” ( A n Λ , φ n Λ ) by re-moving all the atoms ( A na , φ na ), a = , . . . , Λ , from the data.We then start our grand inductive procedure by showing the following proposition. ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 61
Proposition 7.2.
The parameter ε > can be chosen su ffi ciently small depending only on thesize of E crit such that the following holds. Constructing the lowest frequency approximations { ( A n Λ , φ n Λ ) } n ∈ N as described in (7.4) , then there exists a constant C ( E crit ) > such that for allsu ffi ciently large n, the data given by ( A n Λ , φ n Λ ) can be evolved globally in time and the corre-sponding solution satisfies k ( A n Λ , φ n Λ ) k S ( R × R ) ≤ C ( E crit ) . Proof.
The idea is to use a finite number of further low frequency approximations of { ( A n Λ , φ n Λ ) } n ∈ N and to inductively obtain bounds on the S norms of their evolutions. Here it is essential that thenumber of these further approximations is bounded by C ( E crit ) > { ( A n Λ , φ n Λ ) } n ∈ N . To begin with,for some su ffi ciently small δ = δ ( E crit ) >
0, in particular δ ≪ ε , we use decompositions A n Λ = Λ ( δ ) X j = A n j ) Λ + A n Λ ( Λ ) ,φ n Λ = Λ ( δ ) X j = φ n j ) Λ + φ n Λ ( Λ ) , where the frequency atoms ( A n j ) Λ , φ n j ) Λ ) have frequency support in mutually disjoint intervals[( λ n j ) ) − ( R ( j ) n ) − , ( λ n j ) ) − R ( j ) n ]with R ( j ) n → ∞ as n → ∞ , and furthermore, we have the boundlim sup n →∞ (cid:8) k A n Λ ( Λ ) k ˙ B , ∞ × ˙ B , ∞ + k φ n Λ ( Λ ) k ˙ B , ∞ × ˙ B , ∞ (cid:9) < δ . We may again assume that the atoms ( A n j ) Λ , φ n j ) Λ ) have increasing frequency support as j increases.The number of frequency atoms Λ ( δ ) here is potentially extremely large. It is crucial that thenumber of steps, i.e. the number of low frequency approximations of { ( A n Λ , φ n Λ ) } n ∈ N , required inthe inductive procedure is in fact much smaller, of size C = C ( E crit ) ≪ Λ ( δ ). As we shall see, C can be chosen independently of δ and Λ ( δ ). We now pick the low frequency approximationsof the data { ( A n Λ , φ n Λ ) } n ∈ N . For ε fixed as before, we inductively construct O ( E crit ε ) closed frequencyintervals ˜ J l for the variable | ξ | , disjoint up the the endpoints and increasing. The chosen intervalswill also depend on n , but for notational ease we do not indicate this. So consider n and Λ fixednow. Having picked the intervals ˜ J = ( −∞ , b ], ˜ J l = [ a l , b l ] with b l − = a l for l = , . . . , L −
1, wepick an interval [ a L , ˜ b L ] with a L = b L − as follows. First, pick ˜ b L in such a fashion that E ( P [ a L , ˜ b L ] A n Λ , P [ a L , ˜ b L ] φ n Λ ) = ε or else, if this is impossible, then pick ˜ b L = log ( λ n ) − − log R n , i.e. pick the upper endpoint ofthe frequency interval containing the lowest frequency “large atom” ( A n , φ n ). Now, in the formercase assume that ˜ b L ∈ [log ( λ n j ) ) − − log R ( j ) n , log ( λ n j ) ) − + log R ( j ) n ]for some 1 ≤ j ≤ Λ ( δ ), i.e. ˜ b L falls within the frequency support of one of the (finite numberof) “small frequency atoms” ( A n j ) Λ , φ n j ) Λ ) constituting ( A n Λ , φ n Λ ). Then we shift ˜ b L upwards to coincide with the upper limit, that is, we set b L = log ( λ n j ) ) − + log R ( j ) n . Otherwise, we set b L = ˜ b L . Then we define the interval ˜ J L = [ a L , b L ]. Observe that for su ffi ciently large n , we have E ( P ˜ J L A n Λ , P ˜ J L φ n Λ ) . ε . In particular, this implies that for su ffi ciently large n the total number of intervals ˜ J l is C = O ( E crit ε ).We now define the low frequency approximations of the data ( A n Λ , φ n Λ ) by truncating the frequencysupport of ( A n Λ , φ n Λ ) to the intervals J L : = ∪ Ll = ˜ J l . More precisely, for 1 ≤ L ≤ C we define the L -th low frequency approximation of the data( A n Λ , φ n Λ ) by the expression ( P J L A n Λ , P J L φ n Λ ) , where by construction C = C ( E crit ) . E crit ε . In particular, we have( P J C A n Λ , P J C φ n Λ ) = ( A n Λ , φ n Λ ) . We also state the following key lemma, whose proof is a consequence of the preceding construction.
Lemma 7.3.
For L = , . . . , C and for any R > , we have for all su ffi ciently large n that (cid:13)(cid:13)(cid:13) P [ a L − R , a L + R ] ∇ t , x A n Λ (cid:13)(cid:13)(cid:13) L x + (cid:13)(cid:13)(cid:13) P [ a L − R , a L + R ] ∇ t , x φ n Λ (cid:13)(cid:13)(cid:13) L x . R δ . In order to prove Proposition 7.2, we inductively show that for L = , . . . , C and for all su ffi -ciently large n , the evolutions of the data( P J L A n Λ , P J L φ n Λ )exist globally and satisfy the desired global S norm bounds, which of course get larger as L grows.For L = Proposition 7.4.
Let us assume that the evolution of the data (cid:0) P J L − A n Λ , P J L − φ n Λ (cid:1) is globally defined for some ≤ L < C . We denote this evolution by ( A n , ( L − Λ , φ n , ( L − Λ ) . Further-more, assume that for all su ffi ciently large n, it holds that (cid:13)(cid:13)(cid:13)(cid:0) A n , ( L − Λ , φ n , ( L − Λ (cid:1)(cid:13)(cid:13)(cid:13) S ( R × R ) ≤ C < ∞ . Provided δ − ≫ C with δ > as above, there exists C = C ( C ) < ∞ such that for all su ffi cientlylarge n, the data (cid:0) P J L A n Λ , P J L φ n Λ (cid:1) can be evolved globally and for the corresponding evolutions (cid:0) A n , ( L ) Λ , φ n , ( L ) Λ (cid:1) , it holds that (cid:13)(cid:13)(cid:13)(cid:0) A n , ( L ) Λ , φ n , ( L ) Λ (cid:1)(cid:13)(cid:13)(cid:13) S ( R × R ) ≤ C . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 63
Proposition 7.2 is then an immediate consequence of applying Proposition 7.4 C many times.We note that there exists δ > E crit such that choosing δ < δ in each step,Proposition 7.4 can be applied. Since C = C ( E crit ) this results in a bound (cid:13)(cid:13)(cid:13) ( A n Λ , φ n Λ ) (cid:13)(cid:13)(cid:13) S ( R × R ) ≤ C ( E crit ) . (cid:3) Proof of Proposition 7.4.
We proceed in several steps.
Step 1.
The assumed bound on (cid:13)(cid:13)(cid:13)(cid:0) A n , ( L − Λ , φ n , ( L − Λ (cid:1)(cid:13)(cid:13)(cid:13) S ( R × R ) implies an exponential decay for largefrequencies, (cid:13)(cid:13)(cid:13)(cid:0) P k A n , ( L − Λ , P k φ n , ( L − Λ (cid:1)(cid:13)(cid:13)(cid:13) S ( R × R ) . − σ ( k − b L − ) for k ≥ b L − . This will follow once we can show that in fact (cid:13)(cid:13)(cid:13)(cid:0) P k A n , ( L − Λ , P k φ n , ( L − Λ (cid:1)(cid:13)(cid:13)(cid:13) S ( R × R ) . c ( L − k , where (cid:8) c ( L − k (cid:9) k ∈ Z is a su ffi ciently flat frequency envelope covering the initial data (cid:0) A n , ( L − Λ , φ n , ( L − Λ (cid:1) [0]at time t =
0. This in turn is a consequence of Proposition 4.2 whose proof will be given in Subsec-tion 7.4.
Step 2.
Localizing (cid:0) A n , ( L − Λ , φ n , ( L − Λ (cid:1) to suitable space-time slices . In order to ensure that we caninduct on perturbations of size ∼ ε that are not “too small” (such as the δ ), we have to make surethat the S norms of (cid:0) A n , ( L − Λ , φ n , ( L − Λ (cid:1) are not too large. To simplify the notation, we label thesecomponents by ( A , φ ) for the rest of this step. The idea is to localize to suitable space-time slices I × R , whose number may be very large (depending on k ( A , φ ) k S ( R × R ) and E crit ), but such that wehave on each slice k ( A , φ ) k S ( I × R ) ≤ C ( E crit ) , where the function C ( · ) grows at most polynomially. Proposition 7.5.
There exist N = N (cid:0) k ( A , φ ) k S ( R × R ) , E crit (cid:1) many time intervals I , . . . , I N partition-ing the time axis R such that we have for n = , . . . , N a decomposition (referring to the spatialcomponents of the connection form simply by A) (7.5) A | I n = A f ree , ( I n ) + A nonlin , ( I n ) , (cid:3) A f ree , ( I n ) = , where A f ree , ( I n ) and A nonlin , ( I n ) are in Coulomb gauge and satisfy (cid:13)(cid:13)(cid:13) ∇ t , x A f ree , ( I n ) (cid:13)(cid:13)(cid:13) L ∞ t L x ( R × R ) . E / crit , (7.6) (cid:13)(cid:13)(cid:13) A nonlin , ( I n ) (cid:13)(cid:13)(cid:13) ℓ S ( I n × R ) ≪ . (7.7) Moreover, we have for n = , . . . , N that (7.8) k φ k S ( I n × R ) . C ( E crit ) , where C ( · ) grows at most polynomially.Proof. We first define precisely the decompositions A = A f ree + A nonlin that we are using. Thenonlinear structure inherent in A nonlin will be pivotal for controlling the equation for φ . For a timeinterval I ⊂ R , say of the form I = [ t , t ] for some t < t , we define for i = , . . . , A nonlin , ( I ) i : = − χ I X k , j (cid:3) − P k Q j Im P i (cid:0) ( χ I φ ) · ∇ x ( χ I φ ) − χ I iA | φ | (cid:1) , where χ I is a smooth cuto ff to the interval I and (cid:3) − denotes multiplication by the Fourier symbol.Then we define A f ree , ( I ) to be the free wave with initial data at time t given by A [ t ] − A nonlin , ( I ) [ t ].By construction, we then have A = A f ree , ( I ) + A nonlin , ( I ) on I × R . We now describe how to partition the time axis into N = N (cid:0) k ( A , φ ) k S , E crit (cid:1) many suitable timeintervals so that the bounds (7.6) – (7.8) hold on each such interval. For this, we first need thefollowing technical lemma. Lemma 7.6.
Given ε > , there exist M = M (cid:0) k ( A , φ ) k S ( R × R ) , ε (cid:1) many time intervals I , . . . , I M partitioning the time axis R such that for m = , . . . , M and i = , . . . , , (7.10) X k (cid:13)(cid:13)(cid:13)(cid:13) ∇ t , x X j (cid:3) − P k Q j P i (cid:0) ( χ I m φ ) · ∇ x ( χ I m φ ) − χ I m iA | φ | (cid:1)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ t L x ( R × R ) . ε and (7.11) (cid:13)(cid:13)(cid:13) P i (cid:0) ( χ I m φ ) · ∇ x ( χ I m φ ) − χ I m iA | φ | (cid:1)(cid:13)(cid:13)(cid:13) ( ℓ N ∩ ℓ L t ˙ H − x )( R × R ) . ε. In particular, it then holds that (cid:13)(cid:13)(cid:13) ∇ t , x A f ree , ( I m ) i (cid:13)(cid:13)(cid:13) L ∞ t L x ( R × R ) . E / crit + ε, (7.12) (cid:13)(cid:13)(cid:13) ∇ t , x A nonlin , ( I m ) i (cid:13)(cid:13)(cid:13) L ∞ t L x ( R × R ) . ε, (7.13) (cid:13)(cid:13)(cid:13) A nonlin , ( I m ) i (cid:13)(cid:13)(cid:13) ℓ S ( I m × R ) . ε. (7.14) Proof.
We begin with the quadratic interaction term in (7.10) and show that the time axis R can bepartitioned into M = M (cid:0) k ( A , φ ) k S , ε (cid:1) many intervals so that on each such interval I , it holds that(7.15) X k (cid:13)(cid:13)(cid:13)(cid:13) ∇ t , x X j (cid:3) − P k Q j P i (cid:0) ( χ I φ ) · ∇ x ( χ I φ ) (cid:1)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ t L x ( R × R ) . ε. To this end we exploit that there is an inherent null form in the above expression P i (cid:0) φ · ∇ x φ (cid:1) = ∆ − ∇ r N ir ( φ, φ ) , where N ir ( φ, ψ ) = ( ∂ i φ )( ∂ r ψ ) − ( ∂ r φ )( ∂ i ψ ) . We first prove that on suitable intervals I ,(7.16) X k X j ≤ k + C (cid:13)(cid:13)(cid:13) ∇ t , x (cid:3) − P k Q j ∆ − ∇ r N ir (cid:0) Q ≤ j − C ( χ I φ ) , Q ≤ j − C ( χ I φ ) (cid:1)(cid:13)(cid:13)(cid:13) L ∞ t L x ( R × R ) . ε. By a Littlewood-Paley trichotomy we may reduce to the case where both inputs are at frequency ∼ k . The singular operator (cid:3) − costs 2 − j − k , so we need to recover the factor 2 − j . From the nullform we gain 2 j − k , while the inclusion Q j L t L x ֒ → L ∞ t L x gains another 2 j . Finally, we obtaina small power in j − k from the improved Bernstein estimate P k Q j L t L x ֒ → k ( j − k ) L t L x (byinterpolating with the X s , b version of the Strichartz estimate P k Q j L t L x ֒ → k ( j − k ) L t L x ) and that ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 65 L t L x · L ∞ t L x ֒ → L t L x . Thus, we find X k X j ≤ k + C (cid:13)(cid:13)(cid:13) ∇ t , x (cid:3) − P k Q j ∆ − ∇ r N ir (cid:0) Q ≤ j − C ( χ I φ k ) , Q ≤ j − C ( χ I φ k ) (cid:1)(cid:13)(cid:13)(cid:13) L ∞ t L x . (cid:18)X k (cid:16) − k k χ I ∇ x φ k k L t L x (cid:17) (cid:19) / k φ k S and smallness follows from divisibility of the L t L x ( R × R ) norm. Next, we show that on suitableintervals I , it holds that(7.17) X k X j > k + C (cid:13)(cid:13)(cid:13) ∇ t , x (cid:3) − P k Q j ∆ − ∇ r N ir (cid:0) χ I φ, χ I φ (cid:1)(cid:13)(cid:13)(cid:13) L ∞ t L x ( R × R ) . ε. By a Littlewood-Paley trichotomy we may again reduce to the case where both inputs are at fre-quency ∼ k . Then we obtain, using the Bernstein inequality both in time and space, that X k X j > k + C (cid:13)(cid:13)(cid:13) ∇ t , x (cid:3) − P k Q j ∆ − ∇ r N ir (cid:0) χ I φ k , χ I φ k (cid:1)(cid:13)(cid:13)(cid:13) L ∞ t L x . (cid:18)X k (cid:16) − k (cid:13)(cid:13)(cid:13) χ I ∇ x φ k (cid:13)(cid:13)(cid:13) L t L x (cid:17) (cid:19) / k φ k S and smallness follows from the divisibility of the L t L x ( R × R ) norm. In view of (7.16) and (7.17),in order to finish the proof of (7.15) we may assume that one of the two inputs has the leadingmodulation. It therefore su ffi ces to show that on suitable intervals I we have bounds of the form(7.18) X k , j (cid:13)(cid:13)(cid:13)(cid:13) ∇ t , x (cid:3) − P k Q ≤ j − C ∆ − ∇ r N ir (cid:0) Q j ( χ I φ ) , Q ≤ j − C ( χ I φ ) (cid:1)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ t L x ( R × R ) . ε, where we use the convention (cid:3) − P k Q ≤ j − C = P l ≤ j − C (cid:3) − P k Q l . Using that (cid:0) (cid:3) − P k Q < j − C F (cid:1) ( t , · ) = − Z ∞ t sin(( t − s ) |∇| ) |∇| ( P k Q < j − C F )( s , · ) ds , it is enough to show X k , j (cid:13)(cid:13)(cid:13)(cid:13) P k Q ≤ j − C ∆ − ∇ r N ir (cid:0) Q j ( χ I φ ) , Q ≤ j − C ( χ I φ ) (cid:1)(cid:13)(cid:13)(cid:13)(cid:13) L t L x ( R × R ) . ε. By estimate (143) in [22] we may reduce to the case where j = k + O (1) and both inputs are atfrequency ∼ k . Then we find X k (cid:13)(cid:13)(cid:13)(cid:13) P k Q ≤ k − C ∆ − ∇ r N ir (cid:0) Q k ( χ I φ k ) , Q ≤ k − C ( χ I φ k ) (cid:1)(cid:13)(cid:13)(cid:13)(cid:13) L t L x . X k (cid:13)(cid:13)(cid:13) χ I ∇ x φ k (cid:13)(cid:13)(cid:13) X , ∞ − k (cid:13)(cid:13)(cid:13) χ I ∇ x φ k (cid:13)(cid:13)(cid:13) L t L x . k φ k S (cid:18)X k (cid:16) − k (cid:13)(cid:13)(cid:13) χ I ∇ x φ k (cid:13)(cid:13)(cid:13) L t L x (cid:17) (cid:19) / and smallness follows by divisibility. Next, we consider the cubic term in (7.10). Here we have toprove that on suitable intervals I it holds that(7.19) X k (cid:13)(cid:13)(cid:13)(cid:13) ∇ t , x X j (cid:3) − P k Q j P i (cid:0) χ I A | φ | (cid:1)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ t L x ( R × R ) . ε. By similar arguments as above, this reduces to showing X k (cid:13)(cid:13)(cid:13) P k (cid:0) χ I A | φ | (cid:1)(cid:13)(cid:13)(cid:13) L t L x ( R × R ) . ε, which follows from estimate (64) in [22] and a divisibility argument. We note that the bound (7.10)implies that the estimates (7.12) and (7.13) hold on each such interval I .It remains to choose the intervals so that the bound (7.11) also holds. The energy estimatefor the S k and N k spaces together with the bounds (7.10) and (7.11) then also imply the bound(7.14). We pick M (cid:0) k ( A , φ ) k S , ε (cid:1) many time intervals I m , m = , . . . , M , on which the bound(7.10) already holds. We show that, if necessary, each time interval I m can be subdivided into M = M (cid:0) k ( A , φ ) k S , ε (cid:1) many intervals I ma , a = , . . . , M , such that we have(7.20) (cid:13)(cid:13)(cid:13) P i (cid:0) ( χ I ma φ ) · ∇ x ( χ I ma φ ) − χ I ma iA | φ | (cid:1)(cid:13)(cid:13)(cid:13) ( ℓ N ∩ ℓ L t ˙ H − x )( R × R ) . ε. For the rest of the proof of (7.20) we denote an interval I ma just by I and say that it is of the form I = [ t , t ] for some t < t . We only outline how to make the left hand side of (7.20) small in ℓ N for suitable intervals I , the ℓ L t ˙ H − x component being easier. We first estimate the quadraticinteraction term in (7.20), X k (cid:13)(cid:13)(cid:13) P i (cid:0) ( χ I φ ) · ∇ x ( χ I φ ) (cid:1)(cid:13)(cid:13)(cid:13) N k = X k (cid:13)(cid:13)(cid:13) P k ∆ − ∇ r N ir (cid:0) χ I φ, χ I φ (cid:1)(cid:13)(cid:13)(cid:13) N k . By (131) in [22], it su ffi ces to consider the case where both inputs are at frequency ∼ k and haveangular separation ∼ X k (cid:13)(cid:13)(cid:13) P k ∆ − ∇ r N ir (cid:0) χ I φ k , χ I φ k (cid:1) ′ (cid:13)(cid:13)(cid:13) N k . Here, the prime indicates the angular separation. We split into high and low modulation output. X k (cid:13)(cid:13)(cid:13) P k ∆ − ∇ r N ir (cid:0) χ I φ k , χ I φ k (cid:1) ′ (cid:13)(cid:13)(cid:13) N k ≤ X k (cid:13)(cid:13)(cid:13) P k Q > k − C ∆ − ∇ r N ir (cid:0) χ I φ k , χ I φ k (cid:1) ′ (cid:13)(cid:13)(cid:13) N k + X k (cid:13)(cid:13)(cid:13) P k Q ≤ k − C ∆ − ∇ r N ir (cid:0) χ I φ k , χ I φ k (cid:1) ′ (cid:13)(cid:13)(cid:13) N k . The term with high modulation output is estimated by X k (cid:13)(cid:13)(cid:13) P k Q > k − C ∆ − ∇ r N ir (cid:0) χ I φ k , χ I φ k (cid:1) ′ (cid:13)(cid:13)(cid:13) N k . X k − k (cid:13)(cid:13)(cid:13) χ I ∇ x φ k (cid:13)(cid:13)(cid:13) L t L ∞ x (cid:13)(cid:13)(cid:13) χ I ∇ x φ k (cid:13)(cid:13)(cid:13) L ∞ t L x . (cid:18)X k (cid:16) − k (cid:13)(cid:13)(cid:13) χ I ∇ x φ k (cid:13)(cid:13)(cid:13) L t L ∞ x (cid:17) (cid:19) / k φ k S and can be made small on suitable intervals I using the divisibility of the quantity X k (cid:16) − k (cid:13)(cid:13)(cid:13) ∇ x φ k (cid:13)(cid:13)(cid:13) L t L ∞ x ( R × R ) (cid:17) . k φ k S ( R × R ) . For the term with low modulation output we note that the angular separation of the inputs allows usto write schematically P k Q ≤ k − C ∆ − ∇ r N ir (cid:0) χ I φ k , χ I φ k (cid:1) ′ = P k Q ≤ k − C ∆ − ∇ r N ir (cid:0) Q > k − C ( χ I φ k ) , χ I φ k (cid:1) ′ + P k Q ≤ k − C ∆ − ∇ r N ir (cid:0) Q ≤ k − C ( χ I φ k ) , Q > k − C ( χ I φ k ) (cid:1) ′ . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 67
Then we estimate X k (cid:13)(cid:13)(cid:13) P k Q ≤ k − C ∆ − ∇ r N ir (cid:0) Q > k − C ( χ I φ k ) , χ I φ k (cid:1) ′ (cid:13)(cid:13)(cid:13) N k . X k − k (cid:13)(cid:13)(cid:13) Q > k − C (cid:0) χ I ∇ x φ k (cid:1)(cid:13)(cid:13)(cid:13) L t L x (cid:13)(cid:13)(cid:13) χ I ∇ x φ k (cid:13)(cid:13)(cid:13) L t L ∞ x . k φ k S (cid:18)X k (cid:16) − k (cid:13)(cid:13)(cid:13) χ I ∇ x φ k (cid:13)(cid:13)(cid:13) L t L ∞ x (cid:17) (cid:19) / and similarly for the other term. Smallness follows as before by divisibility. The cubic interactionterm in (7.20) is much simpler to treat, it can be made small on suitable intervals I using estimate(64) from [22] and divisibility of the L t ˙ W , x norm. (cid:3) It remains to prove that the bound (7.8) in the statement of Proposition 7.5 holds. For ε > ffi ciently small further below, depending only on the size of k ( A , φ ) k S and E crit , thereexist M (cid:0) k ( A , φ ) k S , ε (cid:1) many intervals I m , m = , . . . , M , partitioning the time axis R on which theconclusion of Lemma 7.6 holds. We pick such an interval I m and now show that, if necessary, it canbe subdivided into M (cid:0) k ( A , φ ) k S , E crit (cid:1) many intervals I ma , a = , . . . , M , such that k φ k S ( I ma × R ) ≤ C ( E crit ) , where C ( · ) grows at most polynomially. Upon renumbering the intervals I ma , we will then havefinished the proof of Proposition 7.5.For the remainder of the proof, we denote an interval I ma just by I and assume that it is of theform I = [ t , t ] for some t < t . From the equation (cid:3) A φ = A | I = A f ree , ( I ) + A nonlin , ( I ) provided by Lemma 7.6, we conclude that on I × R it holds that (cid:3) pA free , ( I ) φ = − i X k (cid:0) P > k − C A f ree , ( I ) j (cid:1) P k ∂ j φ − iA nonlin , ( I ) j ∂ j φ + iA ∂ t φ + i ( ∂ t A ) φ + A α A α φ ≡ M + M , (7.22)where we use the notation (cid:3) pA free , ( I ) φ = (cid:3) φ + i X k (cid:0) P ≤ k − C A f ree , ( I ) j (cid:1) P k ∂ j φ, M = − i X k (cid:0) P > k − C A f ree , ( I ) j (cid:1) P k ∂ j φ − iA nonlin , ( I ) j ∂ j φ + iA ∂ t φ, M = i ( ∂ t A ) φ + A α A α φ. We further split the term M into M ≡ X k N (cid:0) P > k − C A f ree , ( I ) , P k φ (cid:1) + N (cid:0) A nonlin , ( I ) , φ (cid:1) + N (cid:0) A , φ (cid:1) . Since A f ree , ( I ) and A nonlin , ( I ) are in Coulomb gauge, we observe that the terms N (cid:0) P > k − C A f ree , ( I ) , P k φ (cid:1) and N (cid:0) A nonlin , ( I ) , φ (cid:1) exhibit a null structure, N (cid:0) P > k − C A f ree , ( I ) , P k φ (cid:1) = − i X j , r N jr (cid:0) ∆ − ∇ j P > k − C A f ree , ( I ) r , P k φ (cid:1) , N (cid:0) A nonlin , ( I ) , φ (cid:1) = − i X j , r N jr (cid:0) ∆ − ∇ j A nonlin , ( I ) r , φ (cid:1) . We emphasize that the right hand side of (7.22) is defined on the whole space-time, but which onlycoincides with (cid:3) pA free , ( I ) φ on I × R . Using the linear estimate (3.3) for the magnetic wave operator (cid:3) pA free , ( I ) and working with suitable Schwartz extensions, we obtain that k φ k S ( I × R ) . k∇ t , x φ ( t ) k L x + (cid:13)(cid:13)(cid:13) χ I (cid:0) M + M (cid:1)(cid:13)(cid:13)(cid:13) N ∩ ℓ L t ˙ H − x ( R × R ) . E crit + (cid:13)(cid:13)(cid:13) χ I (cid:0) M + M (cid:1)(cid:13)(cid:13)(cid:13) N ∩ ℓ L t ˙ H − x ( R × R ) . We note that by Theorem 3.1, the implicit constant in the above estimate for the magnetic waveoperator depends polynomially on k∇ t , x A f ree , ( I ) k L ∞ t L x and we have k∇ t , x A f ree , ( I ) k L ∞ t L x . E / crit byLemma 7.6. In order to prove the bound (7.8), it therefore su ffi ces to show that we can choose theintervals I such that (cid:13)(cid:13)(cid:13) M + M (cid:13)(cid:13)(cid:13) N ∩ ℓ L t ˙ H − x ( I × R ) . E crit . Our general strategy to achieve this consists in first using the o ff -diagonal decay in the multilinearestimates from [22] to reduce to a situation in which a suitable divisibility argument works.We only outline how to obtain smallness of the term M in N ( I × R ), the estimate of M in ℓ L t ˙ H − x and of M in N ∩ ℓ L t ˙ H − x being easier. We begin with the first term in the definition of M , (cid:13)(cid:13)(cid:13)X k N (cid:0) P > k − C A f ree , ( I ) , P k φ (cid:1)(cid:13)(cid:13)(cid:13) N ( I × R ) . From the estimate (131) in [22], we conclude that it su ffi ces to bound the expression(7.23) X k (cid:13)(cid:13)(cid:13) P k N (cid:0) P k A f ree , ( I ) , P k φ (cid:1) ′ (cid:13)(cid:13)(cid:13) N k ( I × R ) , where k = k = k + O (1) and both inputs have angular separation ∼
1. Similarly to the estimateof (7.21), we bound this term by (cid:18)X k (cid:16) − k (cid:13)(cid:13)(cid:13) χ I P k ∇ x A f ree , ( I ) (cid:13)(cid:13)(cid:13) L t L ∞ x (cid:17) (cid:19) k φ k S and a divisibility argument then yields smallness. To deal with the other two terms in M , we needto achieve (cid:13)(cid:13)(cid:13) N (cid:0) A nonlin , ( I ) , φ (cid:1) + N ( A , φ ) (cid:13)(cid:13)(cid:13) N ( I × R ) . E crit on suitable intervals I . To this end we will make similar reductions as in Section 4 of [22], peelingo ff the “good parts” of N (cid:0) A nonlin , ( I ) , φ (cid:1) and of N (cid:0) A , φ (cid:1) until we are left with three quadrilinearnull form bounds.We introduce the expressions N lowhi (cid:0) A nonlin , ( I ) , φ (cid:1) = X k N (cid:0) P ≤ k − C A nonlin , ( I ) , P k φ (cid:1) and H ∗ N lowhi ( A nonlin , ( I ) , φ ) = X k X k ′ ≤ k − C X j ≤ k ′ + C Q ≤ j − C N (cid:0) Q j P k ′ A nonlin , ( I ) , Q ≤ j − C P k φ (cid:1) . By estimate (53) in [22], we have(7.24) (cid:13)(cid:13)(cid:13) N (cid:0) A nonlin , ( I ) , φ (cid:1) − N lowhi (cid:0) A nonlin , ( I ) , φ (cid:1)(cid:13)(cid:13)(cid:13) N . (cid:13)(cid:13)(cid:13) A nonlin , ( I ) (cid:13)(cid:13)(cid:13) S k φ k S ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 69 and by estimate (54) in [22], it holds that(7.25) (cid:13)(cid:13)(cid:13) N lowhi (cid:0) A nonlin , ( I ) , φ (cid:1) − H ∗ N lowhi (cid:0) A nonlin , ( I ) , φ (cid:1)(cid:13)(cid:13)(cid:13) N . (cid:13)(cid:13)(cid:13) A nonlin , ( I ) (cid:13)(cid:13)(cid:13) ℓ S k φ k S . Fixing ε > ffi ciently small, depending only on the size of k ( A , φ ) k S and E crit , Lemma 7.6ensures that k A nonlin , ( I ) k ℓ S is small enough so that the right hand sides of (7.24) and (7.25) arebounded by E crit . We now define H A nonlin , ( I ) i : = − χ I X k , k , kk ≤ min { k , k }− C X j ≤ k + C (cid:3) − P k Q j Im P i (cid:0) Q ≤ j − C ( χ I φ k ) · ∇ x Q ≤ j − C ( χ I φ k ) (cid:1) . By estimate (55) in [22] it holds that (cid:13)(cid:13)(cid:13) H ∗ N lowhi (cid:0) A nonlin , ( I ) − H A nonlin , ( I ) , φ (cid:1)(cid:13)(cid:13)(cid:13) N . (cid:13)(cid:13)(cid:13) A nonlin , ( I ) − H A nonlin , ( I ) (cid:13)(cid:13)(cid:13) Z k φ k S , so we have to make (cid:13)(cid:13)(cid:13) A nonlin , ( I ) − H A nonlin , ( I ) (cid:13)(cid:13)(cid:13) Z small. We recall the definition of the Z space, k φ k Z = X k k P k φ k Z k , k φ k Z k = sup l < C X ω l k P ω l Q k + l φ k L t L ∞ x . Using estimate (134) in [22] and that one obtains an extra gain for very negative l when estimatingin the Z space, we are reduced to bounding X k (cid:3) − P k Q k + O (1) ∆ − ∇ x N (cid:0) χ I φ k , χ I φ k (cid:1) . We easily find that X k (cid:13)(cid:13)(cid:13) (cid:3) − P k Q k + O (1) ∆ − ∇ x N (cid:0) χ I φ k , χ I φ k (cid:1)(cid:13)(cid:13)(cid:13) L t L ∞ x . (cid:18)X k (cid:16) − k (cid:13)(cid:13)(cid:13) χ I ∇ x φ k (cid:13)(cid:13)(cid:13) L t L x (cid:17) (cid:19) / k φ k S , which can be made small by a divisibility argument. We are thus left with the term H ∗ N lowhi (cid:0) H A nonlin , ( I ) , φ (cid:1) . Carrying out similar reductions as in Section 4 of [22] for the “elliptic term” N ( A , φ ), we arriveat the key remaining term H ∗ N lowhi (cid:0) H A ( I )0 , φ (cid:1) , where H ∗ N lowhi (cid:0) H A ( I )0 , φ (cid:1) = X k X k ′ ≤ k − C X j ≤ k ′ + C Q ≤ j − C N (cid:0) Q j P k ′ H A ( I )0 , Q ≤ j − C P k φ (cid:1) and H A ( I )0 : = − χ I X k , k , kk ≤ min { k , k }− C X j ≤ k + C ∆ − P k Q j Im (cid:0) Q ≤ j − C ( χ I φ k ) · Q ≤ j − C ∂ t ( χ I φ k ) (cid:1) . As in [22], we combine the “hyperbolic term” H ∗ N lowhi (cid:0) H A nonlin , ( I ) , φ (cid:1) and the preceding “ellipticterm” H ∗ N lowhi (cid:0) H A ( I )0 , φ (cid:1) and wind up with the null forms (61) – (63) in [22]. We formulate theseas quadrilinear expressions as in [22] and then prove that smallness can be achieved for each ofthese. First null form ((61) in [22]).
By estimate (148) in [22], it su ffi ces to consider the following twocases. First, we show that X k X k = k + O (1) (cid:12)(cid:12)(cid:12)(cid:10) (cid:3) − P k Q j (cid:0) Q ≤ j − C ( χ I φ k ) · ∂ α Q ≤ j − C ( χ I φ k ) (cid:1) , P k Q j (cid:0) ∂ α Q ≤ j − C φ k · Q ≤ j − C ψ k (cid:1)(cid:11)(cid:12)(cid:12)(cid:12) ≪ k ψ k N ∗ , where k > k + C , j = k + O (1) and k = k + O (1) = k + O (1). Second, we prove that X k X k = k + O (1) (cid:12)(cid:12)(cid:12)(cid:10) (cid:3) − P k Q j (cid:0) Q ≤ j − C ( χ I φ k ) · ∂ α Q ≤ j − C ( χ I φ k ) (cid:1) , P k Q j (cid:0) ∂ α Q ≤ j − C φ k · Q ≤ j − C ψ k (cid:1)(cid:11)(cid:12)(cid:12)(cid:12) ≪ k ψ k N ∗ , where k > k + C , j = k + O (1) and k = k + O (1) = k + O (1).We begin with the first case. Here, the inputs Q ≤ j − C ( χ I φ k ) and ∂ α Q ≤ j − C ( χ I φ k ) have Fouriersupports in identical (or opposite) angular sectors ω of size ∼ k − k . Then we bound X k X k > k + O (1) (cid:12)(cid:12)(cid:12)(cid:10) (cid:3) − P k Q k + O (1) (cid:0) Q ≤ k − C ( χ I φ k ) · ∂ α Q ≤ k − C ( χ I φ k + O (1) ) (cid:1) , P k Q k + O (1) (cid:0) ∂ α Q ≤ k − C φ k + O (1) · Q ≤ k − C ψ k + O (1) (cid:1)(cid:11)(cid:12)(cid:12)(cid:12) . X k X k > k + O (1) ( k − k ) (cid:18)X ω k (cid:13)(cid:13)(cid:13) P ω Q ≤ k − C ( χ I φ k ) (cid:13)(cid:13)(cid:13) L t L x (cid:19) (cid:18)X ω (cid:13)(cid:13)(cid:13) P ω Q ≤ k − C ∇ t , x ( χ I φ k ) (cid:13)(cid:13)(cid:13) L ∞ t L x (cid:19) ×× − k (cid:13)(cid:13)(cid:13) ∇ t , x φ k + O (1) (cid:13)(cid:13)(cid:13) L t L x k ψ k + O (1) k L ∞ t L x . (cid:18)X k sup l < − C X ω k (cid:13)(cid:13)(cid:13) P ω l Q ≤ k + l − C ( χ I φ k ) (cid:13)(cid:13)(cid:13) L t L x (cid:19) k φ k S k ψ k N ∗ . The desired smallness comes from the divisibility of the quantity (cid:18)X k sup l < − C X ω k (cid:13)(cid:13)(cid:13) P ω l Q ≤ k + l − C ( χ I φ k ) (cid:13)(cid:13)(cid:13) L t L x (cid:19) . To see the divisibility, we write(7.26) P ω l Q ≤ k + l − C ( χ I φ k ) = P ω l Q ≤ k + l − C ( χ I P ω l Q ≤ k + l + M φ k ) + P ω l Q ≤ k + l − C ( χ I Q > k + l + M φ k )for some M > ffi ciently large. By disposability of the operator P ω l Q ≤ k + l − C , weestimate the first term on the right hand side of (7.26) by (cid:18)X k sup l < − C X ω k (cid:13)(cid:13)(cid:13) P ω l Q ≤ k + l − C ( χ I P ω l Q ≤ k + l + M φ k ) (cid:13)(cid:13)(cid:13) L t L x (cid:19) . (cid:18)X k sup l < − C X ω k (cid:13)(cid:13)(cid:13) χ I P ω l Q ≤ k + l + M φ k (cid:13)(cid:13)(cid:13) L t L x (cid:19) and smallness can be forced by divisibility of the quantity (cid:18)X k sup l < − C X ω k (cid:13)(cid:13)(cid:13) P ω l Q ≤ k + l + M φ k (cid:13)(cid:13)(cid:13) L t L x (cid:19) . k φ k S . For the second term on the right hand side of (7.26), we use P ω l Q ≤ k + l − C ( χ I Q > k + l + M φ k ) = P ω l Q ≤ k + l − C (cid:0) Q > k + l + M ( χ I ) Q > k + l + M φ k (cid:1) . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 71
By Bernstein’s inequality in space and in time, we then have (cid:13)(cid:13)(cid:13) P ω l Q ≤ k + l − C (cid:0) Q > k + l + M ( χ I ) Q > k + l + M φ k (cid:1)(cid:13)(cid:13)(cid:13) L t L x . ( k + l ) k l (cid:13)(cid:13)(cid:13) Q > k + l + M χ I (cid:13)(cid:13)(cid:13) L t (cid:13)(cid:13)(cid:13) P ω l Q > k + l + M φ k (cid:13)(cid:13)(cid:13) L t L x . ( k + l ) k l − ( k + l + M ) (cid:13)(cid:13)(cid:13) P ω l Q > k + l + M φ k (cid:13)(cid:13)(cid:13) L t L x . k l − M (cid:13)(cid:13)(cid:13) P ω l Q > k + l + M φ k (cid:13)(cid:13)(cid:13) L t L x . Thus, we obtain (cid:18)X k sup l < − C X ω k (cid:13)(cid:13)(cid:13) P ω l Q ≤ k + l − C (cid:0) Q > k + l + M ( χ I ) Q > k + l + M φ k (cid:1)(cid:13)(cid:13)(cid:13) L t L x (cid:19) . (cid:18)X k sup l < − C l − M k∇ x φ k k X , ∞ (cid:19) . − M k φ k S and a smallness factor follows for su ffi ciently large M > X k X k > k + O (1) (cid:12)(cid:12)(cid:12)(cid:10) (cid:3) − P k Q k + O (1) (cid:0) Q ≤ k − C ( χ I φ k + O (1) ) · ∂ α Q ≤ k − C ( χ I φ k + O (1) ) (cid:1) , P k Q k + O (1) (cid:0) ∂ α Q ≤ k − C φ k · Q ≤ k − C ψ k + O (1) (cid:1)(cid:11)(cid:12)(cid:12)(cid:12) . X k X k > k + O (1) − k (cid:13)(cid:13)(cid:13) χ I ∇ x φ k + O (1) (cid:13)(cid:13)(cid:13) L t L ∞ x − k (cid:13)(cid:13)(cid:13) ∇ t , x ( χ I φ k + O (1) ) (cid:13)(cid:13)(cid:13) L t L ∞ x ×× (cid:13)(cid:13)(cid:13) Q ≤ k − C ∇ t , x φ k (cid:13)(cid:13)(cid:13) L ∞ t L x (cid:13)(cid:13)(cid:13) Q ≤ k − C ψ k + O (1) (cid:13)(cid:13)(cid:13) L ∞ t L x . (cid:18)X k (cid:16) − k (cid:13)(cid:13)(cid:13) χ I ∇ x φ k + O (1) (cid:13)(cid:13)(cid:13) L t L ∞ x (cid:17) (cid:19) k φ k S k ψ k N ∗ and immediately obtain smallness by divisibility. Second null form ((62) in [22]) . By the estimates (149) and (150) in [22], we only have to showthat X k X k = k + O (1) (cid:12)(cid:12)(cid:12)(cid:10) ( (cid:3) ∆ ) − P k Q j ∂ t ∂ α (cid:0) Q ≤ j − C ( χ I φ k ) · ∂ α Q ≤ j − C ( χ I φ k ) (cid:1) , P k Q j (cid:0) ∂ t Q ≤ j − C φ k · Q ≤ j − C ψ k (cid:1)(cid:11)(cid:12)(cid:12)(cid:12) ≪ k ψ k N ∗ , where j = k + O (1), k = k + O (1) = k + O (1) and k > k + C . Then we estimate X k X k > k + C (cid:12)(cid:12)(cid:12)(cid:10) ( (cid:3) ∆ ) − P k Q k + O (1) ∂ t ∂ α (cid:0) Q ≤ k − C ( χ I φ k + O (1) ) · ∂ α Q ≤ k − C ( χ I φ k + O (1) ) (cid:1) , P k Q k + O (1) (cid:0) ∂ t Q ≤ k − C φ k · Q ≤ k − C ψ k + O (1) (cid:1)(cid:11)(cid:12)(cid:12)(cid:12) . X k X k > k + C (cid:13)(cid:13)(cid:13) ( (cid:3) ∆ ) − P k Q k + O (1) ∂ t ∂ α (cid:0) Q ≤ k − C ( χ I φ k + O (1) ) · ∂ α Q ≤ k − C ( χ I φ k + O (1) ) (cid:1)(cid:13)(cid:13)(cid:13) L t L ∞ x ×× (cid:13)(cid:13)(cid:13) P k Q k + O (1) (cid:0) ∂ t Q ≤ k − C φ k · Q ≤ k − C ψ k + O (1) (cid:1)(cid:13)(cid:13)(cid:13) L ∞ t L x . X k X k > k + C − k (cid:13)(cid:13)(cid:13) ∇ x ( χ I φ k + O (1) ) (cid:13)(cid:13)(cid:13) L t L ∞ x − k (cid:13)(cid:13)(cid:13) ∇ x ( χ I φ k + O (1) ) (cid:13)(cid:13)(cid:13) L t L ∞ x k ∂ t φ k k L ∞ t L x k ψ k + O (1) k L ∞ t L x . (cid:18)X k (cid:16) − k (cid:13)(cid:13)(cid:13) ∇ x ( χ I φ k + O (1) ) (cid:13)(cid:13)(cid:13) L t L ∞ x (cid:17) (cid:19) / k φ k S k ψ k N ∗ and smallness follows by divisibility. Third null form ((63) in [22]).
By the estimates (152) – (154) in [22], it su ffi ces to consider thefollowing two cases. First, we show that X k X k = k + O (1) (cid:12)(cid:12)(cid:12)(cid:10) ( (cid:3) ∆ ) − P k Q j ∂ i (cid:0) Q ≤ j − C ( χ I φ k ) · ∂ i Q ≤ j − C ( χ I φ k ) (cid:1) , P k Q j ∂ α (cid:0) ∂ α Q ≤ j − C φ k · Q ≤ j − C ψ k (cid:1)(cid:11)(cid:12)(cid:12)(cid:12) ≪ k ψ k N ∗ , where k > k + C , j = k + O (1) and k = k + O (1) = k + O (1). Second, we prove that X k X k = k + O (1) (cid:12)(cid:12)(cid:12)(cid:10) ( (cid:3) ∆ ) − P k Q j ∂ i (cid:0) Q ≤ j − C ( χ I φ k ) · ∂ i Q ≤ j − C ( χ I φ k ) (cid:1) , P k Q j ∂ α (cid:0) ∂ α Q ≤ j − C φ k · Q ≤ j − C ψ k (cid:1)(cid:11)(cid:12)(cid:12)(cid:12) ≪ k ψ k N ∗ , where k > k + C , j = k + O (1) and k = k + O (1) = k + O (1).In the first case we note that the first two inputs have Fourier supports in identical (or opposite)angular sectors ω of size ∼ k − k . Using Bernstein’s inequality, we then place the first input in L t L x ,the second one in L ∞ t L x , the third one in L t L ∞ x and the fourth one in L ∞ t L x . As in the first case ofthe first null form we obtain the desired smallness by divisibility of the quantity (cid:18)X k sup l < − C X ω k (cid:13)(cid:13)(cid:13) P ω l Q ≤ k + l − C ( χ I φ k ) (cid:13)(cid:13)(cid:13) L t L x (cid:19) . The second case is easier to deal with and we omit the details. (cid:3)
Step 3.
Solution of perturbative problems on suitable space-time slices.
This is the crucial technicalstep. We write (cid:0) A n , ( L ) Λ , φ n , ( L ) Λ (cid:1) = (cid:0) A n , ( L − Λ , φ n , ( L − Λ (cid:1) + (cid:0) δ A ( L ) , δφ ( L ) (cid:1) . Then we obtain the following system of equations for the perturbations (cid:0) δ A ( L ) , δφ ( L ) (cid:1) ,(7.27) (cid:3) A n , ( L − Λ + δ A ( L ) (cid:0) φ n , ( L − Λ + δφ ( L ) (cid:1) − (cid:3) A n , ( L − Λ φ n , ( L − Λ = , (cid:3) δ A ( L ) = − Im P (cid:16) φ n , ( L − Λ · ∇ x δφ ( L − + δφ ( L − · ∇ x φ n , ( L − Λ + δφ ( L − · ∇ x δφ ( L − (cid:17) + Im P (cid:16)(cid:0) A n , ( L − Λ + δ A ( L ) (cid:1)(cid:12)(cid:12)(cid:12) φ n , ( L − Λ + δφ ( L ) (cid:12)(cid:12)(cid:12) − A n , ( L − Λ (cid:12)(cid:12)(cid:12) φ n , ( L − Λ (cid:12)(cid:12)(cid:12) (cid:17) . (7.28) ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 73
We have to show that if the initial data ( δ A ( L ) , δφ ( L ) )[0] are less than the absolute constant ε in theenergy sense, then we can prove frequency localized S norm bounds via bootstrap on any space-time slice on which certain “divisible” norms of (cid:0) A n , ( L − Λ , φ n , ( L − Λ (cid:1) are small. Furthermore, thenumber of such space-time slices needed to fill all of space-time depends on the a priori assumed S norm bounds for the components (cid:0) A n , ( L − Λ , φ n , ( L − Λ (cid:1) .One technical di ffi culty is the formulation of the correct frequency localized S norm boundfor the propagation of δφ ( L ) , because there is a contribution from low frequencies of φ n , ( L − Λ , andsimilarly for δ A ( L ) . However, this low frequency contribution can be made arbitrarily small bypicking n large and δ small enough.We note that while (cid:0) A n , ( L − Λ , φ n , ( L − Λ (cid:1) exists globally in time, (cid:0) δ A ( L ) , δφ ( L ) (cid:1) only exists locallyin time and we will have to prove global existence and S norm bounds for it. For now, anystatement we make about (cid:0) δ A ( L ) , δφ ( L ) (cid:1) is meant locally in time on some interval I around t = R into N = N (cid:0) k (cid:0) A n , ( L − Λ , φ n , ( L − Λ (cid:1) k S ( R × R ) (cid:1) manytime intervals { I j } Nj = , on which the smallness conclusions (in terms of E crit ) of Proposition 7.5 hold.We tacitly assume that these intervals are intersected with I and now fix the interval I , which weassume to contain t =
0. All the arguments in this step can be carried out for any of the laterintervals I , . . . , I N . Bootstrap assumptions : Suppose that there exist decompositions δ A ( L ) = δ A ( L ) ① + δ A ( L ) ② , δφ ( L ) = δφ ( L ) ① + δφ ( L ) ② satisfying the following bounds.(i) Let (cid:8) c ( L ) δ A , k (cid:9) k ∈ Z be a frequency envelope controlling the data P k δ A ( L ) [0] at time t = (cid:8) d ( L ) δ A , k (cid:9) k ∈ Z be a frequency envelope that decays exponentially for k > b L but is otherwise notlocalized and satisfies the smallness condition X k d ( L ) δ A , k ≤ δ = δ ( δ ) . Then we assume that for all k ∈ Z , (cid:13)(cid:13)(cid:13) P k δ A ( L ) ① (cid:13)(cid:13)(cid:13) S ( I × R ) ≤ Cc ( L ) δ A , k , (cid:13)(cid:13)(cid:13) P k δ A ( L ) ② (cid:13)(cid:13)(cid:13) S ( I × R ) ≤ Cd ( L ) δ A , k , where C ≡ C (cid:0) E crit (cid:1) is su ffi ciently large.(ii) Let (cid:8) c ( L ) δφ, k (cid:9) k ∈ Z be a frequency envelope controlling the data P k δφ ( L ) [0] at time t = (cid:8) d ( L ) δφ, k (cid:9) be a frequency envelope that decays exponentially for k > b L , but is otherwise not localizedand satisfies the smallness condition (cid:16)X k (cid:0) d ( L ) δφ, k (cid:1) (cid:17) ≤ δ = δ ( δ ) . Then we assume that for all k ∈ Z , (cid:13)(cid:13)(cid:13) P k δφ ( L ) ① (cid:13)(cid:13)(cid:13) S k ( I × R ) ≤ Cc ( L ) δφ, k , (cid:13)(cid:13)(cid:13) P k δφ ( L ) ② (cid:13)(cid:13)(cid:13) S k ( I × R ) ≤ Cd ( L ) δφ, k , where C ≡ C (cid:0) E crit (cid:1) is su ffi ciently large. We now show that we can improve this to a similar decomposition with (cid:13)(cid:13)(cid:13) P k δ A ( L ) ① (cid:13)(cid:13)(cid:13) S k ( I × R ) ≤ C c ( L ) δ A , k , (cid:13)(cid:13)(cid:13) P k δ A ( L ) ② (cid:13)(cid:13)(cid:13) S k ( I × R ) ≤ C d ( L ) δ A , k , (cid:13)(cid:13)(cid:13) P k δφ ( L ) ① (cid:13)(cid:13)(cid:13) S k ( I × R ) ≤ C c ( L ) δφ, k , (cid:13)(cid:13)(cid:13) P k δφ ( L ) ② (cid:13)(cid:13)(cid:13) S k ( I × R ) ≤ C d ( L ) δφ, k , (7.29)provided we make the additional assumption δ ≪ δ with implied constant depending only on E crit .Observe that we have X k (cid:0) c ( L ) δ A , k (cid:1) + (cid:0) c ( L ) δφ, k (cid:1) . ε and that our smallness parameters satisfy δ ≪ δ ≪ δ ≪ ε . For the remainder of this step we simply write I ≡ I and φ ≡ φ n , ( L − Λ , δφ ≡ δφ ( L ) , A ≡ A n , ( L − Λ , δ A ≡ δ A ( L ) . Step 3a.
Reorganizing the key equation (7.27) . We introduce the connection form ( A + δ A ) nonlin , ( I ) analogously to (7.9) by setting for i = , . . . , A + δ A ) nonlin , ( I ) i : = − χ I X k , j (cid:3) − P k Q j P i (cid:0) χ I ( φ + δφ ) · ∇ x (cid:0) χ I ( φ + δφ ) (cid:1) − χ I i ( A + δ A ) | φ + δφ | (cid:1) , and define ( A + δ A ) f ree , ( I ) as the free wave with initial data at time t = A + δ A ) f ree , ( I ) [0] = ( A + δ A )[0] − ( A + δ A ) nonlin , ( I ) [0] . Then we have ( A + δ A ) | I = ( A + δ A ) f ree , ( I ) + ( A + δ A ) nonlin , ( I ) . On I × R we may rewrite the equation (7.27) for δφ into the following frequency localized form (cid:3) p ( A + δ A ) free , ( I ) (cid:0) P δφ (cid:1) = − (cid:2) P , (cid:3) p ( A + δ A ) free , ( I ) (cid:3) δφ − P (cid:16) i X k P > k − C ( A + δ A ) f ree , ( I ) j P k ∂ j δφ (cid:17) − P (cid:16) i ( A + δ A ) nonlin , ( I ) j ∂ j δφ − i ( A + δ A ) ∂ t δφ (cid:17) − P (cid:16) i ( δ A ) j ∂ j φ − i ( δ A ) ∂ t φ (cid:17) + P (cid:16) i ( ∂ t A + ∂ t δ A )( φ + δφ ) − i ( ∂ t A ) φ (cid:17) + P (cid:16) ( A + δ A ) α ( A + δ A ) α φ − A α A α φ (cid:17) . (7.31)We immediately see that compared to (7.22), a qualitatively new feature in (7.31) is the interactionterm(7.32) P (cid:16) ( δ A ) j ∂ j φ − ( δ A ) ∂ t φ (cid:17) . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 75
Step 3b.
Improving the bounds for δφ using (7.31) . In order to obtain bounds on the S ( I × R )norm of P δφ by bootstrap, we work with suitable Schwartz extensions and use the linear estimate(3.3) for the magnetic wave operator (cid:3) p ( A + δ A ) free , ( I ) . We first consider the new interaction term (7.32).As usual the main di ffi culty comes from the low-high interactions, so we begin with this case, i.e.the term P < ( δ A ) j P ∂ j φ − P < ( δ A ) P ∂ t φ. For the spatial components of δ A , we define( δ A ) f ree , ( I ) : = ( A + δ A ) f ree , ( I ) − A f ree , ( I ) , ( δ A ) nonlin , ( I ) : = ( A + δ A ) nonlin , ( I ) − A nonlin , ( I ) and correspondingly have on I × R that( δ A ) | I = ( δ A ) f ree , ( I ) + ( δ A ) nonlin , ( I ) . We can therefore split on I × R , P < ( δ A ) j P ∂ j φ = P < ( δ A ) f ree , ( I ) j P ∂ j φ + P < ( δ A ) nonlin , ( I ) j P ∂ j φ. The first term on the right hand side can in turn be split into two contributions(7.33) P < ( δ A ) f ree , ( I ) j P ∂ j φ = P < ( δ A ) f ree , ( I ) j P ∂ j φ + P < ( δ A ) f ree , ( I ) j P ∂ j φ, where ( δ A ) f ree , ( I ) is the free evolution of the data ( δ A ) ( L ) [0], while ( δ A ) f ree , ( I ) is the free wave withdata (cid:18)X k , j (cid:3) − P k Q j P (cid:0) χ I ( φ + δφ ) · ∇ x (cid:0) χ I ( φ + δφ ) (cid:1) − χ I i ( A + δ A ) | φ + δφ | (cid:1)(cid:19) [0] , − (cid:18)X k , j (cid:3) − P k Q j P (cid:0) ( χ I φ ) · ∇ x ( χ I φ ) − χ I iA | φ | (cid:1)(cid:19) [0] . In order to estimate the terms on the right hand side of (7.33), we will invoke the following estimatefrom [22] for a free wave A f ree in Coulomb gauge for k ≤ k − C ,(7.34) (cid:13)(cid:13)(cid:13) P k A f reej P k ∂ j φ (cid:13)(cid:13)(cid:13) N . (cid:13)(cid:13)(cid:13) P k A f ree (cid:13)(cid:13)(cid:13) S (cid:13)(cid:13)(cid:13) P k φ (cid:13)(cid:13)(cid:13) S . (cid:13)(cid:13)(cid:13) P k A f ree [0] (cid:13)(cid:13)(cid:13) ˙ H x × L x (cid:13)(cid:13)(cid:13) P k φ (cid:13)(cid:13)(cid:13) S . We begin with the first term on the right hand side of (7.33), P < ( δ A ) f ree , ( I ) j P ∂ j φ. Here we have to take advantage of the properties of the Fourier support of the data P < ( δ A ) f ree , ( I ) j [0].It su ffi ces to assume that (cid:13)(cid:13)(cid:13) P k ( δ A ) f ree j [0] (cid:13)(cid:13)(cid:13) ˙ H x × L x ≤ C (cid:0) c δ A , k + d δ A , k (cid:1) for C ≡ C ( E crit ) su ffi ciently large. This is an assumption that will hold inductively at later initialtimes (for the intervals I , . . . , I N ). We observe that the frequency envelope (cid:8) c δ A , k (cid:9) k ∈ Z is “sharplylocalized” to the dyadic frequency interval [ a L , b L ] in the sense that it is exponentially decaying for k < a L and k > b L . By (7.34) we then have(7.35) (cid:13)(cid:13)(cid:13) P < ( δ A ) f ree , ( I ) j P ∂ j φ (cid:13)(cid:13)(cid:13) N ( I × R ) . X k < c δ A , k (cid:13)(cid:13)(cid:13) P φ (cid:13)(cid:13)(cid:13) S ( I × R ) + X k < d δ A , k (cid:13)(cid:13)(cid:13) P φ (cid:13)(cid:13)(cid:13) S ( I × R ) . We begin to estimate the first term on the right hand side of (7.35), where we only consider the casewhen a L <
0. For R > ffi ciently large later on, we split X k < c δ A , k (cid:13)(cid:13)(cid:13) P φ (cid:13)(cid:13)(cid:13) S ( I × R ) = X k ≤ a L − R c δ A , k (cid:13)(cid:13)(cid:13) P φ (cid:13)(cid:13)(cid:13) S ( I × R ) + X a L − R < k ≤ a L + R c δ A , k (cid:13)(cid:13)(cid:13) P φ (cid:13)(cid:13)(cid:13) S ( I × R ) + X a L + R < k < c δ A , k (cid:13)(cid:13)(cid:13) P φ (cid:13)(cid:13)(cid:13) S ( I × R ) . (7.36)To make the first term on the right hand side of (7.36) small, we use the exponential decay of thefrequency envelope (cid:8) c δ A , k (cid:9) k ∈ Z to bound X k ≤ a L − R c δ A , k (cid:13)(cid:13)(cid:13) P φ (cid:13)(cid:13)(cid:13) S ( I × R ) . X k ≤ a L − R − σ ( a L − k ) k δ A [0] k ˙ H x × L x k P φ k S ( I × R ) . E crit − σ R k P φ k S ( I × R ) . Upon replacing the output frequency 0 by general l ∈ Z , square summing over l and choosing R > ffi ciently large, we bound the preceding by ≪ E crit δ , as desired. In order to make thethird term on the right hand side of (7.36) small, we exploit that by Step 1 the S norms of P l φ areexponentially decaying beyond the scale l > a L . We have X a L + R < k < c δ A , k (cid:13)(cid:13)(cid:13) P φ (cid:13)(cid:13)(cid:13) S ( I × R ) . E crit ( | a L | − R ) c , where { c l } l ∈ Z is a su ffi ciently flat frequency envelope covering the initial data (cid:0) A n , ( L − Λ , φ n , ( L − Λ (cid:1) [0]as in Step 1. Then replacing the frequency 0 in P φ by a general dyadic frequency l > a L + R ,square summing over l and choosing R > ffi ciently large, we find X l > a L + R (cid:12)(cid:12)(cid:12)(cid:12) X a L + R < k < l c δ A , k (cid:13)(cid:13)(cid:13) P l φ (cid:13)(cid:13)(cid:13) S ( I × R ) (cid:12)(cid:12)(cid:12)(cid:12) . E crit X l > a L + R ( l − a L − R ) c l . E crit − σ R ≪ E crit δ , which is acceptable. It remains to make the second term on the right hand side of (7.36) small.To this end we exploit the frequency evacuation property of the data (cid:0) A n Λ , φ n Λ (cid:1) at the edges of thefrequency intervals [ a L , b L ] that we established in Lemma 7.3. For su ffi ciently small δ > ffi ciently large n , we then have X a L − R < k ≤ a L + R c δ A , k (cid:13)(cid:13)(cid:13) P φ (cid:13)(cid:13)(cid:13) S ( I × R ) . R δ (cid:13)(cid:13)(cid:13) P φ (cid:13)(cid:13)(cid:13) S ( I × R ) ≪ δ (cid:13)(cid:13)(cid:13) P φ (cid:13)(cid:13)(cid:13) S ( I × R ) . Upon replacing the frequency 0 in P φ by an arbitrary dyadic frequency l ∈ Z and square summing,we obtain the desired smallness ≪ E crit δ for the last estimate.The contribution of the second term on the right hand side of (7.35) is acceptable, because, uponreplacing the output frequency 0 by l ∈ Z , square summing and using the bootstrap assumptions onthe interval I , we obtain the bound (cid:18)X l (cid:12)(cid:12)(cid:12)(cid:12)X k < l d δ A , k (cid:13)(cid:13)(cid:13) P l φ (cid:13)(cid:13)(cid:13) S ( I × R ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) . E crit δ ≪ δ , where the implied constant in . E crit depends at most polynomially on E crit .Next, we estimate the second term on the right hand side of (7.33), P < ( δ A ) f ree , ( I ) j P ∂ j φ. By (7.34) we have(7.37) (cid:13)(cid:13)(cid:13) P < ( δ A ) f ree , ( I ) j P ∂ j φ (cid:13)(cid:13)(cid:13) N ( I × R ) . (cid:13)(cid:13)(cid:13) P < ( δ A ) f ree , ( I ) (cid:13)(cid:13)(cid:13) ℓ S ( I × R ) (cid:13)(cid:13)(cid:13) P φ (cid:13)(cid:13)(cid:13) S ( I × R ) . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 77
We illustrate how to obtain the desired smallness in this case by assuming for simplicity that P < ( δ A ) f ree , ( I ) is just the free evolution of the data X k < X j (cid:3) − P k Q j P (cid:0) χ I φ · ∇ x ( χ I δφ ) (cid:1) [0] . If δφ = ( δφ ) ① , we obtain by similar estimates as in the proof of Lemma 7.6 that (cid:13)(cid:13)(cid:13) P < ( δ A ) f ree , ( I ) (cid:13)(cid:13)(cid:13) ℓ S ( I × R ) . X k < (cid:13)(cid:13)(cid:13) χ I φ (cid:13)(cid:13)(cid:13) S c δφ, k . Then we achieve the desired smallness for (cid:13)(cid:13)(cid:13) P < ( δ A ) f ree , ( I ) j P ∂ j φ (cid:13)(cid:13)(cid:13) N ( I × R ) . E crit X k < c δφ, k (cid:13)(cid:13)(cid:13) P φ (cid:13)(cid:13)(cid:13) S ( I × R ) by proceeding exactly as with the term (7.36). If instead δφ = ( δφ ) ② , we find (cid:13)(cid:13)(cid:13) P < ( δ A ) f ree , ( I ) j P ∂ j φ (cid:13)(cid:13)(cid:13) N ( I × R ) . (cid:13)(cid:13)(cid:13) χ I φ (cid:13)(cid:13)(cid:13) S (cid:18)X k d δφ, k (cid:19) (cid:13)(cid:13)(cid:13) P φ (cid:13)(cid:13)(cid:13) S ( I × R ) . δ (cid:13)(cid:13)(cid:13) χ I φ (cid:13)(cid:13)(cid:13) S (cid:13)(cid:13)(cid:13) P φ (cid:13)(cid:13)(cid:13) S ( I × R ) . Upon replacing the output frequency 0 by l ∈ Z , square summing and using that k φ k S ( I × R ) . C ( E crit ) by the choice of the interval I , we obtain the bound . E crit δ . This is unfortunately not yetenough to close the bootstrap. To gain the extra smallness we partition the interval I further and usedivisibility arguments as in the proof of Lemma 7.6. However, the number of intervals needed forthis partition depends only on E crit (and not on the stage of the induction), which is acceptable.This finishes the estimate of the contribution of P < ( δ A ) f ree , ( I ) j P ∂ j φ and we now have to bound (cid:13)(cid:13)(cid:13) P < ( δ A ) nonlin , ( I ) j P ∂ j φ − P < ( δ A ) P ∂ t φ (cid:13)(cid:13)(cid:13) N ( I × R ) . At this point we can proceed by analogy to the treatment of the φ equation in the proof of Propo-sition 7.5. After a further partitioning of the interval I and similar divisibility arguments, we canreplace the output frequency 0 by l ∈ Z and upon square summing, we obtain a bound of the desiredform ≪ E crit δ .The remaining frequency interactions in the estimate of the term (7.32) as well as all other termson the right hand side of (7.31) are easier to control. We omit the details. Step 3c.
Improving the bounds for δ A using (7.28) . In order to deduce S ( I × R ) norm bounds on P δ A from the perturbation equation (7.28) by bootstrap, we perform the same kind of divisibilityarguments as in the proof of estimate (7.11) in Lemma 7.6 for the terms linear in δφ or δ A . Step 4.
Repetition of the bootstrap on suitable space-time slices; proof that the energy of pertur-bation remains small.
In this final step we show that the crucial assumption on the energy of theperturbation E (cid:0) δ A ( L ) , δφ ( L ) (cid:1) (0) < ε remains in tact along the evolution up to a very small correction. We recall that δ A ( L ) = A n , ( L ) Λ − A n , ( L − Λ , δφ ( L ) = φ n , ( L ) Λ − φ n , ( L − Λ . Lemma 7.7.
Assuming the bounds (7.29) on the evolution of (cid:0) δ A ( L ) , δφ ( L ) (cid:1) on I × R , we have forsu ffi ciently small δ > and all su ffi ciently large n thatE (cid:0) δ A ( L ) , δφ ( L ) (cid:1) ( t ) < ε for t ∈ I . Proof.
By energy conservation for the evolutions of (cid:0) A n , ( L ) Λ , φ n , ( L ) Λ (cid:1) and of (cid:0) A n , ( L − Λ , φ n , ( L − Λ (cid:1) , itsu ffi ces to show that (cid:12)(cid:12)(cid:12)(cid:12) E (cid:0) A n , ( L ) Λ , φ n , ( L ) Λ (cid:1) ( t ) − E (cid:0) A n , ( L − Λ , φ n , ( L − Λ (cid:1) ( t ) − E (cid:0) δ A ( L ) , δφ ( L ) (cid:1) ( t ) (cid:12)(cid:12)(cid:12)(cid:12) can be made arbitrarily small uniformly for all t ∈ I by choosing δ > ffi ciently small and n su ffi ciently large. This reduces to bounding the following expression evaluated at any time t ∈ I , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X i < j Z R (cid:0) ∂ i A n , ( L − Λ , j (cid:1)(cid:0) ∂ i δ A ( L ) j (cid:1) dx + X i Z R (cid:0) ∂ t A n , ( L − Λ , i (cid:1)(cid:0) ∂ t δ A ( L ) i (cid:1) + (cid:0) ∂ i A n , ( L ) Λ , (cid:1)(cid:0) ∂ i δ A ( L − (cid:1) dx + X α Z R (cid:12)(cid:12)(cid:12)(cid:0) A n , ( L − Λ ,α (cid:1)(cid:0) δφ ( L ) (cid:1) + (cid:0) δ A ( L ) α (cid:1)(cid:0) φ n , ( L − Λ (cid:1)(cid:12)(cid:12)(cid:12) dx + X α Re Z R (cid:0) ∂ α φ n , ( L − Λ + iA n , ( L − Λ ,α φ n , ( L − Λ (cid:1)(cid:0) ∂ α δφ ( L ) + i δ A ( L ) α δφ ( L ) (cid:1) + (cid:0) ∂ α δφ ( L ) + i δ A ( L ) α δφ ( L ) (cid:1)(cid:0) iA n , ( L − Λ ,α δφ ( L ) + i δ A ( L ) α φ n , ( L − Λ (cid:1) + (cid:0) ∂ α φ n , ( L − Λ + iA n , ( L − Λ ,α φ n , ( L − Λ (cid:1)(cid:0) iA n , ( L − Λ ,α δφ ( L ) + i δ A ( L ) α φ n , ( L − Λ (cid:1) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . We note that in this expression at least one term of the form A n , ( L − Λ or φ n , ( L − Λ is paired againstat least one term of the form δ A ( L ) or δφ ( L ) . By Plancherel’s theorem (and a Littlewood-Paleytrichotomy to deal with the multilinear interactions), we reduce to estimating a sum of the form (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X k ∈ Z X i < j Z R P k (cid:0) ∂ i A n , ( L − Λ , j (cid:1) P k (cid:0) ∂ i δ A ( L ) j (cid:1) dx + . . . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . By the bounds (7.29) and Step 1, we estimate this by . X k ∈ Z (cid:13)(cid:13)(cid:13) P k ∇ x A n , ( L − Λ ( t ) (cid:13)(cid:13)(cid:13) L x (cid:13)(cid:13)(cid:13) P k ∇ x δ A ( L ) ( t ) (cid:13)(cid:13)(cid:13) L x + . . . . X k ∈ Z c ( L − k (cid:0) c ( L ) δ A , k + d ( L ) δ A , k (cid:1) + . . . To see that this expression can be made arbitrarily small, we split X k ∈ Z c ( L − k c ( L ) δ A , k = X k ≤ a L − R c ( L − k c ( L ) δ A , k + X a L − R < k ≤ a L + R c ( L − k c ( L ) δ A , k + X k > a L + R c ( L − k c ( L ) δ A , k . The first term can be made arbitrarily small for su ffi ciently large R > (cid:8) c ( L ) δ A , k (cid:9) k ∈ Z beyond [ a L , b L ]. Similarly, we achieve smallness for the thirdterm for su ffi ciently large R > (cid:8) c ( L − k (cid:9) k ∈ Z for k > a L established inStep 1. Finally, we gain smallness for the second term for all su ffi ciently large n from the frequencyevacuation property in Lemma 7.3. Moreover, we have by (7.29) that X k ∈ Z c ( L − k d ( L ) δ A , k . E crit δ ( δ ) ≪ . (cid:3) ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 79
We now summarize how the previous steps yield the proof of Proposition 7.4. In order to derive S norm bounds on the evolutions (cid:0) A n , ( L ) Λ , φ n , ( L ) Λ (cid:1) , we first use Proposition 7.5 from Step 2 topartition the time axis R into N = N (cid:16)(cid:13)(cid:13)(cid:13)(cid:0) A n , ( L − Λ , φ n , ( L − Λ (cid:1)(cid:13)(cid:13)(cid:13) S ( R × R ) (cid:17) many time intervals I , . . . , I N ,on which certain “divisible norms” of (cid:0) A n , ( L − Λ , φ n , ( L − Λ (cid:1) are small in terms of E crit . Let I be thetime interval containing t =
0. By construction of the frequency intervals J L , 1 ≤ L ≤ C , theenergy of the perturbation (cid:0) δ A ( L ) , δφ ( L ) (cid:1) [0] = (cid:0) A n , ( L ) Λ − A n , ( L − Λ , φ n , ( L ) Λ − φ n , ( L − Λ (cid:1) [0]at time t = ε . Thus, we can prove frequency localized S normbounds for (cid:0) δ A ( L ) , δφ ( L ) (cid:1) on I × R by bootstrap as in Step 3. Crucially, Lemma 7.7 from Step 4ensures that the energy of the perturbation (cid:0) δ A ( L ) , δφ ( L ) (cid:1) [ t ], t ∈ I , is approximately conserved onthe time interval I , up to a very small error term that is controlled by the size of δ . Hence, we canensure that at the starting points of all later (or earlier) time intervals I , . . . , I N , the energy of theperturbation is still less than the absolute constant ε by choosing δ su ffi ciently small dependingon the number N of “divisibility intervals”, which is bounded by the size of (cid:13)(cid:13)(cid:13)(cid:0) A n , ( L − Λ , φ n , ( L − Λ (cid:1)(cid:13)(cid:13)(cid:13) S ( R × R ) ≤ C . This allows us to repeat the same bootstrap argument from Step 3 on all other time intervals I , . . . , I N . Putting all estimates together, we then obtain the desired S norm bounds (cid:13)(cid:13)(cid:13)(cid:0) A n , ( L ) Λ , φ n , ( L ) Λ (cid:1)(cid:13)(cid:13)(cid:13) S ( R × R ) ≤ C ( C ) , where the bound C only depends on the size of C . (cid:3) Interlude: Proofs of Proposition 4.3, Proposition 5.1, Proposition 5.13 and Proposition 6.1.
Proof of Proposition 4.3.
Let E denote the conserved energy of the admissible solution ( A , φ ).Analogously to the proof of Proposition 7.5, we can partition the time interval ( − T , T ) into N = N (cid:0) k ( A , φ ) k S (( − T , T ) × R ) (cid:1) many intervals I such that A | I = A f ree , ( I ) + A nonlin , ( I ) , (cid:3) A f ree , ( I ) = (cid:13)(cid:13)(cid:13) ∇ t , x A f ree , ( I ) (cid:13)(cid:13)(cid:13) L ∞ t L x ( R × R ) . E / , (cid:13)(cid:13)(cid:13) A nonlin , ( I ) (cid:13)(cid:13)(cid:13) ℓ S ( I × R ) ≪ , k φ k S ( I × R ) . C ( E ) , where C ( · ) grows at most polynomially. For each such interval I , say of the form I = [ t , t ] forsome t < t , we let { c k } k ∈ Z be a su ffi ciently flat frequency envelope covering the data ( A , φ )[ t ] attime t . Then we show that the bootstrap assumption (cid:13)(cid:13)(cid:13) P k A (cid:13)(cid:13)(cid:13) S k ( I × R ) + (cid:13)(cid:13)(cid:13) P k φ (cid:13)(cid:13)(cid:13) S k ( I × R ) ≤ Dc k for D = D ( E ) su ffi ciently large, implies the improved bound (cid:13)(cid:13)(cid:13) P k A (cid:13)(cid:13)(cid:13) S k ( I × R ) + (cid:13)(cid:13)(cid:13) P k φ (cid:13)(cid:13)(cid:13) S k ( I × R ) ≤ D c k . We only discuss the equation for φ , because the equation for A is easier. It su ffi ces to consider thecase k =
0. On I × R we may rewrite the equation for φ into the following frequency localizedform (cid:3) pA free , ( I ) (cid:0) P φ (cid:1) = − (cid:2) P , (cid:3) pA free , ( I ) (cid:3) φ − iP (cid:18)X k P > k − C A f ree , ( I ) j P k ∂ j φ (cid:19) − iP (cid:0) A nonlin , ( I ) j ∂ j φ − A ∂ t φ (cid:1) + P (cid:0) i ( ∂ t A ) φ + A α A α φ (cid:1) . (7.38)In order to close the bootstrap argument we now translate the estimates in the proof of Proposition7.5 into the language of frequency envelopes. For example, to bound the high-high interactions inthe term P (cid:18)X k P > k − C A f ree , ( I ) j P k ∂ j φ (cid:19) , we use estimate (131) from [22] to obtain X k = k + O (1) k > O (1) (cid:13)(cid:13)(cid:13) P (cid:0) P k A f ree , ( I ) j P k ∂ j φ (cid:1)(cid:13)(cid:13)(cid:13) N ( I × R ) . X k = k + O (1) k > O (1) − δ k (cid:13)(cid:13)(cid:13) P k A f ree , ( I ) (cid:13)(cid:13)(cid:13) S ( I × R ) (cid:13)(cid:13)(cid:13) P k φ (cid:13)(cid:13)(cid:13) S ( I × R ) . X k = k + O (1) k > O (1) − δ k (cid:13)(cid:13)(cid:13) P k A f ree , ( I ) (cid:13)(cid:13)(cid:13) S ( I × R ) c k . Summing over all su ffi ciently large k ≫ ≤ D c . This allows us to reduce to the case k = k + O (1) = O (1). Here we gain the necessary smallnessby further partitioning the interval I (where the total number of subintervals depends only on thesize of E ), using exactly the same divisibility argument as for the term (7.23) in the proof of Propo-sition 7.5. All other terms on the right hand side of (7.38) can be treated analogously to the aboveargument. (cid:3) Proof of Proposition 5.1.
Here we are in the situation of Step 3 of the proof of Proposition 7.4. Weobtain the bound (cid:13)(cid:13)(cid:13) ( δ A , δφ ) (cid:13)(cid:13)(cid:13) S ([ − T , T ] × R ) . L δ for su ffi ciently small δ by means of a bootstrap argument performed on a finite number of space-time slices, whose number depends on L . We select these space-time slices as in Proposition 7.5.The main di ffi culty arises from the equation for φ . As in Step 3a of the proof of Proposition 7.4, welocalize the equation for φ to frequency 0 on a suitable space-time slice I × R with 0 ∈ I . Then, asin Step 3b there, the main di ffi culty comes from the new low-high interaction term P < ( δ A ) j P ∂ j φ − P < ( δ A ) P ∂ t φ. Using notation from the proof of Proposition 7.4, the worst contribution comes from(7.39) P < δ A f ree , ( I ) j P ∂ j φ, where we recall that δ A f ree , ( I ) is the free evolution of the data δ A [0]. We observe that for δ A f ree , ( I ) the interaction term (7.39) vanishes by assumption on the frequency support of δ A [0] unless K ≤ A , φ )[0] and by Proposition 4.3, we obtain (cid:13)(cid:13)(cid:13) P φ (cid:13)(cid:13)(cid:13) S ([ − T , T ] × R ) . L σ K . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 81
More generally, replacing the frequency 0 by l ∈ Z with l ≥ K , we have (cid:13)(cid:13)(cid:13) P l φ (cid:13)(cid:13)(cid:13) S ([ − T , T ] × R ) . L − σ ( l − K ) . By estimate (7.34), we then find for l ≥ K that (cid:13)(cid:13)(cid:13) P < l δ A f ree , ( I ) j P l ∂ j φ (cid:13)(cid:13)(cid:13) N l ( I × R ) . L δ | l − K | − σ ( l − K ) , where the extra factor | l − K | arises due to the ℓ summation over the frequencies of P < l δ A f ree , ( I ) .But then we get the bound (cid:18)X l ≥ K (cid:13)(cid:13)(cid:13) P < l δ A f ree , ( I ) j P l ∂ j φ (cid:13)(cid:13)(cid:13) N l ( I × R ) (cid:19) . L δ , which gives the required smallness for this term. Then the argument proceeds as for Proposition 7.4. (cid:3) Proof of Proposition 5.13.
We write for large R ≥ R ≥ R (cid:0) ˜ A R , ˜ φ R (cid:1) = (cid:0) ˜ A R + δ A , ˜ φ R + δφ (cid:1) . Then we analyze the equations for ( δ A , δφ ). In fact, the only new feature occurs for the δφ equationand so we explain this here. We obtain the equation (cid:3) ˜ A R + δ A ( δφ ) + (cid:0) (cid:3) ˜ A R + δ A − (cid:3) ˜ A R (cid:1) ˜ φ R = . Here we only retain the key di ffi cult term that cannot be treated via a perturbative argument, us-ing suitable divisibility properties as for example done in great detail in Step 3 of the proof ofProposition 7.4. This term is given by X k ∈ Z P < k ( δ A f ree ) j P k ∂ j ˜ φ R . However, since we localize to a small time interval [ − T , T ] around t =
0, it will be possible to obtaingood N norm bounds. Note that on account of the estimates in Subsection 5.1, we may assume thatlim sup R →∞ (cid:13)(cid:13)(cid:13) ( ˜ A R , ˜ φ R ) (cid:13)(cid:13)(cid:13) S ([ − T , T ] × R ) < ∞ for all R su ffi ciently large, provided that T is su ffi ciently small. We shall assume, as we may that T <
1. Then write X k P < k ( δ A f ree ) j P k ∂ j ˜ φ R = X k P < min {− log R , k } ( δ A f ree ) j P k ∂ j ˜ φ R + X k P [ − log R , k ] ( δ A f ree ) j P k ∂ j ˜ φ R . (7.40)The last term will be estimated by taking advantage of Huygens’ principle as well as our particularchoice of initial data, namely that ˜ φ R [0] is supported on the set (cid:8) | x | ≤ R (cid:9) , while δ A [0] is supportedon (cid:8) | x | ≥ R (cid:9) up to tails that essentially decay exponentially fast. We now estimate both terms on the right hand side of (7.40). For the first term we find (cid:13)(cid:13)(cid:13) X k P < min {− log R , k } ( δ A f ree ) j P k ∂ j ˜ φ R (cid:13)(cid:13)(cid:13) N ([ − T , T ] × R ) . (cid:18)X k (cid:13)(cid:13)(cid:13) P < min {− log R , k } ( δ A f ree ) j P k ∂ j ˜ φ R (cid:13)(cid:13)(cid:13) L t L x ([ − T , T ] × R ) (cid:19) . sup l (cid:13)(cid:13)(cid:13) P < min {− log R , l } δ A f ree (cid:13)(cid:13)(cid:13) L ∞ t L ∞ x (cid:18)X k (cid:13)(cid:13)(cid:13) P k ∇ x ˜ φ R (cid:13)(cid:13)(cid:13) L ∞ t L x ([ − T , T ] × R ) (cid:19) . R − (cid:13)(cid:13)(cid:13) δ A [0] (cid:13)(cid:13)(cid:13) ˙ H x × L x (cid:13)(cid:13)(cid:13) ˜ φ R (cid:13)(cid:13)(cid:13) S ([ − T , T ] × R ) and so this converges to 0 as R → + ∞ . For the second term we have (cid:13)(cid:13)(cid:13) X k P [ − log R , k ] ( δ A f ree ) j P k ∂ j ˜ φ R (cid:13)(cid:13)(cid:13) N ([ − T , T ] × R ) . (cid:18)X k (cid:13)(cid:13)(cid:13) P [ − log R , k ] ( δ A f ree ) j P k ∂ j ˜ φ R (cid:13)(cid:13)(cid:13) L t L x ([ − T , T ] × R ) (cid:19) . (cid:18)X k (cid:13)(cid:13)(cid:13) χ {| x | < R } P [ − log R , k ] ( δ A f ree ) j P k ∂ j ˜ φ R (cid:13)(cid:13)(cid:13) L t L x ([ − T , T ] × R ) (cid:19) + (cid:18)X k (cid:13)(cid:13)(cid:13) χ {| x |≥ R } P [ − log R , k ] ( δ A f ree ) j P k ∂ j ˜ φ R (cid:13)(cid:13)(cid:13) L t L x ([ − T , T ] × R ) (cid:19) . For the last term but one we use the localization properties of δ A f ree to conclude (cid:18)X k (cid:13)(cid:13)(cid:13) χ {| x | < R } P [ − log R , k ] ( δ A f ree ) j P k ∂ j ˜ φ R (cid:13)(cid:13)(cid:13) L t L x ([ − T , T ] × R ) (cid:19) . sup l > − log( R ) (cid:13)(cid:13)(cid:13) χ {| x | < R } P [ − log R , l ] ( δ A f ree ) (cid:13)(cid:13)(cid:13) L ∞ t L ∞ x ([ − T , T ] × R ) (cid:18)X k (cid:13)(cid:13)(cid:13) P k ∇ x ˜ φ R (cid:13)(cid:13)(cid:13) L ∞ t L x ([ − T , T ] × R ) (cid:19) . R − M (cid:13)(cid:13)(cid:13) ˜ φ R (cid:13)(cid:13)(cid:13) S ([ − T , T ] × R ) . R − M , while for the last term, we get (cid:18)X k (cid:13)(cid:13)(cid:13) χ {| x |≥ R } P [ − log R , k ] ( δ A f ree ) j P k ∂ j ˜ φ R (cid:13)(cid:13)(cid:13) L t L x ([ − T , T ] × R ) (cid:19) . (cid:18)X k (cid:13)(cid:13)(cid:13) P [ − log R , k ] ( δ A f ree ) (cid:13)(cid:13)(cid:13) L ∞ t L ∞ x (cid:13)(cid:13)(cid:13) χ | x |≥ R P k ∇ x ˜ φ R (cid:13)(cid:13)(cid:13) L ∞ t L x ([ − T , T ] × R ) (cid:19) . Here we use the localization properties of ˜ φ R to bound the second factor by (cid:13)(cid:13)(cid:13) χ | x |≥ R P k ∇ x ˜ φ R (cid:13)(cid:13)(cid:13) L ∞ t L x ([ − T , T ] × R ) . (cid:16) max { k , } R (cid:17) − M as long as k > − log R as we may assume and also we have the crude bound (cid:13)(cid:13)(cid:13) P [ − log R , k ] ( δ A f ree ) (cid:13)(cid:13)(cid:13) L ∞ t L ∞ x . k , ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 83 whence we finally obtain the bound (cid:18)X k (cid:13)(cid:13)(cid:13) P [ − log R , k ] ( δ A f ree ) (cid:13)(cid:13)(cid:13) L ∞ t L ∞ x (cid:13)(cid:13)(cid:13) χ | x |≥ R P k ∇ x ˜ φ R (cid:13)(cid:13)(cid:13) L ∞ t L x ([ − T , T ] × R ) (cid:19) . R − M . Letting R → + ∞ then again gives the required smallness. (cid:3) Proof of Proposition 6.1.
In view of Lemma 5.4, it su ffi ces to consider the case I = R . We argueby contradiction. Assume that we have(7.41) k ( A , φ ) k S ( R × R ) < ∞ . Then the idea is that using this ingredient as well as a correct perturbative ansatz for the evolu-tions ( A n , φ n ) for n large enough, we can show that the corresponding S norms of ( A n , φ n ) muststay finite, contradicting the assumption. We introduce the perturbative term δ A n for the magneticpotential by A n = A + δ A n and the perturbative term δφ n by means of φ n = χ I φ + χ I ˜ φ A ,δ A freen + δφ n , where I is a very large time interval centered around t = I represents the complement. Thefunction ˜ φ A ,δ A freen solves the wave equation˜ (cid:3) A + δ A freen ( ˜ φ A ,δ A freen ) = , ˜ φ A ,δ A freen [0] = φ [0] , where ˜ (cid:3) A + δ A freen = (cid:3) + i ( A + δ A f reen ) ν ∂ ν and in this context we let δ A f reen be the actual free evolution of the data δ A n [0] (as usual only thespatial components). We let χ I , χ I be a smooth partition of unity subordinate to dilates of theintervals I , I . We note that in this argument one has to in fact replace the energy class solution( A , φ ) by the evolution of a low frequency approximation of the energy class data very close to itand then show that this implies S norm bounds for ( A n , φ n ) uniformly for all su ffi ciently close lowfrequency approximations.To begin with, observe that we can show by a variant of the proof of Lemma 7.9, proved laterindependently, that given any γ > I suitably large (depending on A , φ, γ ), we canarrange that ˜ φ A ,δ A freen = (cid:0) ˜ φ A ,δ A freen (cid:1) + (cid:0) ˜ φ A ,δ A freen (cid:1) with (cid:13)(cid:13)(cid:13)(cid:0) ˜ φ A ,δ A freen (cid:1) (cid:13)(cid:13)(cid:13) S < γ, (cid:13)(cid:13)(cid:13) χ I (cid:0) ˜ φ A ,δ A freen (cid:1) (cid:13)(cid:13)(cid:13) L ∞ t L ∞ x < γ. Now the equation for δφ n becomes the following (cid:3) A + δ A n δφ n = − χ I (cid:3) A + δ A n φ − χ I (cid:3) A + δ A n ˜ φ A ,δ A freen + ( ∂ t χ I ) (cid:0) φ − ˜ φ A ,δ A freen (cid:1) + ∂ t χ I ) (cid:0) ∂ t φ − ∂ t ˜ φ A ,δ A freen (cid:1) + i ( ∂ t χ I )( A + δ A n ) (cid:0) φ − ˜ φ A ,δ A freen (cid:1) . The error term ∂ t ( χ I )( φ − ˜ φ A ,δ A freen ) is potentially problematic, because we cannot place the factor (cid:0) φ − ˜ φ A ,δ A freen (cid:1) into L ∞ t L x . In fact, the latter is only possible provided we have compact spatial support (precisely,in a ball of radius R with 1 ≪ R ≤ | I | ) according to the Huygens principle, because then the extrafactor | I | − stemming from ∂ t ( χ I ) will counterbalance the factor | I | in (cid:13)(cid:13)(cid:13) φ − ˜ φ A ,δ A freen (cid:13)(cid:13)(cid:13) L ∞ t L x ( I × R ) . | I | (cid:13)(cid:13)(cid:13) ∇ x (cid:0) φ − ˜ φ A ,δ A freen (cid:1)(cid:13)(cid:13)(cid:13) L ∞ t L x ( I × R ) . Here it is natural to truncate the data φ [0] in physical space to force this spatial localization later intime via Huygens’ principle, but one needs to ensure that this does not destroy the good S normbounds for ( A , φ ). In fact, since we use the same A data, the argument for Proposition 5.1 appliesto yield a global S norm bound for the new ( A , φ ). We then incorporate the error due to truncatingthe data φ [0] into δφ n (while δ A n [0] remains unchanged!), and hence infer the desired bound (cid:13)(cid:13)(cid:13) ∂ t ( χ I ) (cid:0) φ − ˜ φ A ,δ A freen (cid:1)(cid:13)(cid:13)(cid:13) N . (cid:13)(cid:13)(cid:13) ∇ x (cid:0) φ − ˜ φ A ,δ A freen (cid:1)(cid:13)(cid:13)(cid:13) L ∞ t L x ( I × R ) . This gives the required smallness provided we can make (cid:13)(cid:13)(cid:13) ∇ x (cid:0) φ − ˜ φ A ,δ A freen (cid:1)(cid:13)(cid:13)(cid:13) L ∞ t L x ( I × R ) small. Forthis observe that (we omit the cubic interaction terms) (cid:3) A (cid:0) φ − ˜ φ A ,δ A freen (cid:1) = i ( δ A f reen ) j ∂ j ˜ φ A ,δ A freen + . . . , and further (cid:13)(cid:13)(cid:13) χ I i ( δ A f reen ) j ∂ j ˜ φ A ,δ A freen (cid:13)(cid:13)(cid:13) N ≤ C ( I , φ, A ) (cid:13)(cid:13)(cid:13) δ A n [0] (cid:13)(cid:13)(cid:13) ˙ H x × L x . This implies (cid:13)(cid:13)(cid:13) ∇ t , x (cid:0) φ − ˜ φ A ,δ A freen ) (cid:13)(cid:13)(cid:13) L ∞ t L x ( I × R ) ≤ C ( I , φ, A ) (cid:13)(cid:13)(cid:13) δ A n [0] (cid:13)(cid:13)(cid:13) ˙ H x × L x and so we conclude that (cid:13)(cid:13)(cid:13) ∂ t ( χ I ) (cid:0) ∂ t φ − ∂ t ˜ φ A ,δ A freen (cid:1)(cid:13)(cid:13)(cid:13) N + (cid:13)(cid:13)(cid:13) ∂ t ( χ I ) (cid:0) φ − ˜ φ A ,δ A freen (cid:1)(cid:13)(cid:13)(cid:13) N . C ( I , φ, A ) (cid:13)(cid:13)(cid:13) δ A n [0] (cid:13)(cid:13)(cid:13) ˙ H x × L x . Furthermore, we can write − χ I (cid:3) A + δ A n φ = − χ I (cid:3) A + δ A nonlinn φ − error , where we use the decomposition δ A n = δ A f reen + δ A nonlinn with the first term on the right hand side the free propagation of δ A n [0]. For the error term we get (cid:13)(cid:13)(cid:13) error (cid:13)(cid:13)(cid:13) N ≤ C ( | I | , A ) (cid:13)(cid:13)(cid:13) δ A n [0] (cid:13)(cid:13)(cid:13) ˙ H x × L x . Furthermore, by using a divisibility argument and subdividing time axis into N (cid:0) k ( A , φ ) k S (cid:1) manytime intervals J , using the argument for Proposition 7.4, we can force (for each such J ) (cid:13)(cid:13)(cid:13) χ I (cid:3) A + δ A nonlinn φ (cid:13)(cid:13)(cid:13) N ( J × R ) ≪ k δφ n k S + k δ A n k S . Similarly, we have (cid:13)(cid:13)(cid:13) χ I (cid:3) A + δ A n ˜ φ A ,δ A freen (cid:13)(cid:13)(cid:13) N ( J × R ) ≪ k δφ n k S + k δ A n k S , which then su ffi ces for the bootstrap for δφ n .Next, we consider the equation for δ A n , which is of the schematic form (cid:3) δ A n = φ · ∇ x φ − (cid:0) χ I φ (cid:1) · ∇ x (cid:0) χ I φ (cid:1) − (cid:0) χ I φ (cid:1) · ∇ x (cid:0) χ I ˜ φ A ,δ A freen + δφ n (cid:1) − (cid:0) χ I ˜ φ A ,δ A freen + δφ n (cid:1) · ∇ x (cid:0) χ I φ + χ I ˜ φ A ,δ A freen + δφ n (cid:1) + ( A + δ A n ) (cid:12)(cid:12)(cid:12) χ I φ + χ I ˜ φ A ,δ A freen + δφ n (cid:12)(cid:12)(cid:12) − A | φ | . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 85
Then we make the following observations. The first line on the right hand side satisfies (cid:13)(cid:13)(cid:13) φ · ∇ x φ − (cid:0) χ I φ (cid:1) · ∇ x (cid:0) χ I φ (cid:1)(cid:13)(cid:13)(cid:13) N ≤ ν for any prescribed ν >
0, provided we pick I su ffi ciently large. The reason for this is that this termis supported around the endpoints of I (which is centered around t = (cid:3) A φ =
0, weobtain similarly to the proof of Lemma 7.9 the dispersive decay for φ at large times, which easilygives the desired smallness for the N norm. For the second and third line on the right, we find (cid:13)(cid:13)(cid:13)(cid:0) χ I φ (cid:1) · ∇ x (cid:0) χ I ˜ φ A ,δ A freen + δφ n (cid:1)(cid:13)(cid:13)(cid:13) N ( J × R ) + (cid:13)(cid:13)(cid:13)(cid:0) χ I ˜ φ A ,δ A freen + δφ n (cid:1) · ∇ x (cid:0) χ I φ + χ I ˜ φ A ,δ A freen + δφ n (cid:1)(cid:13)(cid:13)(cid:13) N ( J × R ) . ν + M − k δφ n k S ( J × R ) + C k δφ n k S ( J × R ) , where J is a member of a suitable partition of the time axis into N (cid:0) k ( A , φ ) k S , M (cid:1) many intervalsand C is a universal constant. Here we exploit the uniform dispersive decay of ˜ φ A ,δ A freen . The lastline is handled similarly, (cid:13)(cid:13)(cid:13) ( A + δ A n ) (cid:12)(cid:12)(cid:12) χ I φ + χ I ˜ φ A ,δ A freen + δφ n (cid:12)(cid:12)(cid:12) − A | φ | (cid:13)(cid:13)(cid:13) N ( J × R ) . ν + C M − (cid:0) k δφ n k S ( J × R ) + k δ A n k S ( J × R ) (cid:1) + C (cid:0) k δφ n k S ( J × R ) + k δ A n k S ( J × R ) (cid:1) + k δ A n k S ( J × R ) k δφ n k S ( J × R ) . Combining these bounds, we then finally infer for the interval J containing t = k δ A n k S ( J × R ) . k δ A n [0] k ˙ H x × L x + ν + C M − (cid:0) k δφ n k S ( J × R ) + k δ A n k S ( J × R ) (cid:1) + C (cid:0) k δφ n k S ( J × R ) + k δ A n k S ( J × R ) (cid:1) + k δ A n k S ( J × R ) k δφ n k S ( J × R ) , which su ffi ces to bootstrap the bound for k δ A n k S on J . The bootstrap on the remaining intervalsfollows by induction (and choosing ν and k δ A n [0] k ˙ H x × L x su ffi ciently small depending on M and k ( A , φ ) k S ). Finally, we observe that the S norm bounds on φ , ˜ φ A ,δ A freen , and δφ n are “inherited” bythe expression χ I φ + χ I ˜ φ A ,δ A freen + δφ n on account of the support properties of the functions φ , ˜ φ A ,δ A freen . (cid:3) Selecting concentration profiles and adding the first large frequency atom.
We recall thatwe decomposed the essentially singular sequence of data (cid:8) ( A n , φ n )[0] (cid:9) n ∈ N into frequency atoms A n [0] = Λ X a = A na [0] + A n Λ [0] ,φ n [0] = Λ X a = φ na [0] + φ n Λ [0] , where Λ was chosen such that X a ≥ Λ + lim sup n →∞ E ( A na , φ na ) < ε . Moreover, we remind the reader that the frequency atoms split the errors (cid:0) A n Λ , φ n Λ (cid:1) [0] into Λ + (cid:0) A n j Λ , φ n j Λ (cid:1) [0], 1 ≤ j ≤ Λ +
1, ordered by the size of | ξ | in their Fourier supports. Having established control over the evolution of the data (cid:0) A n Λ , φ n Λ (cid:1) [0] in the preceding subsec-tions, we now add the components (cid:0) A n , φ n (cid:1) [0], i.e. we pass to the initial data(7.42) (cid:0) A n Λ + A n , φ n Λ + φ n (cid:1) [0] . Here we first have to understand the lack of compactness of the “large” added term (cid:0) A n , φ n (cid:1) [0].To this end we carry out a careful profile decomposition in physical space of the added data (cid:0) A n , φ n (cid:1) [0]. To obtain a profile decomposition for the magnetic potential components A n j [0], j = , . . . ,
4, we just use the standard Bahouri-G´erard method [1] to extract the profiles via the freewave evolution. However, for the φ field, we mimic [20] and select the concentration profiles byevolving the data φ n [0] using the following “covariant” wave operator(7.43) e (cid:3) A n : = (cid:3) + i (cid:0) A n Λ ,ν + A n , f ree ν (cid:1) ∂ ν . Here, the functions A n , f ree ν are defined as the solutions to the free wave equation (cid:3) A n , f reej = , A n , f reej [0] = A n j [0]for j = , . . . ,
4, while we simply put A n , f ree ≡ . It follows from standard results that the solution u to e (cid:3) A n u = u [0] ∈ ˙ H x ( R ) × L x ( R ) exists globally in time. Moreover, the parametrix construc-tion from Section 3 together with suitable divisibility arguments yields that this solution satisfiesthe global S norm bound(7.44) k u k S ( R × R ) . E crit k∇ t , x u (0) k L x . At this point we emphasize that both the influence of the evolution of the low frequency mag-netic potential A n Λ and the influence of the free wave evolution of the data A n [0] are built into the“covariant” wave operator e (cid:3) A n . This is di ff erent from the situation for critical wave maps in [20],where only the corresponding low frequency components are built into the “covariant” wave oper-ator there, see Definition 9.18 in [20]. The reason for this is that the interaction term A n ν ∂ ν φ n where both factors are essentially supported at frequency ∼
1, cannot be bounded due to the con-tribution of the free term A n , f ree ν . Thus, the φ field experiences not only an “asymptotic” twistingdue to the contribution of the extremely low frequency components A n Λ (as is the case for criticalwave maps), but also from the frequency ∼ A n , f ree ν . This needs to be reflected by our choiceof concentration profiles.An important fact about the wave operator e (cid:3) A n is that solutions to e (cid:3) A n u = n → ∞ . By rescaling we may assume that λ n = (cid:0) A n , φ n (cid:1) [0] is uniformly concentrated around | ξ | ∼ ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 87
Lemma 7.8.
Assume that the Schwartz data u [0] is essentially supported at frequency | ξ | ∼ with k u [0] k ˙ H x × L x . . Moreover, assume that A n is -oscillatory and that A n Λ satisfies a uniform S norm bound lim sup n →∞ (cid:13)(cid:13)(cid:13) A n Λ (cid:13)(cid:13)(cid:13) S < ∞ as well as sup n (cid:13)(cid:13)(cid:13) A n j [0] (cid:13)(cid:13)(cid:13) ˙ H x × L x < ∞ for j = , . . . , . Let e (cid:3) A n be defined as in (7.43) . Then thesolutions u ( t , x ) of the linear problem (with implicit n dependence suppressed) e (cid:3) A n u = with fixed initial data u [0] satisfy (7.45) lim R → + ∞ lim n →∞ sup t ∈ R + (cid:12)(cid:12)(cid:12) k∇ t , x u ( R + t , · ) k L x − k∇ t , x u ( R , · ) k L x (cid:12)(cid:12)(cid:12) = . The same holds even when replacing + ∞ by −∞ and R + by R − . Furthermore, assume that u k is asequence of solutions to (again suppressing the n dependence) e (cid:3) A n u k = , supported at frequency | ξ | ∼ (in the sense of -oscillatory), and satisfying S norm bounds uni-form in k, while u is as above (with fixed data u [0] ). Then we have (7.46) lim R → + ∞ lim n →∞ sup t ∈ R + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R (cid:16) ∇ t , x u ( t + R , x ) · ∇ t , x u k ( t , x ) − ∇ t , x u ( R , x ) · ∇ t , x u k (0 , x ) (cid:17) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = uniformly in k, and the same holds when replacing + ∞ by −∞ , and R + by R − . In the proof of Lemma 7.8 we shall need the following uniform dispersive type bounds. Thesewill also play a crucial role to control the interactions of the concentration profiles to be discussedbelow. Note that this is an analogue of Proposition 9.20 in [20] and is proved in an analogousfashion.
Lemma 7.9.
Let u [0] ∈ ˙ H x ( R ) × L x ( R ) be fixed initial data and consider the solution u ( t , x ) ofthe linear problem e (cid:3) A n u = with given data u [0] at time t = . Then for any γ > , there exists a decompositionu = u + u such that k u k S < γ and there exists a time t = t (cid:0) u [0] , γ (cid:1) such that for any | t | > t , k u ( t , · ) k L ∞ x < γ. Proof.
We first prove the dispersive type bounds for solutions to the microlocalized magnetic waveequation (cid:3) pA u ≡ (cid:3) u + i X k ∈ Z P ≤ k − C A f reej P k ∂ j u = u [0] = ( f , g ) ∈ ˙ H x ( R ) × L x ( R ). Here, the spatial components A f reej of themagnetic potential are in Coulomb gauge and are solutions to the free wave equation. We recallthat the magnetic wave operator (cid:3) pA was treated in detail in Section 3. The asserted dispersive typebounds for solutions to e (cid:3) A n u = The main di ff erence over the argument for wave maps in [20, Proposition 9.20] is that we needto use a nested double iteration, on account of the fact that our parametrix for (cid:3) pA u = ψ ± ( t , x , ξ ) defined in (3.17) forthe construction of the parametrix for, say, the frequency 0 mode is truncated to low frequencies k ≤ − C σ . This generates the additional error terms2 i X − C σ ≤ k < P k A f reej P ∂ j u . These can only be iterated away by using divisibility, i.e. by restricting to a finite number of suitabletime intervals. In fact, due to the summation over k ∈ [ − C σ , C (and also depends on the energy and σ , of course). Now we formallydenote the (exact) Duhamel propagator for the equation (cid:3) pA u = F by u ( t , · ) = Z t ˜ U ( t − s ) F ( s ) ds . Moreover, we denote by J , J , . . . , J N the partition of the forward time axis [0 , ∞ ) into consecutivetime intervals on which the error terms N lh ( u ) : = i X m ∈ Z X − C σ + m ≤ k < m P k A f reej P m ∂ j u as well as the remaining errors generated by the parametrix ˜ U need to be handled by divisibility. Asobserved before, their number depends linearly on C and implicitly on the energy and σ . We write J i = [ t i , t i + ] for 0 ≤ i ≤ N − t = J N = [ t N , ∞ ). Then, proceeding by exact analogy tothe proof of Proposition 9.20 in [20], we can write for u ( i ) : = u | J i , u ( i ) = ∞ X l = u ( J i , l ) , u ( J i , ( t ) = ˜ S ( t − t i ) u ( i − [ t i ] , u ( J i , l ) ( t ) = − Z tt i ˜ U ( t − s ) N lh ( u ( J i , l − )( s ) ds , where ˜ S is the homogeneous data propagator for (cid:3) pA , while ˜ U is the homogeneous propagator fordata of the special form (0 , g ). Then the inductive nature of the construction is revealed by therelation (see (9.74) in [20]) u ( J i , ( t ) = ˜ S ( t )( f , g ) − i − X k = ∞ X l = Z t k + t k ˜ U ( t − s ) N lh ( u ( J k , l ) )( s ) ds . The new aspect of our setting is that the propagators ˜ U , ˜ S themselves are only obtained as infiniteconvergent sums of further terms, which need to be analyzed. Our strategy is to reduce preciselyto the situation treated in [20], by using the error analysis in [22]. Thus, denote the approximateinhomogeneous Duhamel parametrix by Z t ˜ U ( app ) ( t − s ) F ( s ) ds . Note that due to Proposition 7 in [22], the parametrix ˜ U ( app ) ( t − s ) is given by an integral kernelthat satisfies the same decay estimates as the standard d’Alembertian propagator, independent of ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 89 the precise potential A f ree used (but with implicit constants depending on its energy, of course).Then recall from the proof of Theorem 4 in [22] that we may write Z t ˜ U ( t − s ) F ( s ) ds = ∞ X j = Z t ˜ U ( app ) ( t − s ) F j ( s ) ds , where we have F = F and writing inductively B j : = Z t ˜ U ( app ) ( t − s ) F j ( s ) ds , we have for j ≥ F j = F j + F j + F j + F j with (schematic notation following [22]) P F j = (cid:16) (cid:3) pA < e − i ψ ± < ( t , xD ) − e − i ψ ± < ( t , xD ) (cid:3) (cid:17) P B j − , P F j = (cid:16) e − i ψ ± < ( t , x , D ) e i ψ ± < ( D , y , t ) − (cid:17) P F j − , P F j = (cid:16) e − i ψ ± < ( t , x , D ) | D | − e i ψ ± < ( D , y , t ) − | D | − (cid:17) P ∂ t F j − , P F j = (cid:16) e − i ψ ± < ( t , x , D ) | D | − ∂ t e i ψ ± < ( D , y , t ) − | D | − (cid:17) P F j − . Here the first term, which is treated in Section 10.2 in [22], gains a smallness factor of the form2 − σ C , which of course overwhelms any losses polynomial in C for C ≫
1. However, the re-maining three terms do not gain smallness from C , but rather by divisibility, and so we have to bemore careful to force smallness for them (we cannot make the number of intervals depend on theprescribed smallness threshold γ ). Here we exploit the fact that due to Proposition 6 in [22], thekernels of the operators12 (cid:16) e − i ψ ± < ( t , x , D ) e i ψ ± < ( D , y , t ) − (cid:17) , (cid:16) e − i ψ ± < ( t , x , D ) | D | − e i ψ ± < ( D , y , t ) − | D | − (cid:17) , (cid:16) e − i ψ ± < ( t , x , D ) | D | − ∂ t e i ψ ± < ( D , y , t ) − | D | − (cid:17) are rapidly decaying away from the diagonal x = y . This means that up to small errors (which maybe incorporated into the small energy part of u ), we may think of these operators as local ones, andthen the estimates in the proof of Proposition 9.20 in [20] which rely on the inductive bound (9.81)there, go through for the error terms F rj , r = , ,
4, as long as F rj − , r = , ,
4, satisfy these bounds.This means that the inductive argument in [20] goes through here as well. (cid:3)
We are now in a position to prove the asymptotic energy conservation for solutions to e (cid:3) A n u = Proof of Lemma 7.8.
We consider the natural energy functional E A n ( u )( t ) = Z R X α = (cid:12)(cid:12)(cid:12)(cid:0) ∂ α u + i (cid:0) A n Λ ,α + A n , f ree α (cid:1) u (cid:1) ( t , x ) (cid:12)(cid:12)(cid:12) dx , where it is to be kept in mind that the potential A is in Coulomb gauge. Di ff erentiating this energyfunctional with respect to t and using that e (cid:3) A n u =
0, we infer the following relation E A n ( u )( R + T ) − E A n ( u )( R ) = Re Z R + TR Z R (cid:0) ∂ t A n Λ , (cid:1) u (cid:0) ∂ t + iA n Λ , (cid:1) u dx dt + Re Z R + TR Z R (cid:16) − (cid:0) A n Λ , (cid:1) + X j (cid:0) A n Λ , j + A n , f reej (cid:1) (cid:17) u (cid:0) ∂ t + iA n Λ , (cid:1) u dx dt + X j Re Z R + TR Z R (cid:0) ∂ j + i (cid:0) A n Λ , j + A n , f reej (cid:1)(cid:1) u i (cid:0) ∂ t A n Λ , j + ∂ t A n , f reej − ∂ j A n Λ , (cid:1) u dx dt . (7.47)We now show that uniformly in T ≥
0, the terms on the right hand side converge to zero as n → ∞ and then R → + ∞ .The quartic and quintic terms are all expected to be straightforward and so we focus on the moredi ffi cult cubic interaction terms. Here we note that the cubic interaction terms Z R + TR Z R (cid:0) ∂ t A n Λ , (cid:1) u ∂ t u dx dt and X j Z R + TR Z R ∂ j u i (cid:0) ∂ j A n Λ , (cid:1) u dx dt are also easier to treat due to the inherent quadratic nonlinear struture of the temporal components A n Λ , as solutions to the elliptic compatibility equation of MKG-CG.So we now consider the delicate cubic interaction terms X j Re Z R + TR Z R ∂ j u i (cid:0) ∂ t A n Λ , j (cid:1) u dx dt = − X j Z R + TR Z R Im (cid:0) ∂ j uu (cid:1) (cid:0) ∂ t A n Λ , j (cid:1) dx dt , (7.48) X j Re Z R + TR Z R ∂ j u i (cid:0) ∂ t A n , f reej (cid:1) u dx dt = − X j Z R + TR Z R Im (cid:0) ∂ j uu (cid:1) (cid:0) ∂ t A n , f reej (cid:1) dx dt . (7.49)We begin with the first term (7.48). The Coulomb condition satisfied by ∂ t A n Λ , j allows us to projectthe term Im (cid:0) ∂ j uu (cid:1) onto its divergence-free part, which means that we can replace this by a nullform of the schematic type Im (cid:0) ∂ j uu (cid:1) −→ ∆ − ∂ i N i j (cid:0) u , u (cid:1) . Thus we reduce to bounding uniformly the following schematic integral X j Z R + TR Z R ∆ − ∂ i N i j (cid:0) u , u (cid:1) (cid:0) ∂ t A n Λ , j (cid:1) dx dt . Now we claim the microlocalized bound (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R + TR Z R ∆ − ∂ i N i j (cid:0) P k u , P k u (cid:1) P k (cid:0) ∂ t A n Λ , j (cid:1) dx dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . σ (min { k , k , k }− max { k , k , k } ) (cid:13)(cid:13)(cid:13) P k u (cid:13)(cid:13)(cid:13) S (cid:13)(cid:13)(cid:13) P k u (cid:13)(cid:13)(cid:13) S (cid:13)(cid:13)(cid:13) P k A n Λ (cid:13)(cid:13)(cid:13) S for suitable σ >
0. Since there are at least two comparable frequencies in the above, this is enoughto give the desired result in view of the frequency localizations of u and A n Λ . In order to prove ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 91 this, we localize the above expression further and also omit the localization to the time interval[ R , R + T ], as we may get rid of it via a suitable cuto ff (which is compatible with the S norms), Z R + ∆ − ∂ i N i j (cid:0) P k u , P k u (cid:1) P k (cid:0) ∂ t A n Λ , j (cid:1) dx dt = Z R + ∆ − ∂ i N i j (cid:0) P k u , P k u (cid:1) P k Q > k (cid:0) ∂ t A n Λ , j (cid:1) dx dt + Z R + ∆ − ∂ i N i j (cid:0) P k u , P k u (cid:1) P k Q ≤ k (cid:0) ∂ t A n Λ , j (cid:1) dx dt . Here we only estimate the more di ffi cult second term on the right hand side. We write this term as Z R + ∆ − ∂ i N i j (cid:0) P k u , P k u (cid:1) P k Q ≤ k (cid:0) ∂ t A n Λ , j (cid:1) dx dt = X l ≤ k Z R + ∆ − ∂ i N i j (cid:0) P k u , P k u (cid:1) P k Q l (cid:0) ∂ t A n Λ , j (cid:1) dx dt . By symmetry we may assume k ≤ k . Then we distinguish the following cases. Case 1: k = k + O (1) > k + O (1). Since the Q l transfers to the null form N i j , we save2 k − k + ( l − k ) . Thus, we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R + ∆ − ∂ i N i j (cid:0) P k u , P k u (cid:1) P k Q l (cid:0) ∂ t A n Λ , j (cid:1) dx dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . k − k + ( l − k ) − k (cid:13)(cid:13)(cid:13) P k ∇ x u (cid:13)(cid:13)(cid:13) L t L x (cid:13)(cid:13)(cid:13) P k ∇ x u (cid:13)(cid:13)(cid:13) L t L x (cid:13)(cid:13)(cid:13) P k Q l (cid:0) ∂ t A n Λ , j (cid:1)(cid:13)(cid:13)(cid:13) L t L x , where we observe that the exponent pair (4 ,
3) is Strichartz admissible in four space dimensions.Then we use the improved Sobolev type bound (cid:13)(cid:13)(cid:13) P k Q l (cid:0) ∂ t A n Λ , j (cid:1)(cid:13)(cid:13)(cid:13) L t L x . k γ ( l − k ) (cid:13)(cid:13)(cid:13) P k Q l (cid:0) ∂ t A n Λ , j (cid:1)(cid:13)(cid:13)(cid:13) L t L x for suitable γ > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R + ∆ − ∂ i N i j (cid:0) P k u , P k u (cid:1) P k Q l (cid:0) ∂ t A n Λ , j (cid:1) dx dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . k − k + ( l − k ) − k k k − l k γ ( l − k ) (cid:13)(cid:13)(cid:13) P k u (cid:13)(cid:13)(cid:13) S (cid:13)(cid:13)(cid:13) P k u (cid:13)(cid:13)(cid:13) S (cid:13)(cid:13)(cid:13) P k A n Λ (cid:13)(cid:13)(cid:13) S , which in turn can be bounded by . ( k − k ) γ ( l − k ) (cid:13)(cid:13)(cid:13) P k u (cid:13)(cid:13)(cid:13) S (cid:13)(cid:13)(cid:13) P k u (cid:13)(cid:13)(cid:13) S (cid:13)(cid:13)(cid:13) P k A n Λ (cid:13)(cid:13)(cid:13) S . Summing over l ≤ k yields the desired bound in this case. Case 2: k = k + O (1) > k . We distinguish between the cases l ≤ k and l > k . Case 2a: l ≤ k . Here we estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R + ∆ − ∂ i N i j (cid:0) P k u , P k u (cid:1) P k Q l (cid:0) ∂ t A n Λ , j (cid:1) dx dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . − k ( l − k ) (cid:13)(cid:13)(cid:13) ∇ t , x P k u (cid:13)(cid:13)(cid:13) L t L x (cid:13)(cid:13)(cid:13) ∇ t , x P k u (cid:13)(cid:13)(cid:13) L t L x (cid:13)(cid:13)(cid:13) P k Q l (cid:0) ∂ t A n Λ (cid:1)(cid:13)(cid:13)(cid:13) L t L x . − k k ( l − k ) − l (cid:13)(cid:13)(cid:13) P k u (cid:13)(cid:13)(cid:13) S (cid:13)(cid:13)(cid:13) P k u (cid:13)(cid:13)(cid:13) S (cid:13)(cid:13)(cid:13) P k A n Λ (cid:13)(cid:13)(cid:13) S . To get summability over l one can replace the norm k · k L t L x by k · k L t L + x and then use k · k L t L − x insteadfor the second factor. Case 2b: l > k . Here we simply get the bound (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R + ∆ − ∂ i N i j (cid:0) P k u , P k u (cid:1) P k Q l (cid:0) ∂ t A n Λ , j (cid:1) dx dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . ( k − k ) (cid:13)(cid:13)(cid:13) P k u (cid:13)(cid:13)(cid:13) S (cid:13)(cid:13)(cid:13) P k u (cid:13)(cid:13)(cid:13) S (cid:13)(cid:13)(cid:13) P k A n Λ (cid:13)(cid:13)(cid:13) S , which can then be summed over k > l > k to give the desired bound. This in essence finishes theestimate of the cubic interaction term (7.48).Next, we consider the other delicate cubic interaction term (7.49). Using that ∂ t A n , f reej alsosatisfies the Coulomb condition, we reduce as before to bounding uniformly the expression X j Z R + TR Z R ∆ − ∂ i N i j (cid:0) P k u , P k u (cid:1) P k (cid:0) ∂ t A n , f reej (cid:1) dx dt . Compared with the treatment of the previous cubic term, the issue here is how to deal with theinteractions of u and A n , f ree , which are now both 1-oscillatory. We may assume that all frequencies2 k , , ∼
1, otherwise smallness follows from the treatment of the previous cubic interaction term(7.48). Choosing R > ffi ciently large, we obtain from the dispersive decay from Lemma 7.9and interpolation with the endpoint Strichartz estimate that (cid:13)(cid:13)(cid:13) P k , u (cid:13)(cid:13)(cid:13) L t L + x ([ R , R + T ] × R ) ≪ T ≥ n (recalling that the implicit dependence of u on n is suppressed). Onthe other hand, for the factor P k (cid:0) ∂ t A n , f reej (cid:1) , we can use L t L − x instead.The last statement of the lemma follows similarly, by expressing the inner product in terms ofthe energies of u and u k , and reducing to bounding expressions such as X j Z R + TR Z R ∆ − ∂ i N i j (cid:0) P k u , P k u k (cid:1) P k (cid:0) ∂ t A n , f reej (cid:1) dx dt . (cid:3) We now begin to quantify the lack of compactness for the functions (cid:8) ( A n , φ n )[0] (cid:9) n ∈ N . To clarifythe notation and make it adapted to the ensuing induction procedure, we replace the superscript 1in ( A n , φ n )[0] by a to indicate the frequency level of the large frequency atom, although we areonly considering a = (cid:8) φ na [0] (cid:9) n ∈ N . We evolveeach of these using the flow of the covariant wave operator e (cid:3) A na and extract concentration profiles.The method for this follows along the lines of the modified Bahouri-G´erard profile extraction pro-cedure of Lemma 9.23 in [20]. However, we have to use the asymptotic energy conservation fromLemma 7.8 instead of the stronger asymptotic energy conservation in [20, Lemma 9.19], whichforces us to modify the asymptotic orthogonality relation for the free energies of the profiles. Wefirst introduce the following terminology. Definition 7.10.
Given initial data u [0] ∈ ˙ H x ( R ) × L x ( R ) , we denote byS A na (cid:0) u [0] (cid:1) the solution to the initial value problem e (cid:3) A na u = with data u [0] at time t = . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 93
Following [20], which in turn mimics [1], we introduce the set U A na ( φ na [0]), which consists ofall functions that can be extracted as weak limits in the following fashion U A na ( φ na [0]) = n V ∈ L t , loc H x ∩ C L x : ∃ (cid:8) ( t n , x n ) (cid:9) n ≥ ⊂ R × R s.t. S A na (cid:0) φ na [0] (cid:1) ( t + t n , x + x n ) ⇀ V ( t , x ) o . Here the weak limit is in the sense of L t , loc H x . We emphasize that the sequences (cid:8) ( t n , x n ) (cid:9) n ≥ ⊂ R × R are completely arbitrary. We observe that for a non-trivial profile V ∈ U A na ( φ na [0]) withassociated sequence of space-time translations (cid:8) ( t abn , x abn ) (cid:9) n ≥ , by passing to a further subsequence,we may assume that either S (cid:0) A na [0] (cid:1) ( t + t abn , x + x abn ) ⇀ S (cid:0) A na [0] (cid:1) ( t + t abn , x + x abn ) ⇀ A ab ( t , x ).Here, S (cid:0) · (cid:1) ( t , x ) denotes the free wave propagator and A ab are free waves, see Proposition 7.12 below.Noting that the contribution of A n Λ in the definition of e (cid:3) A na vanishes in the limit, then in the formercase we have (cid:3) V =
0, i.e. V is actually a weak solution to the free wave equation, while in the lattersituation, V solves the linear magnetic wave equation (cid:0) (cid:3) + iA abj ∂ j (cid:1) V = η A na (cid:0) φ na [0] (cid:1) : = sup n E ( V ) : V ∈ U A na (cid:0) φ na [0] (cid:1)o < ∞ , where E refers to the functional E ( V ) = Z R (cid:12)(cid:12)(cid:12) ∇ t , x V (0 , x ) (cid:12)(cid:12)(cid:12) dx . Observe that for temporally unbounded sequences, i.e. | t n | → ∞ , the energy E ( V ) is identi-cal to the “asymptotic free energy” associated with solutions to (cid:0) (cid:3) + iA abj ∂ j (cid:1) V = (cid:8) φ na [0] (cid:9) n ∈ N , which is at the core of the second stage of the modified Bahouri-G´erardprocedure for MKG-CG. Recall that we consider a = Proposition 7.11.
There exists a collection of sequences { ( t abn , x abn ) } n ∈ N ⊂ R × R , b ≥ , as well asa corresponding family of concentration profiles φ ab [0] ∈ ˙ H x ( R ) × L x ( R ) , b ≥ , with the following properties: Introducing the space-time translated gauge potentials ˜ A nab ν ( t , x ) : = A na Λ ,ν ( t + t abn , x + x abn ) + A na , f ree ν ( t + t abn , x + x abn ) , ν = , , . . . , , we have • For any B ≥ , there exists a decomposition (7.50) S A na (cid:0) φ na [0] (cid:1) ( t , x ) = B X b = S ˜ A nab (cid:0) φ ab [0] (cid:1) ( t − t abn , x − x abn ) + φ naB ( t , x ) , where each of the functions ˜ φ nab ( t , x ) : = S ˜ A nab (cid:0) φ ab [0] (cid:1) ( t − t abn , x − x abn ) , φ naB ( t , x ) solves the covariant wave equation e (cid:3) A na u = . Moreover, the error satisfies the crucial asymptotic vanishing condition (7.51) lim B →∞ η A na (cid:0) φ naB [0] (cid:1) = . • The sequences are mutually divergent, by which we mean that for b , b ′ , (7.52) lim n →∞ (cid:0) | t abn − t ab ′ n | + | x abn − x ab ′ n | (cid:1) = ∞ . • There is asymptotic energy partition (7.53) E ( φ na [0]) = B X b = E ( ˜ φ nab [0]) + E ( φ naB [0]) + o (1) , where the meaning of o (1) here is lim sup n →∞ o (1) = . • All profiles φ ab [0] as well as all errors φ naB [0] are -oscillatory. Before we begin with the proof of Proposition 7.11, we introduce the following important dis-tinction between two possible types of profiles. • Temporally unbounded profiles:
Those profiles for whichlim n →∞ | t abn | = ∞ . • Temporally bounded profiles:
Those profiles for whichlim inf n →∞ | t abn | < ∞ . By passing to a subsequence we may then as well assume that for all n ∈ N , t abn = . For two distinct such profiles corresponding to b , b ′ , we must havelim n →∞ | x abn − x ab ′ n | = ∞ . Proof of Proposition 7.11.
There is nothing to do if η A na (cid:0) φ na [0] (cid:1) = . Let us therefore assume that this quantity is strictly greater than 0. Then we pick a profile φ a ∈ L t , loc H x ∩ C L x and an associated sequence (cid:8) ( t a n , x a n ) (cid:9) n ∈ N ⊂ R × R such that(7.54) S A na (cid:0) φ na [0] (cid:1) ( t + t a n , x + x a n ) ⇀ φ a ( t , x )with E ( φ a ) ≥ η A na (cid:0) φ na [0] (cid:1) . Then we have S A na (cid:0) φ na [0] (cid:1) ( t + t a n , x + x a n ) − S A na (cid:16) S ˜ A na (cid:0) φ a [0] (cid:1) (0 − t a n , · − x a n ) (cid:17) ( t + t a n , x + x a n ) = S A na (cid:0) φ na [0] (cid:1) ( t + t a n , x + x a n ) − S ˜ A na (cid:0) φ a [0] (cid:1) ( t , x ) ⇀ n → ∞ by the construction. Furthermore, it holds that E (cid:0) φ na [0] (cid:1) = E (cid:16) ˜ φ na [0] (cid:17) + E (cid:16) φ na [0] − ˜ φ na [0] (cid:17) + Z R ∇ t , x (cid:16) S ˜ A na (cid:0) φ a [0] (cid:1) (0 − t a n , x − x a n ) (cid:17) ·· ∇ t , x (cid:16) φ na (0 , x ) − S ˜ A na (cid:0) φ a [0] (cid:1) (0 − t a n , x − x a n ) (cid:17) dx , (7.55) ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 95 where in the last term we ignored that the φ field is complex-valued. If φ a [0] is a temporallyunbounded profile, we may without loss of generality assume that t a n → + ∞ . In view of (7.46)from Lemma 7.8 the last term on the right hand side of (7.55) can be arbitrarily well approximatedby 2 Z Z R ∇ t , x S A na (cid:16) S ˜ A na (cid:0) φ a [0] (cid:1) (0 − t a n , · − x a n ) (cid:17) ( t − R + t a n , x + x a n ) ·· ∇ t , x S A na (cid:16) φ na [0] − S ˜ A na (cid:0) φ a [0] (cid:1) (0 − t a n , · − x a n ) (cid:17) ( t − R + t a n , x + x a n ) dx dt (7.56)as n → ∞ by choosing R > ffi ciently large. Then we observe that the first factor in the integrandin (7.56) satisfies ∇ t , x S A na (cid:16) S ˜ A na (cid:0) φ a [0] (cid:1) (0 − t a n , · − x a n ) (cid:17) ( t − R + t a n , x + x a n ) = ∇ t , x φ a ( t − R , x ) + o L x (1)as n → ∞ , while by construction S A na (cid:16) φ na [0] − S ˜ A na (cid:0) φ a [0] (cid:1) (0 − t a n , · − x a n ) (cid:17) ( · + t a n , · + x a n ) ⇀ L t , loc H x as n → ∞ . Thus, we conclude that E (cid:0) φ na [0] (cid:1) = E (cid:16) ˜ φ na [0] (cid:17) + E (cid:16) φ na [0] − ˜ φ na [0] (cid:17) + o (1)as n → ∞ . If instead φ a [0] is a temporally bounded profile, we may and shall have t a n = n ∈ N . Then the last term on the right hand side of (7.55) is given by2 Z R ∇ t , x φ a (0 , x − x a n ) · ∇ t , x (cid:0) φ na (0 , x ) − φ a (0 , x − x a n ) (cid:1) dx , which vanishes as n → ∞ by the weak convergence (7.54) and therefore yields the desired asymp-totic energy partition (7.53).Now we repeat this procedure, but replace φ na [0] by φ na [0] − ˜ φ na [0] . Thus, if η A na (cid:0) φ na [0] − ˜ φ na [0] (cid:1) >
0, we select a sequence (cid:8) ( t a n , x a n ) (cid:9) n ∈ N and a concentration profile φ a ( t , x ) such that E ( φ a ) ≥ η A na (cid:0) φ na [0] − ˜ φ na [0] (cid:1) and S A na (cid:0) φ na [0] − ˜ φ na [0] (cid:1) ( t + t a n , x + x a n ) ⇀ φ a ( t , x ) . We observe that we must necessarily havelim n →∞ (cid:0) | t a n − t a n | + | x a n − x a n | (cid:1) = ∞ . Iterating this process yields the decomposition (7.50) together with (7.52) and (7.53).Finally, we turn to proving the crucial asymptotic vanishing conditionlim B →∞ η A na (cid:0) φ naB [0] (cid:1) = . Here we observe that the fixed profiles φ ab [0] satisfy φ ab (0 , x ) = S A na (cid:0) ˜ φ nab [0] (cid:1) (0 + t abn , x + x abn ) . Then the global S norm bounds (7.44) for solutions to the covariant wave equation e (cid:3) A na u = k∇ t , x φ ab (0 , · ) k L x . (cid:13)(cid:13)(cid:13) S A na (cid:0) ˜ φ nab [0] (cid:1)(cid:13)(cid:13)(cid:13) S . E crit k∇ t , x ˜ φ nab (0 , · ) k L x . E crit E ( ˜ φ nab [0]) , where the implied constant is independent of n . From the asymptotic energy partition (7.53) weconclude that for any B ≥
1, by passing to a subsequence in n , if necessary, we have B X b = lim sup n →∞ E ( ˜ φ nab [0]) ≤ lim sup n →∞ E ( φ na [0]) . E crit . Thus, we have that uniformly in B , B X b = E ( φ ab [0]) . E crit . By construction, the error η A na ( φ naB ) must therefore vanish as B → ∞ . This finishes the proof ofProposition 7.11. (cid:3) We emphasize that in the preceding linear profile decomposition for the φ na fields, the asymp-totic energy partition (7.53) does not yield a sharp energy bound for the actual profiles φ ab [0] oftemporally unbounded character, which is in contrast to the standard Bahouri-G´erard profile decom-position [1] and the modified Bahouri-G´erard profile decomposition in the context of critical wavemaps [20, Lemma 9.23]. Fortunately, this will not doom the construction of the nonlinear concen-tration profiles, because there is a kind of “asymptotic orthogonality statement”, see Lemma 7.13,in particular (7.59). This will allow us to circumvent the problem.Having selected the linear concentration profiles for the φ na fields, it remains to pick correspond-ing profiles for the magnetic potential components A naj for j = , . . . ,
4. In fact, for the latter, wesimply use the standard Bahouri-G´erard method [1] to extract the profiles via the free wave evolu-tion. By passing to suitable subsequences, one obtains an intertwined linear profile decompositionfor ( A na , φ na )[0]. Thus, the same sequences of space-time shifts { ( t abn , x abn ) } n ≥ , b ≥ , are being usedfor the linear concentration profiles for A na [0] and for φ na [0]. This will be crucial later on whenwe construct the associated nonlinear profiles, as the truly nonlinear behavior of both ( A , φ ) willbe exhibited in space-time boxes centered around the points ( t abn , x abn ), see Step 1 in the proof ofTheorem 7.14. We quote Proposition 7.12.
There exists a collection of sequences { ( t abn , x abn ) } n ∈ N ⊂ R × R , b ≥ , as well asa corresponding family of concentration profilesA abj [0] ∈ ˙ H x ( R ) × L x ( R ) , b ≥ for j = , . . . , with the following properties: • For any B ≥ , we have a decompositionS (cid:0) A naj [0] (cid:1) ( t , x ) = B X b = S (cid:0) A abj [0] (cid:1) ( t − t abn , x − x abn ) + A naBj ( t , x ) , where S ( · )( t , x ) denotes the free wave propagator. Then each of the functionsS (cid:0) A abj [0] (cid:1) ( t − t abn , x − x abn ) , A naBj ( t , x ) solves the linear wave equation (cid:3) u = . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 97
Moreover, the error satisfies the crucial asymptotic vanishing condition (7.57) lim B →∞ η (cid:0) A naBj [0] (cid:1) = . • The sequences are mutually divergent, by which we mean that for b , b ′ , lim n →∞ (cid:0) | t abn − t ab ′ n | + | x abn − x ab ′ n | (cid:1) = ∞ . • There is asymptotic energy partitionE ( A naj [0]) = B X b = E ( A abj [0]) + E ( A naBj [0]) + o (1) , where the meaning of o (1) is lim sup n →∞ o (1) = . • All profiles A abj [0] as well as all errors A naBj [0] are -oscillatory. Moreover, they all satisfy theCoulomb condition. In the preceding propositions on the linear profile decompositions for the φ na fields and forthe spatial components A naj of the connection form, we established an asymptotic orthogonalityof the profiles with respect to the standard free energy functional. However, for our inductionon energy procedure, we have to use the energy functional of the Maxwell-Klein-Gordon system,which involves nonlinear interactions between the φ field and the connection form A . In the nextproposition we carefully analyze the asymptotic orthogonality relations of the linear profiles withrespect to this proper energy functional. Lemma 7.13.
Given any δ > , there exists B = B ( δ ) such that (7.58) lim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ( A na , φ na )(0) − B X b = E ( ˜ A nab , ˜ φ nab )(0) − E ( A naB , φ naB )(0) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < δ , where E refers to the energy functional of the Maxwell-Klein-Gordon system. Here we denote ˜ φ nab : = S ˜ A nab (cid:0) φ ab [0] (cid:1) (0 − t abn , x − x abn ) , ˜ A nabj : = S (cid:0) A abj [0] (cid:1) (0 − t abn , x − x abn ) , j = , . . . , , and the temporal components ˜ A nab (0) are determined in terms of ˜ φ nab [0] via the elliptic compati-bility equation, and similarly for A naB (0) . In particular, if there are at least two non-zero concen-tration profiles ( A ab , φ ab )[0] (corresponding to two distinct values of b), then there exists δ > suchthat for all b, lim sup n →∞ E ( ˜ A nab , ˜ φ nab )(0) < E crit − δ. Moreover, for a temporally unbounded profile ( ˜ A nab , ˜ φ nab ) with, say, t abn → + ∞ as n → ∞ , we have (7.59) E ( ˜ A nab , ˜ φ nab )(0) = E ( ˜ A nab , ˜ φ nab )( t abn − R b ) + κ ab ( n , R b ) , where lim R b → + ∞ lim sup n →∞ κ ab ( n , R b ) = . The B also depends on the sequence of linear concentration profiles, but we omit this dependency here. Proof.
We check the various interaction terms and show that they become small when choosing B as well as n su ffi ciently large. (1) Two temporally bounded profiles. This is straightforward since lim n →∞ | x abn − x ab ′ n | = ∞ . In fact,we immediately infer that schematicallylim n →∞ X temporally bounded profiles , b , b ′ (cid:12)(cid:12)(cid:12)(cid:12) Z R Re (cid:0) ( ∂ α ˜ φ nab + i ˜ A nab α ˜ φ nab ) · ( ∂ α ˜ φ nab ′ + i ˜ A nab ′ α ˜ φ nab ′ ) (cid:1) dx (cid:12)(cid:12)(cid:12)(cid:12) + lim n →∞ X temporally bounded profiles , b , b ′ X j = (cid:12)(cid:12)(cid:12)(cid:12) Z R ∇ t , x ˜ A nabj · ∇ t , x ˜ A nab ′ j dx (cid:12)(cid:12)(cid:12)(cid:12) + lim n →∞ X temporally bounded profiles , b , b ′ X j = (cid:12)(cid:12)(cid:12)(cid:12) Z R ∇ x ˜ A nab · ∇ x ˜ A nab ′ dx (cid:12)(cid:12)(cid:12)(cid:12) = . (2) One temporally bounded and one temporally unbounded profile. Here we exploit that the ampli-tude of the temporally unbounded profile vanishes asymptotically (at time t =
0) as n → ∞ , whilethe temporally bounded profile has bounded support. We conclude that schematicallylim n →∞ X b temporally bounded b ′ temporally unbounded (cid:12)(cid:12)(cid:12)(cid:12) Z R Re (cid:0) ( ∂ α ˜ φ nab + i ˜ A nab α ˜ φ nab ) · ( ∂ α ˜ φ nab ′ + i ˜ A nab ′ α ˜ φ nab ′ ) (cid:1) dx (cid:12)(cid:12)(cid:12)(cid:12) + lim n →∞ X b temporally bounded b ′ temporally unbounded 4 X j = (cid:12)(cid:12)(cid:12)(cid:12) Z R ∇ t , x ˜ A nabj · ∇ t , x ˜ A nab ′ j dx (cid:12)(cid:12)(cid:12)(cid:12) + lim n →∞ X b temporally bounded b ′ temporally unbounded 4 X j = (cid:12)(cid:12)(cid:12)(cid:12) Z R ∇ x ˜ A nab · ∇ x ˜ A nab ′ dx (cid:12)(cid:12)(cid:12)(cid:12) = . (3) Two temporally unbounded profiles. Here we exploit the asymptotic energy conservation andthat the functions φ ab [0] , S ˜ A nab ′ (cid:0) φ ab ′ [0] (cid:1) ( t abn − t ab ′ n , x − x ab ′ n )are asymptotically orthogonal. Similarly, we argue for the interaction terms between the compo-nents of the profiles ˜ A nab and ˜ A nab ′ . (4) Weakly small error φ naB and profiles. This is handled like the interaction of a temporallybounded and a temporally unbounded profile. One uses the fact that we get φ naB = φ naB + φ naB , where we have the bounds (cid:13)(cid:13)(cid:13) φ naB (cid:13)(cid:13)(cid:13) L ∞ t L ∞ x < δ , (cid:13)(cid:13)(cid:13) ∇ t , x φ naB (cid:13)(cid:13)(cid:13) L ∞ t L x < δ , provided B is su ffi ciently large. Of course, choosing B large means that more and more inter-actions have to be controlled, and we can no longer simply use the choice of extremely large n to ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 99 “asymptotically kill” all such interactions as in the preceding cases. Thus, one has to argue carefullyas follows: Given δ >
0, we pick ˜ B su ffi ciently large such that for any B ≥ ˜ B , we havelim sup n →∞ B X b = ˜ B (cid:16) E ( ˜ φ nab ) + E ( ˜ A nab ) (cid:17) ≪ δ , where E indicates the standard free energy. Then, passing to the interaction terms in the Maxwell-Klein-Gordon energy functional corresponding to φ naB and A naB with the sum B X b = ˜ B ˜ φ nab , B X b = ˜ B ˜ A nab leads to terms bounded by ≪ δ for any B ≥ ˜ B , provided n is chosen su ffi ciently large (dependingon B ). But then picking B large enough, we can also ensure that the sum of all the interactionsin E ( A , φ ) generated by the profiles ˜ φ nab , ˜ A nab , 1 ≤ b ≤ ˜ B are small, since B ≥ ˜ B can be chosenindependently.The last assertion (7.59) is again a consequence of the asymptotic energy conservation fromLemma 7.8 and the asymptotic vanishing of the amplitude of a temporally unbounded profile at t = n → ∞ . (cid:3) We now begin with the construction of the nonlinear concentration profiles. In what follows, weassume that the linear concentration profiles (cid:0) A ab , φ ab (cid:1) [0], b ≥
1, have been chosen, as well as theparameter sequences (cid:8) ( t abn , x abn ) (cid:9) n ≥ . We recall that when the profile is temporally bounded, i.e.lim sup n →∞ | t abn | < ∞ , we may and shall have t abn = φ nab ( t , x ) : = S ˜ A nab (cid:0) φ ab [0] (cid:1) ( t − t abn , x − x abn ) , ˜ A nabj ( t , x ) : = S (cid:0) A abj [0] (cid:1) ( t − t abn , x − x abn ) , j = , . . . , . Thus, if the profile is temporally bounded, it holds that˜ A nab [0] = A ab [0] , ˜ φ nab [0] = φ ab [0] . We can now state the key result of this subsection.
Theorem 7.14.
Let a = . Assume that there exist at least two non-zero profiles (cid:0) A ab , φ ab (cid:1) [0] , orall such profiles are zero, or else there exists only one such profile but with lim inf n →∞ E ( ˜ A nab , ˜ φ nab )(0) < E crit . Then the initial data (cid:0) A n Λ + A n , φ n Λ + φ n (cid:1) [0] can be evolved globally in time, resulting in a solutionwith finite S norm bounds uniformly for all su ffi ciently large n.Proof. We proceed in several steps.
Step 1:
Construction of the nonlinear concentration profiles.
We distinguish between temporallybounded and unbounded ( ˜ A nab , ˜ φ nab ). In what follows we shall use the notation A na , low : = A n Λ , φ na , low : = φ n Λ .
00 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION
Temporally bounded case:
Here we have ( ˜ A nab [0] , ˜ φ nab [0]) = ( A ab [0] , φ ab [0]) with A ab as usual inthe Coulomb gauge. Then we define the nonlinear concentration profile (cid:0) A nab , Φ nab (cid:1) as follows. Pick a large time T b > − T b , T b ] × R , we definethe profiles to be the solutions ( A ab , Φ ab ) to the MKG-CG system with data ( A ab , φ ab )[0] at time t =
0, which exist globally in time by Lemma 7.13 and the assumption of the theorem with a globalfinite S norm bound (cid:13)(cid:13)(cid:13)(cid:0) A ab , Φ ab (cid:1)(cid:13)(cid:13)(cid:13) S < ∞ . Here the profiles do not depend on n , but we include this superscript since the profiles on the restof space-time will be n -dependent. On the complement [ − T b , T b ] c × R , we define the profiles asfollows. On [ T b , ∞ ) × R , we let (cid:3) A nab = A ab [ T b ] given by the profile constructed on [ − T b , T b ] × R , and we proceed analogouslyon ( −∞ , − T b ] × R . As for the Φ -field, we postulate on [ − T b , T b ] c × R the linear equation (cid:3) A na , low + P Bb ′ = A nab ′ + A naB Φ nab = T b , respectively − T b , by the profile on [ − T b , T b ] × R . Note that in orderfor this to make sense, we also need to know the definition of the temporally unbounded A nab ′ ,which is, of course, accomplished below without knowing the temporally bounded Φ nab to avoidcircularity. Temporally unbounded case:
Assume, for example, that lim n →∞ t abn = + ∞ . Using Lemma 7.13 andthe assumption of the theorem, we can pick R b > ffi ciently large such that˜ φ nab ( t abn − R b , · ) = S ˜ A nab ( φ ab [0])( − R b , · − x abn )satisfies E (cid:16) S ( A ab [0])( − R b , · − x abn ) , S ˜ A nab ( φ ab [0])( − R b , · − x abn ) (cid:17) < E crit . Then we use the data (cid:16) S ( A ab [0])[ − R b ]( · − x abn ) , S ˜ A nab ( φ ab [0])[ − R b ]( · − x abn ) (cid:17) at time t = t abn − R b , and evolve them forward in time using the MKG-CG system up to time t abn + R b ,say, resulting in the nonlinear profiles (cid:0) A nab , Φ nab (cid:1) on [ t abn − R b , t abn + R b ] × R . Observe that this construction does not require knowledge of the otherprofiles (cid:0) A nab ′ , Φ nab ′ (cid:1) . Finally, on the complement [ t abn − R b , t abn + R b ] c × R , we evolve A nab viathe free equation (cid:3) A nab =
0, and Φ nab via the linear evolution (cid:3) A na , low + P Bb ′ = A nab ′ + A naB Φ nab = , with data given at time t abn − R b , respectively t abn + R b , by the profiles constructed on [ t abn − R b , t abn + R b ] × R . Step 2:
Making an ansatz for the evolution ( A n , Φ n ) of the full data (cid:0) A n Λ + A n , φ n Λ + φ n (cid:1) [0] . Wenow assemble the pieces that we have constructed. We shall write(7.60) A n : = A na , low + B X b = A nab + A naB + δ nA , ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 101 where A naB is actually simply given by A naB from Proposition 7.12. We immediately observe thecrucial fact that δ nA [0] = , i.e. the choice of profiles matches the data. We proceed analogously for Φ n , writing(7.61) Φ n : = φ na , low + B X b = Φ nab + Φ naB + δ n Φ , where Φ naB is actually simply given by φ naB from Proposition 7.11. We finally observe that bytruncating the frequency support of the data of the Φ nab to a set {| ξ | ≤ K } for some very large K andincorporating the error into δ n Φ , we may assume that the Φ nab have frequency support in | ξ | ≤ K upto (slowly) exponentially decaying tails. This will be of use later on when controlling the errors. Step 3:
Showing accuracy of the ansatz.
Here we finally prove the following key proposition.
Proposition 7.15.
Assuming the conditions of Theorem 7.14 and given any δ > , there exists Bsu ffi ciently large (depending on the bounds on ( A na , low , φ na , low ) , the actual concentration profilesand on δ ) such that for all su ffi ciently large n, (cid:13)(cid:13)(cid:13) δ n Φ (cid:13)(cid:13)(cid:13) S + (cid:13)(cid:13)(cid:13) δ nA (cid:13)(cid:13)(cid:13) ℓ S < δ . In light of the immediately verified facts thatlim sup n →∞ B X b = (cid:13)(cid:13)(cid:13) Φ nab (cid:13)(cid:13)(cid:13) S + lim sup n →∞ B X b = (cid:13)(cid:13)(cid:13) A nab (cid:13)(cid:13)(cid:13) S < ∞ and lim sup n →∞ (cid:13)(cid:13)(cid:13) Φ naB (cid:13)(cid:13)(cid:13) S + lim sup n →∞ (cid:13)(cid:13)(cid:13) A naB (cid:13)(cid:13)(cid:13) S < ∞ , this proposition then implies Theorem 7.14. (cid:3) Proof of Proposition 7.15.
For the most part, this consists in checking that the (very large numberof) interaction terms sum up to something negligible upon correct choice of B and n . We start withthe equation for δ n Φ . To begin with, we note that δ n Φ [0] is not necessarily 0, since the asymptoticevolution of the profiles Φ nab given by (cid:3) A na , low + P Bb ′ = A nab ′ + A naB Φ nab = ff erent than the one used to extract the concentration profiles, i.e. e (cid:3) A na u =
0. But we alsoobserve that each profile A nab ′ di ff ers from the corresponding linear component in Proposition 7.12given by S ( A ab ′ [0])( t − t ab ′ n , x − x ab ′ n )by a possibly large term, which however lives in a better space (cid:13)(cid:13)(cid:13) A nab ′ ( t , x ) − S ( A ab ′ [0])( t − t ab ′ n , x − x ab ′ n ) (cid:13)(cid:13)(cid:13) ℓ S < ∞ . Denote this di ff erence by B nab ′ ( t , x ). Then it su ffi ces to show Lemma 7.16.
For any temporally unbounded profile Φ nab we have lim n →∞ X ≤ b ′ ≤ B , b ′ , b (cid:13)(cid:13)(cid:13) i B nab ′ ν ∂ ν Φ nab (cid:13)(cid:13)(cid:13) N = .
02 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION
Proof.
We proceed as in the proof of Proposition 7.5, expressing the di ff erence B nab ′ in the schematicform X k , j (cid:3) − P k Q j P (cid:0) φ · ∇ x φ + A | φ | ) , or else as a free wave satisfying a Besov ℓ -bound for the data instead of the weaker energy bound.Using the multilinear estimates from [22] and that all factors as well as Φ nab are 1-oscillatory, wereduce to a diagonal situation, where the frequency of all factors as well as the output modulationare essentially restricted to ∼
1, and have generic position, i.e. the Fourier supports do not haveangular alignment. Then, using that the profiles A nab ′ disperse away from t ab ′ n uniformly in n byLemma 7.9, we easily infer the claim.To be more precise, we first consider the case when A nj , j = , , ,
4, are free waves, which are1-oscillatory, obey the Coulomb condition, and satisfy (cid:13)(cid:13)(cid:13) A nj (cid:13)(cid:13)(cid:13) ℓ S < ∞ . Moreover, assume that Φ n is 1-oscillatory and satisfiessup n (cid:13)(cid:13)(cid:13) Φ n (cid:13)(cid:13)(cid:13) S < ∞ and in view of the dispersive bounds from Lemma 7.9 alsolim n →∞ (cid:13)(cid:13)(cid:13) Φ n (cid:13)(cid:13)(cid:13) L ∞ t L ∞ x = . We now prove that lim n →∞ (cid:13)(cid:13)(cid:13) iA nj ∂ j Φ n (cid:13)(cid:13)(cid:13) N = . By the 1-oscillatory character of the inputs and the ℓ -Besov bound for A n , one may restrict tofrequencies ∼ ∼ (cid:0) , (cid:1) for the first factor, andan interpolate of ( ∞ , ∞ ) with that same space for the second factor to place the output into L t L x .Next, consider the case where A nj is of the schematic form X k , j (cid:3) − P k Q j P i (cid:0) φ · ∇ φ + A | φ | ) . We only consider the most di ffi cult case, where the space-time frequency localizations have beenimplemented and the null form structure revealed as in [22, Theorem 12.1]. For example, consideran expression (cid:3) − P k Q j (cid:0) Q ≤ j − C P k φ n ∂ α Q ≤ j − C P k φ n (cid:1) ∂ α Q ≤ j − C P k Φ n , where the k j indicate frequency localizations, all inputs are 1-oscillatory, and satisfy uniform S norm bounds, and Φ n satisfies the same vanishing relation as above. Also, from [22] we havethe alignments k = k + O (1), k ≥ k + O (1), j ≤ k + O (1). One may then in fact assume j = k + O (1), since else one gets smallness, and the 1-oscillatory character allows us to assume k , , = k + O (1) = O (1). Then one places the output into L t L x by using the Strichartz exponents (cid:0) , (cid:1) for the first two factors, and an interpolate of (cid:0) − , + (cid:1) with (cid:0) ∞ , ∞ (cid:1) for the last factor. Theremaining null forms (see (62) and (63) in [22]) are handled similarly. (cid:3) From the preceding lemma, we infer that we can force (cid:13)(cid:13)(cid:13) δ n Φ [0] (cid:13)(cid:13)(cid:13) ˙ H x × L x ≪ δ , ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 103 provided we pick n su ffi ciently large. The equation for δ n Φ is given by(7.62) (cid:3) A na , low + P Bb ′ = A nab ′ + A naB + δ nA (cid:16) φ na , low + B X b = Φ nab + Φ naB + δ n Φ (cid:17) = . We rewrite this in the following form(7.63) (cid:3) A na , low + P Bb ′ = A nab ′ + A naB + δ nA δ n Φ = − I − II − III , where we put I : = (cid:3) A na , low + P Bb ′ = A nab ′ + A naB + δ nA (cid:16) φ na , low (cid:17) , II : = (cid:3) A na , low + P Bb ′ = A nab ′ + A naB + δ nA (cid:16) B X b = Φ nab (cid:17) , III : = (cid:3) A na , low + P Bb ′ = A nab ′ + A naB + δ nA (cid:16) Φ naB (cid:17) . Now the idea is to show smallness of all these terms (in the N norm sense) provided B and then n are chosen su ffi ciently large. Of course, one needs to be careful with the fact that increasing B alsoleads to more and more terms in the sums B X b ′ = A nab ′ , B X b = Φ nab . To deal with this, we use
Lemma 7.17.
Given δ > , there is a B > such that for all B ≥ B and all su ffi ciently large n(depending on B), it holds that (cid:13)(cid:13)(cid:13)(cid:13) B X b = B A nab (cid:13)(cid:13)(cid:13)(cid:13) S < δ , (cid:13)(cid:13)(cid:13)(cid:13) B X b = B Φ nab (cid:13)(cid:13)(cid:13)(cid:13) S < δ . Proof.
By construction we have (cid:3) A nab = − χ I nb P Im (cid:0) Φ nab D x Φ nab (cid:1) , where I nb = [ − T b , T b ] for temporally bounded profiles and I nb = [ t abn − R b , t abn + R b ] for temporallyunbounded ones. By picking B su ffi ciently large, so that E ( A nab , Φ nab ) ≪ , b ≥ B , we get B X b = B (cid:13)(cid:13)(cid:13)(cid:13) χ I nb P Im (cid:0) Φ nab D Φ nab (cid:1)(cid:13)(cid:13)(cid:13)(cid:13) N . B X b = B E ( ˜ A nab , ˜ φ nab ) , where we recall the notation˜ φ nab = S ˜ A nab (cid:0) φ ab [0] (cid:1) (0 − t abn , x − x abn ) , ˜ A nab = S (cid:0) A abj [0] (cid:1) (0 − t abn , x − x abn ) , j = , . . . , . But then since lim sup n →∞ (cid:13)(cid:13)(cid:13)(cid:13) B X b = B A nab [0] (cid:13)(cid:13)(cid:13)(cid:13) ˙ H x × L x < δ , lim sup n →∞ B X b = B E ( ˜ A nab , ˜ φ nab ) < δ ,
04 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION upon choosing B large enough, the first bound of the lemma follows. To get the second bound, oneuses that for B and n large enough, as well as making some small additional assumption on the data φ ab [0] (see Remark 7.18 below),(7.64) lim n →∞ (cid:13)(cid:13)(cid:13)(cid:13) (cid:3) A na , low + P Bb ′ = A nab ′ + A naB (cid:18) B X b = B χ I nb Φ nab + (1 − χ I nb ) ˜ Φ nab (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) N ≪ δ , where now χ I nb are suitable smooth time cuto ff s and Φ nab is as in Step 1 with I nb = [ − T b , T b ] or I nb = [ t abn + R b , t abn − R b ], while ˜ Φ nab is as in Step 1 but on the complement ( I nb ) c . Againlim sup n →∞ (cid:13)(cid:13)(cid:13)(cid:13) B X b = B Φ nab [0] (cid:13)(cid:13)(cid:13)(cid:13) ˙ H x × L x < δ , provided B is chosen su ffi ciently large. We then infer (cid:13)(cid:13)(cid:13)(cid:13) B X b = B Φ nab (cid:13)(cid:13)(cid:13)(cid:13) S ≪ δ , provided δ is su ffi ciently small. Note that there are small error terms due to the cuto ff , whichhowever are harmless and can be made arbitrarily small by picking the cuto ff suitably, see [20]. Infact, we make the Remark 7.18.
To ensure smallness of the errors generated by the cuto ff s χ I nb and − χ I nb , it su ffi cesto localize each φ ab [0] in physical space to a ball of radius | I nb | , and each A ab [0] to a ball ofradius | I nb | , say. The errors committed thereby may be included in Φ naB , respectively A naB . Observe that for the term (cid:3) A na , low + P Bb ′ = A nab ′ + A naB P Bb = B χ I nb Φ nab , one generates errors of the schematicform χ ′′ I nb Φ nab , χ ′ I nb ( A na , low + A nab ′ + A naB ) Φ nab , χ I nb (cid:0) (cid:3) A na , low + P Bb ′ = A nab ′ + A naB − (cid:3) A nab (cid:1) Φ nab . Then by using the crude bound (cid:13)(cid:13)(cid:13) χ I nb A nab ′ ∇ t , x Φ nab (cid:13)(cid:13)(cid:13) L t L x . | I nb | (cid:13)(cid:13)(cid:13) χ C bn A nab ′ (cid:13)(cid:13)(cid:13) L ∞ t L x (cid:13)(cid:13)(cid:13) ∇ t , x Φ nab (cid:13)(cid:13)(cid:13) L ∞ t L x , where C bn is a suitable space-time cube of width ∼ | I nb | centered around ( t abn , x abn ), and the impliedconstant depends on the frequency support cuto ff for the Φ nab (see the end of Step 2), we see thatin light of the decay properties of the A nab ′ for b ′ , b , the norm converges to zero as n → ∞ . Oneargues similarly for (cid:13)(cid:13)(cid:13) χ I nb A nab ′ A nab ′′ Φ nab (cid:13)(cid:13)(cid:13) L t L x , b ′ , b , as well as those terms generated when we replace A nab ′ by A na , low or A naB , which then takes careof the third expression χ I nb (cid:0) (cid:3) A na , low + P Bb ′ = A nab ′ + A naB − (cid:3) A nab (cid:1) Φ nab . Note that since we can force things to be arbitrarily small here if we simply choose n large enough,we can also sum over b ∈ [ B , B ], while maintaining smallness. The terms χ ′′ I nb Φ nab , χ ′ I nb ( A na , low + A nab ′ + A naB ) Φ nab ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 105 almost cancel the corresponding ones generated by (cid:3) A na , low + P Bb ′ = A nab ′ + A naB (cid:16) B X b = B (1 − χ I nb ) ˜ Φ nab (cid:17) , except the ˜ Φ nab used in the latter di ff ers from Φ nab by a term δ Φ nab whose energy is bounded by (cid:13)(cid:13)(cid:13) ∇ t , x δ Φ nab (cid:13)(cid:13)(cid:13) L ∞ t L x ( I nb × R ) . (cid:13)(cid:13)(cid:13) χ I nb (cid:0) (cid:3) A na , low + P Bb ′ = A nab ′ + A naB − (cid:3) A nab (cid:1) Φ nab (cid:13)(cid:13)(cid:13) N , and the expression on the right here, even when summed over b ∈ [ B , B ], is ≪ δ if we pick n su ffi ciently large. This then su ffi ces to bound B X b = B (cid:13)(cid:13)(cid:13) χ ′′ I nb δ Φ nab (cid:13)(cid:13)(cid:13) L t L x + B X b = B (cid:13)(cid:13)(cid:13) χ ′ I nb ( A na , low + A nab ′ + A naB ) δ Φ nab (cid:13)(cid:13)(cid:13) L t L x ≪ δ for su ffi ciently large n , where we take advantage of the spatial support properties that we assumeabout the data for Φ nab . (cid:3) Next, we show that each of the terms I – III can be made arbitrarily small up to certain errorterms by picking B and then n su ffi ciently large. The contribution of I.
One writes schematically (cid:3) A na , low + P Bb ′ = A nab ′ + A naB + δ nA φ na , low = (cid:3) A na , low + P Bb ′ = A nab ′ + A naB φ na , low + i ( δ nA ) ν ∂ ν φ na , low + (cid:16) A na , low + B X b ′ = A nab ′ + A naB + δ nA (cid:17) δ nA φ na , low . Then one has for any B , lim n →∞ (cid:13)(cid:13)(cid:13) (cid:3) A na , low + P Bb ′ = A nab ′ + A naB φ na , low (cid:13)(cid:13)(cid:13) N = P Bb ′ = A nab ′ , A naB and φ na , low as well as due to the fact that by construction we have (cid:3) A na , low φ na , low = . More precisely, one uses an argument as in the proof of Proposition 7.4. We then still have the errorterms(7.65) 2 i ( δ nA ) ν ∂ ν φ na , low and (cid:16) A na , low + B X b ′ = A nab ′ + A naB + δ nA (cid:17) δ nA φ na , low . The second term here shall be straightforward to treat by means of a simple divisibility argument,while the first will require the equation satisfied by δ nA in conjunction with a divisibility argument.
06 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION
The contribution of II.
We write schematically (cid:3) A na , low + P Bb ′ = A nab ′ + A naB + δ nA (cid:16) B X b = Φ nab (cid:17) = B X b = χ I nb (cid:16) (cid:3) A na , low + P Bb ′ = A nab ′ + A naB − (cid:3) A nab (cid:17) Φ nab + B X b = i ( δ nA ) ν ∂ ν Φ nab + B X b = (cid:16) A na , low + B X b ′ = A nab ′ + A naB + δ nA (cid:17) δ nA Φ nab . Here the time intervals I nb correspond to [ − T b , T b ] for the temporally bounded profiles and to [ t abn − R b , t abn + R b ] for the temporally unbounded ones. We shall henceforth make the following additionalassumption that | I nb | = M ∀ b chosen very large (eventually depending on δ and the profiles). Then we observe that given any δ >
0, we can pick B large enough such that for any su ffi ciently large n , we have (cid:13)(cid:13)(cid:13)(cid:13) B X b = χ I nb (cid:0) (cid:3) A na , low + P Bb ′ = A nab ′ + A naB − (cid:3) A nab (cid:1) Φ nab (cid:13)(cid:13)(cid:13)(cid:13) N ≪ δ . To show this, we need (cid:13)(cid:13)(cid:13)(cid:13) B X b = χ I nb i (cid:16) A na , low + B X b ′ = b ′ , b A nab ′ + A naB (cid:17) ν ∂ ν Φ nab (cid:13)(cid:13)(cid:13)(cid:13) N ≪ δ , (cid:13)(cid:13)(cid:13)(cid:13) B X b = χ I nb (cid:16) ( A na , low + B X b ′ = A nab ′ + A naB ) − ( A nab ) (cid:17) Φ nab (cid:13)(cid:13)(cid:13)(cid:13) N ≪ δ . For the first expression, observe that the interactions of A nab ′ , b ′ , b , with Φ nab are easily seento vanish as n → ∞ , using crude bounds, due to the time localization from χ I nb , and the divergingsupports of these profiles or their dispersive decay. Similarly, the interaction of A na , low with Φ nab isseen to vanish asymptotically as n → ∞ , due to the divergent frequency supports and again takingadvantage of the extra cuto ff χ I nb . Note that at this point we have not yet used the parameter B .Finally, we also need to bound (cid:13)(cid:13)(cid:13)(cid:13) B X b = χ I nb A naB ν ∂ ν Φ nab (cid:13)(cid:13)(cid:13)(cid:13) N , and it is here that we shall take advantage of the size of B . Precisely, we divide the above term intotwo. First, pick B very large, depending on the parameter M (which controls the I nb via | I nb | ≤ M ),such that we have for any B ≥ B ,lim sup n →∞ (cid:13)(cid:13)(cid:13)(cid:13) B X b = B χ I nb A naB ν ∂ ν Φ nab (cid:13)(cid:13)(cid:13)(cid:13) N ≪ δ . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 107
That this is possible follows from Lemma 7.17. Then, with B chosen, pick B ≥ B su ffi cientlylarge such that lim sup n →∞ (cid:13)(cid:13)(cid:13)(cid:13) B X b = χ I nb A naB ν ∂ ν Φ nab (cid:13)(cid:13)(cid:13)(cid:13) N ≪ δ . Here we take advantage of the fact that we essentially havelim sup n →∞ (cid:13)(cid:13)(cid:13) A naB ν (cid:13)(cid:13)(cid:13) L ∞ t L ∞ x + L ∞ t ˙ H x → B → ∞ . In fact, we have to be a bit careful here, because in Remark 7.18 we assume thatwe have incorporated some extra errors into the tail terms A naB and Φ naB , which do not vanish as B → ∞ . However, considering the term corresponding to a fixed b ∈ [1 , B ], we have that the extracontribution to A naB (coming from truncating A nab [0]) interacts weakly with Φ nab (in the sensethat it vanishes as | I b | → ∞ ), see e.g. the proof of Proposition 5.13. The remaining contributionsfrom truncating A nab ′ [0] are easily seen to result in interactions vanishing as n → ∞ . The cubicterm (cid:13)(cid:13)(cid:13)(cid:13) B X b = χ I nb (cid:16) ( A na , low + B X b ′ = A nab ′ + A naB ) − ( A nab ) (cid:17) Φ nab (cid:13)(cid:13)(cid:13)(cid:13) N is actually simpler, because the temporal cuto ff s χ I nb are not even necessary to get the desired bound.This completes the estimate for II except for the error terms(7.66) (cid:3) A na , low + P Bb ′ = A nab ′ + A naB + δ nA (cid:16) B X b = Φ nab (cid:17) − (cid:3) A na , low + P Bb ′ = A nab ′ + A naB (cid:16) B X b = Φ nab (cid:17) . The contribution of III.
Here we take advantage of the fact that Φ naB satisfies the equation e (cid:3) A na u = B large enough such that for all su ffi ciently large n , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:3) A na , low + P Bb ′ = A nab ′ + A naB Φ naB (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N ≪ δ . Of course, the profiles A nab ′ are not free waves, but they di ff er from free waves by terms that arenegligible as far as interactions with Φ naB are concerned. In fact, we recall that (cid:13)(cid:13)(cid:13) A nab ′ ( t , x ) − S ( A ab ′ [0])( t − t ab ′ n , x − x ab ′ n ) (cid:13)(cid:13)(cid:13) ℓ S < ∞ . Using Lemma 7.17, we can refine this to a tail estimate as follows. There exists B su ffi ciently largesuch that denoting B nab ′ : = A nab ′ ( t , x ) − S ( A ab ′ [0])( t − t ab ′ n , x − x ab ′ n ) , we have for any B ≥ B , lim sup n →∞ B X b ′ = B (cid:13)(cid:13)(cid:13) i B nab ′ ν ∂ ν Φ naB (cid:13)(cid:13)(cid:13) N ≪ δ . On the other hand, with this B fixed, we can use the argument for Lemma 7.16 to conclude thatthere exists B ≥ B such that we havelim sup n →∞ B X b ′ = (cid:13)(cid:13)(cid:13) i B nab ′ ν ∂ ν Φ naB (cid:13)(cid:13)(cid:13) N ≪ δ . Finally, we are still left with the error terms(7.67) (cid:3) A na , low + P Bb ′ = A nab ′ + A naB + δ nA Φ naB − (cid:3) A na , low + P Bb ′ = A nab ′ + A naB Φ naB .
08 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION
We have now shown smallness of the terms I – III up to errors that are at least linear in δ nA givenby (7.65) – (7.67).Having dealt with the equation for δ n Φ , we now come to the equation for δ nA given by (cid:3) (cid:16) A na , lowj + B X b ′ = A nab ′ j + A naBj + ( δ nA ) j (cid:17) = −P j Im (cid:18)(cid:16) φ na , low + B X b = Φ nab + Φ naB + δ n Φ (cid:17) D x (cid:16) φ na , low + B X b = Φ nab + Φ naB + δ n Φ (cid:17)(cid:19) , (7.68)where the covariant derivative D x uses the underlying connection form A na , low + B X b ′ = A nab ′ + A naB + ( δ nA ) . We rewrite this in the form (cid:3) ( δ nA ) = − IV − V , (7.69)where we put schematically IV : = Im (cid:18)(cid:16) φ na , low + B X b = Φ nab + Φ naB + δ n Φ (cid:17) D x (cid:16) φ na , low + B X b = Φ nab + Φ naB + δ n Φ (cid:17)(cid:19) − Im (cid:18)(cid:16) B X b = Φ nab + Φ naB + δ n Φ (cid:17) D x (cid:16) B X b = Φ nab + Φ naB + δ n Φ (cid:17)(cid:19) − Im (cid:16) φ na , low D x φ na , low (cid:17) , V : = Im (cid:18)(cid:16) B X b = Φ nab + Φ naB + δ n Φ (cid:17) D x (cid:16) B X b = Φ nab + Φ naB + δ n Φ (cid:17)(cid:19) − B X b = (cid:3) A nab . The term IV can be written in terms of null forms as well as cubic terms involving at least one lowfrequency factor φ na , low as well as at least one high frequency term from B X b = Φ nab + Φ naB , or else error terms involving at least one factor δ n Φ . The former type of interaction is easily seen toconverge to zero with respect to k · k N as n → ∞ , and so only the latter type of error term needs tobe kept. As for term V , again ignoring the terms involving at least one factor δ n Φ , we reduce this toIm (cid:18)(cid:16) B X b = Φ nab + Φ naB (cid:17) D x (cid:16) B X b = Φ nab + Φ naB (cid:17)(cid:19) − B X b = (cid:3) A nab . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 109
Then from the definition of the profiles Φ nab , we can write this for some large B and B ≥ B as B X b = χ ( I nb ) c Im (cid:16) Φ nab D x Φ nab (cid:17) + Im (cid:18)(cid:16) B X b = Φ nab (cid:17) D x (cid:16) B X b = Φ nab (cid:17)(cid:19) − B X b = Im (cid:16) Φ nab D x Φ nab (cid:17) + Im (cid:18)(cid:16) B X b = B Φ nab (cid:17) D x (cid:16) B X b = Φ nab + Φ naB (cid:17)(cid:19) + Im (cid:18) Φ naB D x (cid:16) B X b = B Φ nab (cid:17)(cid:19) − B X b = B χ I nb Im (cid:16) Φ nab D x Φ nab (cid:17) + Im (cid:16) Φ naB D x Φ naB (cid:17) ≡ ( V ) + ( V ) + ( V ) − ( V ) + ( V ) . Then given a δ > B su ffi ciently large such that for all su ffi cientlylarge n we have (cid:13)(cid:13)(cid:13) ( V ) (cid:13)(cid:13)(cid:13) N + (cid:13)(cid:13)(cid:13) ( V ) (cid:13)(cid:13)(cid:13) N ≪ δ , using Lemma 7.17. Then one picks n large enough such that (cid:13)(cid:13)(cid:13) ( V ) (cid:13)(cid:13)(cid:13) N ≪ δ . Further, with B fixed, pick B ≥ B su ffi ciently large such that (cid:13)(cid:13)(cid:13) ( V ) (cid:13)(cid:13)(cid:13) N ≪ δ . Finally, with B fixed, we choose M = | I nb | large enough (depending on the profiles Φ nab , b = , . . . , B , where these of course depend on the n -independent φ ab [0]), such that (cid:13)(cid:13)(cid:13) ( V ) (cid:13)(cid:13)(cid:13) N ≪ δ . This is then the M that needs to be used in the analysis of the δ n Φ equation in the “additionalassumption” there. (cid:3) Conclusion of the induction on frequency process.
In the preceding subsection we obtainedglobal S norm bounds for the MKG-CG evolution of the data (cid:0) A n Λ + A n , φ n Λ + φ n (cid:1) [0]under the assumption that (cid:0) A n , φ n (cid:1) [0] has at least two non-zero concentration profiles, or all suchprofiles are zero, or else there exists only one such profile ( ˜ A n b , ˜ φ n b ) but withlim inf n →∞ E ( ˜ A n b , ˜ φ n b )(0) < E crit . We now make this assumption and continue the process by considering the data(7.70) (cid:0) A n Λ + A n + A n Λ , φ n Λ + φ n + φ n Λ (cid:1) [0]at time t =
0. Proceeding almost identically to Subsection 7.3, we prove that the MKG-CG evolu-tion of this data exists globally and satisfies a priori S norm bounds. These bounds depend on E crit
10 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION and the a priori bounds on the evolution of the data (cid:0) A n Λ + A n , φ n Λ + φ n (cid:1) [0]. The only di ff erencehere is that in the decompositions (see Subsection 7.3) A n Λ [0] = Λ ( δ ) X j = A n j ) Λ [0] + A n Λ ( Λ ) [0] ,φ n Λ [0] = Λ ( δ ) X j = φ n j ) Λ [0] + φ n Λ ( Λ ) [0] , we now have to make sure that (cid:13)(cid:13)(cid:13) A n Λ ( Λ ) [0] (cid:13)(cid:13)(cid:13) ˙ B , ∞ × ˙ B , ∞ + (cid:13)(cid:13)(cid:13) φ n Λ ( Λ ) [0] (cid:13)(cid:13)(cid:13) ˙ B , ∞ × ˙ B , ∞ is small enough depending both on E crit and the a priori bounds for the MKG-CG evolution of thedata (cid:0) A n Λ + A n , φ n Λ + φ n (cid:1) [0]. Then we continue by adding the second frequency atom (cid:0) A n , φ n (cid:1) [0]to the data at time t = e (cid:3) A n : = (cid:3) + i (cid:0) A n Λ ,ν + A n ν + A n Λ ,ν + A n , f ree ν (cid:1) ∂ ν , where A n Λ ,ν + A n ν + A n Λ ,ν is given by the global MKG-CG evolution of the data (7.70).All in all we may carry out this process Λ many times in order to finally conclude that if eitherthere are at least two frequency atoms, or else there is only one frequency atom but withlim inf n →∞ E ( A n , φ n ) < E crit , or if we do have lim n →∞ E ( A n , φ n ) = E crit , but such that there are at least two concentration profiles, or finally if there is only one frequencyatom of asymptotic energy E crit and only one concentration profile ( ˜ A n b , ˜ φ n b ) withlim inf n →∞ E ( ˜ A n b , ˜ φ n b )(0) < E crit , then the sequence ( A n , φ n ) cannot possibly have been essentially singular, resulting in a contradic-tion to our assumption. We can then formulate the following Corollary 7.19.
Assume that ( A n , φ n ) is an essentially singular sequence. Then by re-scaling wemay assume that the sequence of data ( A n , φ n )[0] is -oscillatory, and that there exist sequences { ( t n , x n ) } n ∈ N ⊂ R × R and fixed profiles ( A , φ )[0] ∈ ( ˙ H x × L x ) × ( ˙ H x × L x ) with A satisfying the Coulomb condition, such that we have for j = , . . . , ,A nj [0] = S (cid:0) A j [0] (cid:1) ( · − t n , · − x n )[0] + o ˙ H x × L x (1) as n → ∞ . Here, S ( · )( t , x ) denotes the standard free wave propagator. Furthermore, define for j = , . . . , , ˜ A j ( t , x ) = S (cid:0) A j [0] (cid:1) ( t , x ) and denote by S ˜ A (cid:0) u [0] (cid:1) ( t , x ) the solution to (cid:0) (cid:3) + i ˜ A j ∂ j (cid:1) u = ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 111 with data u [0] ∈ ˙ H x × L x at time t = . Then we have φ n [0] = S ˜ A (cid:0) φ [0] (cid:1) ( · − t n , · − x n )[0] + o ˙ H x × L x (1) as n → ∞ . If the sequence ( t n ) n ∈ N admits a subsequence that is bounded, then by passing to this subse-quence, we may as well replace t n by t n = n , and correspondingly obtain up to rescalingand spatial translations that ( A n , φ n )[0] = ( A , φ )[0] + o ˙ H x × L x (1) . Then Proposition 6.1 implies that the evolution ( A ∞ , Φ ∞ ) of ( A , φ )[0] is in fact a minimal energyblowup solution. In the case that t n → + ∞ (or t n → −∞ ), we need to introduce the concept of aminimum regularity MKG-CG evolution associated with scattering data, or “a solution at infinity”.Here we have the following Proposition 7.20.
Let ( A , φ )[0] be Coulomb energy class data and let { ( t n , x n ) } n ∈ N ⊂ R × R witht n → + ∞ . We introduce the scattering data A nj [0] = S (cid:0) A j [0] (cid:1) ( · − t n , · − x n )[0] , j = , . . . , , Φ n [0] = S ˜ A (cid:0) φ [0] (cid:1) ( · − t n , · − x n )[0] , (7.71) where we use the notation ˜ A j ( t , x ) = S (cid:0) A j [0] (cid:1) ( t , x ) for j = , . . . , . Moreover, we denote by ( A n , Φ n )( t , x ) the MKG-CG evolution (in the sense of Section 5) of the Coulomb data ( A n , Φ n )[0] .Then there exists a su ffi ciently large C ∈ R + such that there exists an energy class solution (cid:0) A ∞ , Φ ∞ (cid:1) to MKG-CG on ( −∞ , − C ) × R , which is the limit of admissible solutions as in Section 5 with (cid:13)(cid:13)(cid:13)(cid:0) A ∞ , Φ ∞ (cid:1)(cid:13)(cid:13)(cid:13) S (( −∞ , − C ] × R ) < ∞ ∀ C > C , and such that for any t ∈ ( −∞ , − C ) we have in the energy topology lim n →∞ (cid:0) A n , Φ n (cid:1) ( t + t n , x + x n ) = (cid:0) A ∞ , Φ ∞ (cid:1) ( t , x ) . In particular, the expressions on the left are well-defined (in the sense of Section 5) for n su ffi cientlylarge.Proof. This is a perturbative argument, which exploits the dispersive behaviour as evidenced byamplitude decay of the functions A n [0] and Φ n [0]. We write A n ( t , x ) = A n ( t , x ) + δ A n ( t , x ) , Φ n ( t , x ) = Φ n ( t , x ) + δ Φ n ( t , x ) , where we use the notation A nj ( t , x ) = S (cid:0) A j [0] (cid:1) ( t − t n , x − x n ) , j = , . . . , , Φ n ( t , x ) = S ˜ A (cid:0) φ [0] (cid:1) ( t − t n , x − x n ) . Also, keep in mind that ( A n , Φ n )( t , x ) denotes the MKG-CG evolution (in the sense of Section 5) ofthe data ( A n , Φ n )[0]. Then we show that ( δ A n , δ Φ n ) satisfy good S -bounds on ( −∞ , t n − C ) × R for some C > ffi ciently large, and all n large enough. This means that the evolutions ( A n , Φ n )are well-defined on ( −∞ , t n − C ) × R . Furthermore, assuming as we may that t n is monotonouslyincreasing, we will show that for n ′ > n , we havelim n , n ′ →∞ n ′ > n (cid:13)(cid:13)(cid:13) A n ′ [ t n ′ − t n ] − A n [0] (cid:13)(cid:13)(cid:13) ℓ ˙ H x × ℓ L x + lim n , n ′ →∞ n ′ > n (cid:13)(cid:13)(cid:13) Φ n ′ [ t n ′ − t n ] − Φ n [0] (cid:13)(cid:13)(cid:13) ˙ H x × L x = ,
12 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION which together with standard perturbation theory then results in the fact thatlim n →∞ (cid:0) A n , Φ n (cid:1) ( t + t n , x + x n ) = (cid:0) A ∞ , Φ ∞ (cid:1) ( t , x ) , provided t ∈ ( −∞ , − C ), and the right hand side is a solution to MKG-CG in the sense of Section 5.To get the desired bounds on ( δ A n , δ Φ n ), we record the schematic system of equations that theysatisfy 0 = (cid:3) A n Φ n + (cid:0) (cid:3) A n + δ A n − (cid:3) A n (cid:1) Φ n + (cid:3) A n + δ A n δ Φ n , (7.72) (cid:3) ( δ A nj ) = P j (cid:0) Φ n D x Φ n (cid:1) + P j (cid:0) δ Φ n D x Φ n (cid:1) + P j (cid:0) Φ n D x δ Φ n (cid:1) + P j (cid:0) δ Φ n D x δ Φ n (cid:1) . (7.73)We then show that given δ >
0, there exists C = C ( δ, A [0] , φ [0]) such that we have (cid:13)(cid:13)(cid:13) δ A n (cid:13)(cid:13)(cid:13) ℓ S (( −∞ , t n − C ] × R ) + (cid:13)(cid:13)(cid:13) δ Φ n (cid:13)(cid:13)(cid:13) S (( −∞ , t n − C ] × R ) < δ. This follows as usual via a bootstrap argument. We show here how to obtain smallness of thenon-perturbative source terms on the right hand side, i.e. the terms (cid:3) A n Φ n , P i (cid:0) Φ n D x Φ n (cid:1) , while the remaining terms are handled either via the smallness of δ (provided they are quadratic in δ A n , δ Φ n ), or else via a standard divisibility argument, just as in the proof of Proposition 7.4. Nowthe first term on the right is in e ff ect equal to A n ν A n ,ν Φ n . To treat it, we note that we may reduce all inputs as well as the output to frequency ∼
1, since elsewe gain smallness for the L t L x -norm of the output by using standard Strichartz norms. Then weestimate the remainder by (cid:13)(cid:13)(cid:13) P O (1) A n ν P O (1) A n ,ν P O (1) Φ n (cid:13)(cid:13)(cid:13) L t L x (( −∞ , t n − C ] × R ) . (cid:13)(cid:13)(cid:13) P O (1) A n ν (cid:13)(cid:13)(cid:13) L t L x (( −∞ , t n − C ] × R ) (cid:13)(cid:13)(cid:13) P O (1) A n ,ν (cid:13)(cid:13)(cid:13) L t L x (( −∞ , t n − C ] × R ) (cid:13)(cid:13)(cid:13) P O (1) Φ n (cid:13)(cid:13)(cid:13) L t L ∞ x (( −∞ , t n − C ] × R ) . Then by exploiting the L ∞ x decay and interpolation, for example, we get (cid:13)(cid:13)(cid:13) P O (1) A n ,ν (cid:13)(cid:13)(cid:13) L t L x (( −∞ , t n − C ] × R ) ≪ δ for C su ffi ciently large, uniformly in n , and this su ffi ces to get the necessary smallness on accountof the fact that uniformly in n , (cid:13)(cid:13)(cid:13) P O (1) A n ν (cid:13)(cid:13)(cid:13) L t L x (( −∞ , t n − C ] × R ) + (cid:13)(cid:13)(cid:13) P O (1) Φ n (cid:13)(cid:13)(cid:13) L t L ∞ x (( −∞ , t n − C ] × R ) . (cid:13)(cid:13)(cid:13) ( A [0] , φ [0]) (cid:13)(cid:13)(cid:13) ˙ H x × L x . As for the quadratic term P j (cid:0) Φ n D x Φ n (cid:1) , its inherent null structure allows to reduce to the case of frequencies ∼ ∼
1. In that situationwe have (cid:13)(cid:13)(cid:13) P j (cid:0) Φ n D x Φ n (cid:1)(cid:13)(cid:13)(cid:13) N (( −∞ , t n − C ] × R ) . (cid:13)(cid:13)(cid:13) P j (cid:0) Φ n D x Φ n (cid:1)(cid:13)(cid:13)(cid:13) L t L x (( −∞ , t n − C ] × R ) , which can be estimated by placing one input into L t L x (( −∞ , t n − C ] × R ) and the other one into L t L x (( −∞ , t n − C ] × R ). The latter norm is small uniformly in n for C su ffi ciently large on accountof (a variant of) Lemma 7.9. (cid:3) ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 113
Remark 7.21.
The preceding proof implies in particular that if ( A , φ )[0] is Coulomb energy classdata, then there exists t > su ffi ciently large such that the initial data (cid:16) S (cid:0) A [0] (cid:1) ( · − t , · ) , S ˜ A (cid:0) φ [0] (cid:1) ( · − t , · ) (cid:17) [0] , where for j = , . . . , , ˜ A j ( t , x ) = S (cid:0) A j [0] (cid:1) ( t , x ) , can be evolved in the sense of Section 5 on ( −∞ , × R and satisfies a global S -bound there. Thisis the analogue of Proposition 7.15 in [20]. Extracting a minimal blowup solution in the case of one temporally unbounded profile is still nota direct consequence of the preceding proposition on account of the somewhat delicate perturbationtheory, but follows by a slightly indirect argument. Here we state
Proposition 7.22.
Assume that the essentially singular sequence ( A n , φ n ) satisfiesA n [0] = S (cid:0) A [0] (cid:1) ( · − t n , · − x n )[0] + o ˙ H x × L x (1) ,φ n [0] = S ˜ A (cid:0) φ [0] (cid:1) ( · − t n , · − x n )[0] + o ˙ H x × L x (1) , where t n → + ∞ , say, and we use the same notation as in the preceding Corollary 7.19. Thendenoting the corresponding MKG-CG evolution of these data by ( A n , φ n )( t , x ) , its lifespan comprises ( −∞ , t n − C ) for C su ffi ciently large, uniformly in n. Also, the sequence (cid:8) ( A n , φ n )[ t n − C ] (cid:9) n ∈ N forms a pre-compact set in the energy topology. Denoting a limit point (any such satisfies theCoulomb condition) by ( A ∞ , Φ ∞ )[0] , we have E ( A ∞ , Φ ∞ ) = E crit , and moreover, denoting thelifespan of its MKG-CG evolution by I, we get sup J ⊂ I (cid:13)(cid:13)(cid:13) ( A ∞ , Φ ∞ ) (cid:13)(cid:13)(cid:13) S ( J × R ) = ∞ . Proof.
The fact that the evolution of ( A n , φ n )( t , x ) is defined and has finite S -bounds on ( −∞ , t n − C )follows by exactly the same method as in the proof of the preceding proposition. We set A n ( t , x ) = A n ( t , x ) + δ A n ( t , x ) ,φ n ( t , x ) = S ˜ A (cid:0) φ [0] (cid:1) ( t − t n , x − x n ) + δφ n ( t , x ) , where we let A n be the free wave evolution of A n [0], i.e. for j = , . . . , A nj ( t , x ) = S (cid:0) A j [0] (cid:1) ( t − t n , x − x n ) + o ˙ H x × L x (1) , and ˜ A j ( t , x ) = S (cid:0) A j [0] (cid:1) ( t , x ) for j = , . . . ,
4. Also, note that δ A n [0] =
0. Then choosing C largeenough, we infer the bounds (cid:13)(cid:13)(cid:13) ( δ A n , δφ n ) (cid:13)(cid:13)(cid:13) ( ℓ S × S )( −∞ , t n − C ) × R ≪ A n , φ n ) is essentially singular, we know by the preceding results that the data( A n , φ n )[ t n − C ]are concentrated at fixed frequency ∼ (cid:8) ( A n , φ n )[ t n − C ] (cid:9) n ≥ is pre-compact in the energy topology. Extracting a limiting profile ( A ∞ , Φ ∞ )[0], the last statementof the proposition follows directly from Proposition 6.1. (cid:3)
14 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION
To conclude this section, we finally state the following crucial compactness property of theminimal blowup solution ( A ∞ , Φ ∞ ) extracted in the preceding. Theorem 7.23.
Denote the lifespan of ( A ∞ , Φ ∞ ) by I. There exist continuous functions x : I → R , λ : I → R + , so that each of the family of functions ((cid:18) λ ( t ) A ∞ j (cid:18) t , · − x ( t ) λ ( t ) (cid:19) , λ ( t ) ∂ t A ∞ j (cid:18) t , · − x ( t ) λ ( t ) (cid:19)(cid:19) : t ∈ I ) for j = , . . . , and ((cid:18) λ ( t ) Φ ∞ (cid:18) t , · − x ( t ) λ ( t ) (cid:19) , λ ( t ) ∂ t Φ ∞ (cid:18) t , · − x ( t ) λ ( t ) (cid:19)(cid:19) : t ∈ I ) is pre-compact in ˙ H x ( R ) × L x ( R ) . The proof of this follows exactly as for Corollary 9.36 in [20], using the preceding Remark 7.21.8. R igidity argument
In this final section we rule out the existence of a minimal blowup solution ( A ∞ , Φ ∞ ) with thecompactness property from Theorem 7.23. To this end we largely follow the scheme of the rigidityargument by Kenig-Merle [10].In Subsection 8.1 we derive several energy and virial identities for energy class solutions toMKG-CG. Then we prove some preliminary properties of the minimal blowup solution ( A ∞ , Φ ∞ ),in particular that its momentum must vanish. Denoting by I the lifespan of ( A ∞ , Φ ∞ ), we distin-guish between I + : = I ∩ [0 , ∞ ) being a finite or an infinite time interval. In the next Subsection 8.2,we exclude the existence of a minimal blowup solution ( A ∞ , Φ ∞ ) with infinite time interval I + using the virial identities, the fact that the momentum of ( A ∞ , Φ ∞ ) must vanish and an additionalVitali covering argument introduced in [20]. Moreover, we reduce the case of finite lifespan I + to a self-similar blowup scenario. In the last Subsection 8.3, we then derive a suitable Lyapunovfunctional for the Maxwell-Klein-Gordon system in self-similar variables, which will finally enableus to also rule out the self-similar case.8.1. Preliminary properties of minimal blowup solutions with the compactness property.
Wewill sometimes use the following notation for the covariant derivatives D α = ∂ α + i A ∞ α and the curvature components F ∞ αβ = ∂ α A ∞ β − ∂ β A ∞ α associated with the minimal blowup solution ( A ∞ , Φ ∞ ). Lemma 8.1.
Let ( A , φ ) be an energy class solution to MKG-CG in the sense of Definition 5.3 withlifespan I containing . For given ε > , let R > be such that Z {| x |≥ R } (cid:18) X α,β F αβ (0 , x ) + X α | D α φ (0 , x ) | (cid:19) dx ≤ ε. Then we have for any t ∈ I + that Z {| x |≥ R + t } (cid:18) X α,β F αβ ( t , x ) + X α | D α φ ( t , x ) | (cid:19) dx ≤ ε. ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 115
Proof.
Let ( A , φ ) be an admissible solution to MKG-CG with lifespan I containing 0. For R > t ∈ I + , we define E R ( t ) = Z {| x |≥ R + t } (cid:18) X α,β F αβ ( t , x ) + X α | D α φ ( t , x ) | (cid:19) dx . Using that the energy-momentum tensor for the Maxwell-Klein-Gordon system T αβ = F αγ F βγ − m αβ F γδ F γδ + Re (cid:0) D α φ D β φ (cid:1) − m αβ D γ φ D γ φ, with m αβ denoting the Minkowski metric, is divergence free ∂ α T αβ = , we easily obtain from the divergence theorem that for any t , t ∈ I + with t < t ,(8.1) E R ( t ) = E R ( t ) + Z M t t (cid:18) T ( t , x ) + x j | x | T j ( t , x ) (cid:19) d σ ( t , x ) . Here, M t t denotes the part of the mantle of the forwards light cone { ( t , x ) ∈ I + × R : | x | ≤ R + t } enclosed by the time slices { t } × R and { t } × R , and d σ denotes the standard surface measure.One easily verifies that the flux T ( t , x ) + x j | x | T j ( t , x )is non-negative using the general identity X j , k (cid:0) ω j r k − ω k r j (cid:1) = (cid:0) r − ( r · ω ) (cid:1) ≤ r for r , ω ∈ R with | ω | =
1. We conclude that(8.2) E R ( t ) ≤ E R ( t ) . Since an energy class solution to MKG-CG in the sense of Definition 5.3 is a locally uniform limitof admissible solutions, the corresponding inequality (8.2) follows by passing to the limit. Thisimplies the claim. (cid:3)
Next, we prove the following energy and virial identities for energy class solutions to MKG-CG.
Proposition 8.2.
Let ( A , φ ) be an energy class solution to MKG-CG in the sense of Definition 5.3.Then the following identities hold. • Energy conservation (8.3) ddt Z R (cid:18) X α,β F αβ + X α | D α φ | (cid:19) dx = . • Momentum conservation (8.4) ddt Z R (cid:16) F j F k j + Re (cid:0) D φ D k φ (cid:1)(cid:17) dx = for k = , . . . , . • Weighted energy (8.5) ddt Z R x k ϕ R (cid:18) X α,β F αβ + X α | D α φ | (cid:19) dx = − Z R (cid:16) F j F k j + Re (cid:0) D φ D k φ (cid:1)(cid:17) dx + O ( r ( R )) for k = , . . . , .
16 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION • Weighted momentum monotonicityddt Z R x k ϕ R (cid:16) F j F k j + Re (cid:0) D φ D k φ (cid:1)(cid:17) dx + ddt Z R ϕ R Re (cid:0) φ D φ (cid:1) dx = − Z R (cid:18)X k F k + | D φ | (cid:19) dx + O ( r ( R )) . (8.6) Here, ϕ ∈ C ∞ c ( R ) is a smooth cuto ff with ϕ ( x ) = for | x | ≤ and ϕ ( x ) = for | x | ≥ . Moreover,for R > we define ϕ R ( x ) = ϕ (cid:0) xR (cid:1) and (8.7) r ( R ) : = Z {| x |≥ R } (cid:18)X α,β F αβ + X α | D α φ | + | φ | | x | (cid:19) dx . Proof.
It su ffi ces to verify these identities for admissible solutions to MKG-CG. Since energy classsolutions in the sense of Definition 5.3 are locally uniform limits of admissible solutions, the cor-responding identities follow by passing to the limit in an integrated formulation.So let ( A , φ ) be an admissible solution to MKG-CG. Then the energy conservation (8.3) andmomentum conservation (8.4) identities follow immediately from the divergence theorem and thefact that the energy-momentum tensor of the Maxwell-Klein-Gordon system T αβ = F αγ F βγ − m αβ F γδ F γδ + Re (cid:0) D α φ D β φ (cid:1) − m αβ D γ φ D γ φ for α, β ∈ { , , . . . , } is divergence free(8.8) ∂ α T αβ = . To prove the weighted energy identity (8.5), we also use the divergence-free property (8.8) of T αβ and compute for k = , . . . , ddt Z R x k ϕ R ( x ) T dx = Z R x k ϕ R ( x ) ∂ j T j dx = − Z R ϕ R ( x ) T k dx − Z R x k R ( ∂ j ϕ )( xR ) T j dx = − Z R T k dx + O ( r ( R )) , where we integrated by parts in the second to last step. This yields (8.5). Finally, to show theweighted momentum monotonicity identity (8.6), we compute ddt Z R x k ϕ R ( x ) T k dx = Z R x k ϕ R ( x ) ∂ j T jk dx = − Z R ϕ R ( x ) (cid:18) X k = T kk (cid:19) dx − Z R x k R ( ∂ j ϕ )( xR ) T jk dx = − Z R (cid:18) X k = F k + | D φ | − X k = | D k φ | (cid:19) dx + O ( r ( R )) . (8.9) ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 117
Since the right hand side of (8.9) does not yet exhibit the desired monotonicity, we also consider ddt Z R ϕ R ( x ) Re (cid:0) φ D φ (cid:1) dx = Z R ϕ R ( x ) Re (cid:0) ∂ t φ D φ (cid:1) dx + Z R ϕ R ( x ) Re (cid:0) φ∂ t D φ (cid:1) dx = Z R ϕ R ( x ) | D φ | dx + Z R ϕ R ( x ) Re (cid:16) φ D φ (cid:17) dx . Inserting the equation for φ and integrating by parts leads to ddt Z R ϕ R ( x ) Re (cid:0) φ D φ (cid:1) dx = Z R ϕ R ( x ) | D φ | dx − X k = Z R ϕ R ( x ) | D k φ | dx − Z R R ( ∂ k ϕ )( xR ) Re (cid:16) ϕ D k φ (cid:17) dx = Z R (cid:18) | D φ | − X k = | D k φ | (cid:19) dx + O ( r ( R )) . (8.10)Putting together (8.9) and (8.10), we obtain (8.6). (cid:3) If I + is a finite time interval, we obtain a lower bound on λ ( t ) from Theorem 7.23. Lemma 8.3.
Assume that I + is finite and after re-scaling that I + = [0 , . Let λ : I + → R + be as inTheorem 7.23. Then there exists a constant C ( K ) > such that < C ( K )1 − t ≤ λ ( t ) for all ≤ t < .Proof. The proof follows exactly as in [20, Lemma 10.4] by combining Corollary 6.3 and Theo-rem 7.23. (cid:3)
Moreover, when I + is a finite time interval, we conclude the following sharp support propertiesof Φ ∞ and the curvature components F ∞ αβ . Lemma 8.4.
Under the same assumptions as in Lemma 8.3 there exists x ∈ R such thatsupp (cid:16) F ∞ αβ ( t , · ) , Φ ∞ ( t , · ) (cid:17) ⊂ B ( x , − t ) for all ≤ t < and all α, β ∈ { , , . . . , } .Proof. We follow the proof of Lemma 4.8 in [10]. Consider a sequence { t n } n ⊂ [0 ,
1) with t n → n → ∞ . From the preceding Lemma 8.3 we know that λ ( t n ) → ∞ as n → ∞ . Together with thecompactness property expressed in Theorem 7.23, we obtain for every R > ε > ffi ciently large n , it holds that Z(cid:8) | x + x ( tn ) λ ( tn ) |≥ R (cid:9)(cid:18)X α (cid:12)(cid:12)(cid:12) ∇ t , x A ∞ α ( t n , x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ∇ t , x Φ ∞ ( t n , x ) (cid:12)(cid:12)(cid:12) (cid:19) dx ≤ ε Z(cid:8) | x + x ( tn ) λ ( tn ) |≥ R (cid:9)(cid:18)X α (cid:12)(cid:12)(cid:12) A ∞ α ( t n , x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Φ ∞ ( t n , x ) (cid:12)(cid:12)(cid:12) (cid:19) dx ≤ ε .
18 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION
Applying Lemma 8.1 backwards in time, we conclude for every R > ε >
0, and s ∈ [0 ,
1) thatwe have for all su ffi ciently large n ,(8.11) Z(cid:8) | x + x ( tn ) λ ( tn ) |≥ R + t n − s (cid:9)(cid:18) X α,β F ∞ αβ ( s , x ) + X α (cid:12)(cid:12)(cid:12) D α Φ ∞ ( s , x ) (cid:12)(cid:12)(cid:12) (cid:19) dx ≤ ε . Next, we show that there exists M > (cid:12)(cid:12)(cid:12)(cid:12) x ( t ) λ ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ M for all 0 ≤ t <
1. Suppose not. Thenit su ffi ces to consider a sequence t n → (cid:12)(cid:12)(cid:12)(cid:12) x ( t n ) λ ( t n ) (cid:12)(cid:12)(cid:12)(cid:12) → ∞ . For all R >
0, we have for su ffi cientlylarge n that (cid:8) x : | x | ≤ R (cid:9) ⊂ (cid:26) x : (cid:12)(cid:12)(cid:12)(cid:12) x + x ( t n ) λ ( t n ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ R + t n (cid:27) . But then we obtain from (8.11) with s = R > Z {| x |≤ R } (cid:18) X α,β F ∞ αβ (0 , x ) + X α (cid:12)(cid:12)(cid:12) D α Φ ∞ (0 , x ) (cid:12)(cid:12)(cid:12) (cid:19) dx ≤ ε . Since ε > t n → x ( t n ) λ ( t n ) → − x ∈ R . Now observe that for every η > s ∈ [0 , ffi ciently large n that (cid:8) x : | x − x | ≥ η + − s (cid:9) ⊂ (cid:26) x : (cid:12)(cid:12)(cid:12)(cid:12) x + x ( t n ) λ ( t n ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ η + t n − s (cid:27) . Hence, we obtain from (8.11) that for every ε > η > s ∈ [0 , Z {| x − x |≥ η + − s } (cid:18) X α,β F ∞ αβ ( s , x ) + X α (cid:12)(cid:12)(cid:12) D α Φ ∞ ( s , x ) (cid:12)(cid:12)(cid:12) (cid:19) dx ≤ ε . We conclude that supp (cid:16) F ∞ αβ ( t , · ) , (cid:0) D α Φ ∞ (cid:1) ( t , · ) (cid:17) ⊂ B ( x , − t )for all 0 ≤ t < α, β = , , . . . ,
4. The claim then follows from the diamagnetic inequality. (cid:3)
In the next key proposition we prove that the momentum of the minimal blowup solution ( A ∞ , Φ ∞ )must vanish. This will later allow us to control the movement of the “center of mass”, or more pre-cisely a weighted energy of ( A ∞ , Φ ∞ ). For technical reasons we have to distinguish between thecase of finite and infinite lifespan. Proposition 8.5.
Let ( A ∞ , Φ ∞ ) be as above. Assume that I + is a finite interval. Then we have fork = , . . . , and all t ∈ I + that (8.12) Z R (cid:18) X j = F ∞ j F ∞ k j + Re (cid:0) D Φ ∞ D k Φ ∞ (cid:1)(cid:19) ( t , x ) dx = . As for the critical focusing nonlinear wave equation [10] and for critical wave maps [20], theLorentz invariance of the Maxwell-Klein-Gordon system and transformational properties of theenergy under Lorentz transformations are essential ingredients in the proof of Proposition 8.5. Webegin by considering the relativistic invariance properties of our system. Assume that L : R + → R + is a Lorentz transformation, acting on column vectors via multiplication with the matrix L . Then φ transforms according to(8.13) φ φ L : = φ (cid:0) L ( t , x ) (cid:1) , which results in ∇ t , x φ L t ∇ t , x φ (cid:0) L ( t , x ) (cid:1) . Then the potential A α needs to transform accordingly, i.e. writing this as a column vector indexedby α , we transform(8.14) A A L : = L t A (cid:0) L ( t , x ) (cid:1) . Then the expression ∂ β F αβ , when interpreted as a column vector in α , also transforms according tomultiplication with L t , as does the expressionIm (cid:0) φ D α φ (cid:1) . Under these transformations, the Maxwell-Klein-Gordon system is then invariant. However, theconserved energy does not remain invariant under general Lorentz transformations, and our firststep is to quantify this. In the sequel we only consider very specific Lorentz transformations of theform(8.15) L = √ − d − d √ − d − d √ − d √ − d for small d ∈ R . Lemma 8.6.
Let ( A , φ ) be an admissible global solution to MKG-CG and let L : R + → R + bea Lorentz transformation of the form (8.15) for some d ∈ R . Then we have for all t ∈ R thatE (cid:0) A L , φ L (cid:1) ( t ) = Z R (cid:18) X α,β F αβ + X α | D α φ | (cid:19)(cid:0) L ( t , x ) (cid:1) dx + d − d Z R (cid:18) X j = ( F j + F j ) + X α = | D α φ | (cid:19)(cid:0) L ( t , x ) (cid:1) dx − d − d Z R (cid:18) X j = F j F j + Re (cid:0) D φ D φ (cid:1)(cid:19)(cid:0) L ( t , x ) (cid:1) dx . (8.16) Proof.
The potential A is transformed into A L as follows A L ( t , x ) = √ − d A ( L ( t , x )) − d √ − d A ( L ( t , x )) , A L ( t , x ) = − d √ − d A ( L ( t , x )) + √ − d A ( L ( t , x )) , A Lj ( t , x ) = A j ( L ( t , x )) , j = , , .
20 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION
Then we compute the corresponding curvature components F L = ∂ t A L − ∂ A L = − d √ − d (cid:18) √ − d ∂ t A − d √ − d ∂ A (cid:19) + √ − d (cid:18) √ − d ∂ t A − d √ − d ∂ A (cid:19) − √ − d (cid:18) − d √ − d ∂ t A + √ − d ∂ A (cid:19) + d √ − d (cid:18) − d √ − d ∂ t A + √ − d ∂ A (cid:19) = F . Here the right hand side has to be evaluated at L ( t , x ). We use this convention for the remainder ofthe proof. Further, we obtain F L = √ − d ∂ t A − d √ − d ∂ A − ∂ (cid:18) √ − d A − d √ − d A (cid:19) = √ − d F − d √ − d F as well as F L = √ − d F − d √ − d F , F L = √ − d F − d √ − d F . Similarly, we compute F L = − d √ − d ∂ t A + √ − d ∂ A + d √ − d ∂ A − √ − d ∂ A = − d √ − d F + √ − d F and F L = − d √ − d F + √ − d F , F L = − d √ − d F + √ − d F . Finally, we have for i , j ≥ F Li j = F i j . In summary, we have found that X α,β (cid:0) F L αβ (cid:1) = X α,β F αβ + d − d X j = (cid:0) F j + F j (cid:1) − d − d X j = F j F j . (8.17)We have to carry out the analogous computations for the part of the energy associated with thescalar field φ . Here we have (cid:12)(cid:12)(cid:12)(cid:0) ∂ t + iA L (cid:1) φ L (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:0) ∂ + iA L (cid:1) φ L (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ − d ∂ t φ − d √ − d ∂ φ + i (cid:18) √ − d A − d √ − d A (cid:19) φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − d √ − d ∂ t φ + √ − d ∂ φ + i (cid:18) − d √ − d A + √ − d A (cid:19) φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = + d − d (cid:16) | D φ | + | D φ | (cid:17) − d − d Re (cid:0) D φ D φ (cid:1) ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 121 and for j = , , (cid:12)(cid:12)(cid:12)(cid:0) ∂ j + iA Lj (cid:1) φ L (cid:12)(cid:12)(cid:12) = | D j φ | . Thus, we obtain that X α (cid:12)(cid:12)(cid:12)(cid:0) ∂ α + iA L α (cid:1) φ L (cid:12)(cid:12)(cid:12) = X α | D α φ | + d − d (cid:0) | D φ | + | D φ | (cid:1) − d − d Re (cid:0) D φ D φ (cid:1) . (8.18)The assertion now follows from (8.17) and (8.18). (cid:3) The identity (8.16) strongly suggests that if it is impossible to lower the energy by means ofa Lorentz transform of the form (8.15) for very small d with a suitable sign, then the momentummust vanish. To make this observation rigorous, we also need to establish a relation between the S norm of an admissible global solution ( A , φ ) to MKG-CG and the S norm of a suitable evolutionof the data ( A L , φ L )[0] obtained from the Lorentz transformed solution ( A L , φ L ). Here we firstobserve that for an admissible global solution ( A , φ ) to MKG-CG, the Lorentz transformed solution( A L , φ L ) is actually globally defined. We can therefore consider the data pair ( A L , φ L )[0] and notethat ( A L , φ L )[0] is C ∞ -smooth, but not in Coulomb gauge. Moreover, if ( t , x ) ∈ R + are restrictedto a space-like hyperplane containing the origin, then we have (cid:12)(cid:12)(cid:12) F jk ( t , x ) (cid:12)(cid:12)(cid:12) . (cid:0) + | t | + | x | (cid:1) − N for j , k ∈ { , . . . , } and any N ≥
1. From the equation satisfied by F αβ we obtain after integrationin time that (cid:12)(cid:12)(cid:12) F k ( t , x ) (cid:12)(cid:12)(cid:12) . (cid:0) + | t | + | x | (cid:1) − for k = , . . . ,
4. Thus, the curvature components of ( A L , φ L )[0] decay like h x i − as | x | → ∞ , whichensures L x -integrability, and the components ∇ t , x φ L decay rapidly with respect to x . In particu-lar, upon transforming ( A L , φ L )[0] into Coulomb gauge, it is meaningful to consider its MKG-CGevolution and its S norm. Then we prove the following technical Proposition 8.7.
Let ( A , φ ) be an admissible global solution to MKG-CG and let L : R + → R + be a Lorentz transformation of the form (8.15) for su ffi ciently small | d | . Let ( A L , φ L )[0] be thedata pair obtained from the Lorentz transformed solution ( A L , φ L ) . Assume that ( A L , φ L )[0] , whentransformed into the Coulomb gauge, results in a smooth global solution (cid:0) ˜ A L , ˜ φ L (cid:1) to MKG-CGsatisfying (cid:13)(cid:13)(cid:13)(cid:0) ˜ A L , ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) S < ∞ . Then we have for the original evolution ( A , φ ) that (cid:13)(cid:13)(cid:13) ( A , φ ) (cid:13)(cid:13)(cid:13) S ≤ C (cid:16)(cid:13)(cid:13)(cid:13)(cid:0) ˜ A L , ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) S , L (cid:17) . We defer the technical proof of Proposition 8.7 to the end of this subsection and first proveProposition 8.5 by combining Lemma 8.6 and Proposition 8.7.
Proof of Proposition 8.5.
In order to be able to apply Proposition 8.7, we have to use smooth solu-tions that are globally defined, because otherwise we cannot meaningfully apply a Lorentz trans-formation. In fact, we may exploit that by the preceding Lemma 8.4, the function Φ ∞ is compactlysupported, which means that its Fourier transform cannot also be compactly supported (we may ofcourse assume Φ ∞ [0] to be non-vanishing, since otherwise, the solution extends trivially in a globalfashion and cannot be singular). But then, truncating the data ( A ∞ , Φ ∞ )[0] in Fourier space as inProposition 5.1 and the discussion following it, we may construct a sequence of smooth Coulomb
22 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION data ( A n , φ n )[0] converging to ( A ∞ , Φ ∞ )[0], and if necessary, multiplying the φ n [0] in the resulting( A n , φ n )[0] by a small scalar λ n ∈ [0 ,
1] with λ n → n → ∞ , we may force that for all n ≥ E ( A n , φ n ) < E crit . Note that then the perturbation theory developed in Proposition 5.1 still applies in relation to( A ∞ , Φ ∞ ), since we have not changed the data for A n . This means that the data ( A n , φ n )[0] doadmit a global MKG-CG evolution by definition of E crit , and can thus be Lorentz transformed. Inorder to justify various conservation laws for the Lorentz transformed ( A n , φ n ), we observe that wemay also localize the data ( A n , φ n )[0] in physical space to a su ffi ciently large ball, using the argu-ment in Subsection 5.2 as well as [7] such that the Lorentz transformed solution also has compactsupport on bounded time slices, and we still have the above inequality (8.19) for the energy.We make the hypothesis that the momentum of ( A ∞ , Φ ∞ ) does not vanish. Then without loss ofgenerality, there exists γ > ffi ciently large n , we have(8.20) Z R (cid:18) X j = F n , j F n , j + Re (cid:0) ( ∂ t + iA n , ) φ n ( ∂ + iA n , ) φ n (cid:1)(cid:19) ( t , x ) dx ≥ γ, where F n ,αβ denote the curvature components of ( A n , φ n ). It su ffi ces to show that a suitable Lorentztransformation L of the form (8.15) exists such that the transformed solutions ( A Ln , φ Ln ) to theMaxwell-Klein-Gordon system have energies(8.21) E ( A Ln , φ Ln ) ≤ E crit − κ ( γ, A ∞ , Φ ∞ )uniformly in n for some κ ( γ, A ∞ , Φ ∞ ) >
0. Then, upon transforming ( A Ln , φ Ln )[0] into the Coulombgauge, we obtain a global solution to MKG-CG with a finite global S norm bound, and usingProposition 8.7, we can infer a global S norm bound for ( A n , φ n ) uniformly in n , which contradictsthat ( A ∞ , Φ ∞ ) is a singular solution. To implement this strategy, we combine the argument forProposition 4.10 in [10] with Lemma 8.6.By energy conservation for ( A Ln , φ Ln ), we have the relation14 E (cid:0) A Ln , φ Ln (cid:1) (0) = Z E (cid:0) A Ln , φ Ln (cid:1) ( t ) dt , where we recall that for a solution ( A , φ ) to the Maxwell-Klein-Gordon system the energy at time t ∈ R is given by E (cid:0) A , φ (cid:1) ( t ) = Z R (cid:18) X α,β F αβ + X α | D α φ | (cid:19) ( t , x ) dx . According to Lemma 8.6, we can write14 E (cid:0) A Ln , φ Ln (cid:1) (0) = I + I , where I = Z Z R (cid:18) X α,β F n ,αβ + X α (cid:12)(cid:12)(cid:12) ( ∂ α + iA n ,α ) φ n (cid:12)(cid:12)(cid:12) (cid:19)(cid:0) L ( t , x ) (cid:1) dx dt + d − d Z Z R (cid:18) X j = (cid:0) F n , j + F n , j (cid:1) + X α = (cid:12)(cid:12)(cid:12) ( ∂ α + iA n ,α ) φ n (cid:12)(cid:12)(cid:12) (cid:19)(cid:0) L ( t , x ) (cid:1) dx dt ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 123 and I = − d − d Z Z R (cid:18) X j = F n , j F n , j + Re (cid:0) ( ∂ t + iA n , ) φ n ( ∂ + iA n , ) φ n (cid:1)(cid:19)(cid:0) L ( t , x ) (cid:1) dx dt . We recall that the above integrands are evaluated at L ( t , x ) = (cid:18) t − dx √ − d , x − dt √ − d , x , x , x (cid:19) . Next we compute the derivative of I + I with respect to d . To this end we note that for a regularfunction f of compact support, it holds that (see page 173 in [10]) ∂∂ d Z R f (cid:0) L ( t , x ) (cid:1) dx = − − d ∂∂ t Z R x f (cid:0) L ( t , x ) (cid:1) dx . Using our assumption that the spatial components of A n as well as φ n are compactly supported onfixed time slices, we thus obtain ∂∂ d I ( d ) = − − d Z ∂∂ t Z R x (cid:18) X α,β F n ,αβ + X α (cid:12)(cid:12)(cid:12) ( ∂ α + iA n ,α ) φ n (cid:12)(cid:12)(cid:12) (cid:19)(cid:0) L ( t , x ) (cid:1) dx dt + d (1 − d ) Z Z R (cid:18) X j = (cid:0) F n , j + F n , j (cid:1) + X α = (cid:12)(cid:12)(cid:12) ( ∂ α + iA n ,α ) φ n (cid:12)(cid:12)(cid:12) (cid:19)(cid:0) L ( t , x ) (cid:1) dx dt − d (1 − d ) Z ∂∂ t Z R x (cid:18) X j = (cid:0) F n , j + F n , j (cid:1) + X α = (cid:12)(cid:12)(cid:12) ( ∂ α + iA n ,α ) φ n (cid:12)(cid:12)(cid:12) (cid:19)(cid:0) L ( t , x ) (cid:1) dx dt and ∂∂ d I ( d ) = − + d (1 − d ) Z Z R (cid:18) X j = F n , j F n , j + Re (cid:0) ( ∂ t + iA n , ) φ n ( ∂ + iA n , ) φ n (cid:1)(cid:19)(cid:0) L ( t , x ) (cid:1) dx dt + d (1 − d ) Z ∂∂ t Z R x (cid:18) X j = F n , j F n , j + Re (cid:0) ( ∂ t + iA n , ) φ n ( ∂ + iA n , ) φ n (cid:1)(cid:19)(cid:0) L ( t , x ) (cid:1) dx dt . But then ∂∂ d ( I + I ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d = = − Z R x (cid:18) X α,β F n ,αβ + X α (cid:12)(cid:12)(cid:12) ( ∂ α + iA n ,α ) φ n (cid:12)(cid:12)(cid:12) (cid:19) ( t , x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = t = − Z Z R (cid:18) X j = F n , j F n , j + Re (cid:0) ( ∂ t + iA n , ) φ n ( ∂ + iA n , ) φ n (cid:1)(cid:19) ( t , x ) dx dt . Using the weighted energy identity (8.5) (for R → ∞ ) and (8.20), we conclude that ∂∂ d ( I + I ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d = = − Z Z R (cid:18) X j = F n , j F n , j + Re (cid:0) ( ∂ t + iA n , ) φ n ( ∂ + iA n , ) φ n (cid:1)(cid:19) ( t , x ) dx dt ≤ − γ uniformly for all su ffi ciently large n . Also, by energy conservation for ( A n , φ n ), we have for all n that ( I + I )( d = = E ( A n , φ n ) < E crit .
24 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION
Hence, we can find a small d > E (cid:0) A Ln , φ Ln (cid:1) (0) ≤ E crit − κ uniformly for all su ffi ciently large n for some κ ≡ κ ( γ, A ∞ , Φ ∞ ) >
0, which yields (8.21) and thusfinishes the proof of Proposition 8.5. (cid:3)
We next state the analogous result to Proposition 8.5 when I + is infinite. Its proof essentiallyfollows the argument of the proof of Proposition 4.11 in [10] using the same modifications as in thepreceding proof of Proposition 8.5. Proposition 8.8.
Let ( A ∞ , Φ ∞ ) be as above. Assume that I + = [0 , ∞ ) . Suppose in addition that λ ( t ) ≥ λ > for all t ≥ . Then we have for k = , . . . , and all t ≥ that (8.22) Z R (cid:18) X j = F ∞ j F ∞ k j + Re (cid:0) D Φ ∞ D k Φ ∞ (cid:1)(cid:19) ( t , x ) dx = . It remains to give the proof of Proposition 8.7.
Proof of Proposition 8.7.
We are given an admissible global solution ( A , φ ) to MKG-CG and aLorentz transformation L : R + → R + of the form (8.15) for small d ∈ R . Applying the Lorentztransformation L to ( A , φ ), we obtain a global solution ( A L , φ L ) to the Maxwell-Klein-Gordon sys-tem. Next we define the gauge transform γ = X l = ∆ − (cid:0) ∂ l A Ll (cid:1) = ∆ − (cid:0) ∂ l A Ll (cid:1) and set ˜ φ L = e i γ φ L , ˜ A L α = A L α − ∂ α γ, α = , , . . . , . Then ( ˜ A L , ˜ φ L ) is in Coulomb gauge and a global solution to MKG-CG. By assumption we have that (cid:13)(cid:13)(cid:13)(cid:0) ˜ A L , ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) S < ∞ . Now the di ffi culty in controlling the S norm of ( A , φ ) is that this norm is far from invariant underthe operation of Lorentz transformations. Nonetheless, one can establish control over a certain setof norms of ( A , φ ) that are essentially invariant under Lorentz transformations, and which in turnimply control over the full S norm of ( A , φ ). We do this in the following observations. Observation 1:
For C = C ( L ) with C ( L ) → ∞ as L → Id, i.e. as d → , we have the bounds (cid:18)X k ∈ Z (cid:13)(cid:13)(cid:13) ∇ x P k Q [ k + C , k + C ] c φ L (cid:13)(cid:13)(cid:13) X , ∞ (cid:19) . (cid:13)(cid:13)(cid:13)(cid:0) ˜ A L , ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) S , (8.23) (cid:18)X k ∈ Z − ν k (cid:13)(cid:13)(cid:13) ∇ x P k Q [ k + C , k + C ] φ L (cid:13)(cid:13)(cid:13) L t L + x (cid:19) . (cid:13)(cid:13)(cid:13)(cid:0) ˜ A L , ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) S (8.24) for some ν > . Similarly for A L , we have the bounds (cid:18)X k ∈ Z (cid:13)(cid:13)(cid:13) ∇ x P k Q [ k + C , k + C ] c A L (cid:13)(cid:13)(cid:13) X , ∞ (cid:19) . (cid:13)(cid:13)(cid:13)(cid:0) ˜ A L , ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) S , (8.25) (cid:18)X k ∈ Z − (1 + ) k (cid:13)(cid:13)(cid:13) ∇ x P k Q [ k + C , k + C ] A L (cid:13)(cid:13)(cid:13) L t L + x (cid:19) . (cid:13)(cid:13)(cid:13)(cid:0) ˜ A L , ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) S . (8.26) ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 125
Moreover, we have the bounds (cid:18)X k ∈ Z (cid:13)(cid:13)(cid:13) ∇ x P k Q ≤ k + C φ L (cid:13)(cid:13)(cid:13) L ∞ t L x (cid:19) . (cid:13)(cid:13)(cid:13)(cid:0) ˜ A L , ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) S , (8.27) (cid:18)X k ∈ Z − k (cid:13)(cid:13)(cid:13) P k Q ≤ k + C φ L (cid:13)(cid:13)(cid:13) L t L ∞ x (cid:19) . (cid:13)(cid:13)(cid:13)(cid:0) ˜ A L , ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) S , (8.28) Here the implicit constants may also depend on (cid:13)(cid:13)(cid:13)(cid:0) ˜ A L , ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) S .Proof of Observation 1. We first derive suitable estimates on the gauge transform γ , which will thenallow us to obtain the claimed bounds in the statement of Observation 1. To this end we compute γ in terms of ˜ A L , for which we already have good bounds by assumption. Note that in view of (8.14),we have (cid:0) ˜ A L (cid:1) L − = A − ( ∇ t , x γ ) L − = A − ∇ t , x (cid:0) γ ( L − · ) (cid:1) and so we get γ ( L − · ) = − ∆ − ∂ l (cid:0)(cid:0) ˜ A L (cid:1) L − (cid:1) l . Thus, we find γ ( t , x ) = − (cid:16) ∆ − ∂ l (cid:0)(cid:0) ˜ A L (cid:1) L − (cid:1) l (cid:17) L ( t , x ) = − (cid:16) ∆ − ∂ l (cid:0)(cid:0) ˜ A L (cid:1) L − (cid:1) l (cid:17) ( L ( t , x )) . Now for any fixed dyadic frequency k ∈ Z , we can write (cid:16) P k ∆ − ∂ l (cid:0)(cid:0) ˜ A L (cid:1) L − (cid:1) l (cid:17) ( t , x ) = Z R m lk ( a ) (cid:0)(cid:0) ˜ A L (cid:1) L − (cid:1) l ( t , x − a ) da for suitable L x -functions m lk ( a ) with L x -mass ∼ − k , and further (cid:16) P k ∆ − ∂ l (cid:0)(cid:0) ˜ A L (cid:1) L − (cid:1) l (cid:17) L ( t , x ) = Z R m lk ( a ) (cid:16)(cid:0) L − (cid:1) t ˜ A L (cid:17) l (cid:0) ( t , x ) − L − (0 , a ) (cid:1) da . Also, if j ≤ k + C for suitable C = C ( L ), then Fourier localization to dyadic modulation 2 j andspatial frequency 2 k essentially commute with the Lorentz transformation, provided C is not toolarge depending on d , and so we have (cid:16) P k Q j ∆ − ∂ l (cid:0)(cid:0) ˜ A L (cid:1) L − (cid:1) l (cid:17) L ( t , x ) = P k Q j (cid:18)Z R m lk ( a ) (cid:16) P k + O (1) Q j + O (1) (cid:16)(cid:0) L − (cid:1) t ˜ A L (cid:17) l (cid:0) ( t , x ) − L − (0 , a ) (cid:1) da (cid:19) , where we note that the right hand side is a linear combination of all the components (cid:0) ˜ A L (cid:1) α . Thisimmediately implies for j ≤ k + C that2 j (cid:13)(cid:13)(cid:13) P k Q j ∇ x γ (cid:13)(cid:13)(cid:13) L t L x = j (cid:13)(cid:13)(cid:13) P k Q j ∇ x (cid:16) ∆ − ∂ l (cid:0)(cid:0) ˜ A L (cid:1) L − (cid:1) l (cid:17) L (cid:13)(cid:13)(cid:13) L t L x . (cid:13)(cid:13)(cid:13) P k ˜ A L (cid:13)(cid:13)(cid:13) X , ∞ . Similarly, one shows that for j > k + C we have2 j (cid:13)(cid:13)(cid:13) P k Q j ∇ x γ (cid:13)(cid:13)(cid:13) L t L x . k − j (cid:13)(cid:13)(cid:13) P j ∇ t , x ˜ A L (cid:13)(cid:13)(cid:13) X , ∞ . In fact, here, the very large modulation j then gets transferred to the frequency after Lorentz trans-form. Finally, for the expression P k Q [ k + C , k + C ] γ , the Lorentz transformation may lead to smallfrequencies . k , which is why we can only place the expression into L t L + x then via Bernstein, i.e. (cid:18)X k ∈ Z − (1 + ) k (cid:13)(cid:13)(cid:13) P k Q [ k + C , k + C ] ∇ t , x γ (cid:13)(cid:13)(cid:13) L t L + x (cid:19) . (cid:13)(cid:13)(cid:13) ˜ A L (cid:13)(cid:13)(cid:13) S .
26 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION
One also infers by similar reasoning that (cid:18)X k ∈ Z (cid:13)(cid:13)(cid:13) P k ∇ t , x γ (cid:13)(cid:13)(cid:13) L t L x ∩ L t ˙ W , x (cid:19) . (cid:13)(cid:13)(cid:13) ˜ A L (cid:13)(cid:13)(cid:13) S as well as (cid:13)(cid:13)(cid:13) P k Q ≤ k + C ∇ t , x γ (cid:13)(cid:13)(cid:13) L ∞ t L x . (cid:13)(cid:13)(cid:13) P k + O (1) ˜ A L (cid:13)(cid:13)(cid:13) L ∞ t L x . With these bounds on γ in hand, we can now start the derivation of the bounds for φ L = e − i γ ˜ φ L .For j ≤ k + C we write P k Q j (cid:0) e − i γ ˜ φ L (cid:1) = P k Q j (cid:0) P ≤ j − Q ≤ j − ( e − i γ ) ˜ φ L (cid:1) + P k Q j (cid:0) P ≤ j − Q > j − ( e − i γ ) ˜ φ L (cid:1) + P k Q j (cid:0) P > j − ( e − i γ ) ˜ φ L (cid:1) . (8.29)For the first term on the right, we have P k Q j (cid:0) P ≤ j − Q ≤ j − ( e − i γ ) ˜ φ L (cid:1) = P k Q j (cid:0) P ≤ j − Q ≤ j − ( e − i γ ) P k + O (1) Q j + O (1) ˜ φ L (cid:1) , and so we infer(8.30) 2 j (cid:13)(cid:13)(cid:13) ∇ t , x P k Q j (cid:0) P ≤ j − Q ≤ j − ( e − i γ ) ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) L t L x . (cid:13)(cid:13)(cid:13) P k ∇ t , x ˜ φ L (cid:13)(cid:13)(cid:13) X , ∞ . For the second term on the right hand side of (8.29), we write schematically P k Q j (cid:0) P ≤ j − Q > j − ( e − i γ ) ˜ φ L (cid:1) = − j P k Q j (cid:0) P ≤ j − Q > j − ( ∂ t γ e − i γ ) ˜ φ L (cid:1) and so we get from the preceding2 j (cid:13)(cid:13)(cid:13) ∇ t , x P k Q j (cid:0) P ≤ j − Q > j − ( e − i γ ) ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) L t L x . j · − j (cid:13)(cid:13)(cid:13) P ≤ j − Q > j − ( ∂ t γ e − i γ ) (cid:13)(cid:13)(cid:13) L t L x (cid:13)(cid:13)(cid:13) P k ∇ t , x ˜ φ L (cid:13)(cid:13)(cid:13) L ∞ t L x . (cid:13)(cid:13)(cid:13) ˜ A L (cid:13)(cid:13)(cid:13) S (cid:13)(cid:13)(cid:13) P k ∇ t , x ˜ φ L (cid:13)(cid:13)(cid:13) L ∞ t L x . (8.31)For the last term on the right hand side of (8.29), write it as P k Q j (cid:0) P > j − ( e − i γ ) ˜ φ L (cid:1) = P k Q j (cid:0) P [ j − , k − ( e − i γ ) ˜ φ L (cid:1) + P k Q j (cid:0) P [ k − , k + ( e − i γ ) ˜ φ L (cid:1) + P k Q j (cid:0) P > k + ( e − i γ ) ˜ φ L (cid:1) . (8.32)The first term on the right is bounded by(8.33) 2 j (cid:13)(cid:13)(cid:13) ∇ x P k Q j (cid:0) P [ j − , k − ( e − i γ ) ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) L t L x . j k − j (cid:13)(cid:13)(cid:13) P [ j − , k − ( ∇ x γ e − i γ ) (cid:13)(cid:13)(cid:13) L t L x (cid:13)(cid:13)(cid:13) ∇ x P k ˜ φ L (cid:13)(cid:13)(cid:13) L ∞ t L x . ( k − j ) (cid:13)(cid:13)(cid:13) ˜ A L (cid:13)(cid:13)(cid:13) S (cid:13)(cid:13)(cid:13) P k ∇ x ˜ φ L (cid:13)(cid:13)(cid:13) L ∞ t L x . For the second term on the right hand side of (8.32), we write it schematically as P k Q j (cid:0) P [ k − , k + ( e − i γ ) ˜ φ L (cid:1) = − k P k Q j (cid:0) P [ k − , k + ( ∇ x γ P ≤ k − Q ≤ k − ( e − i γ )) P ≤ k + ˜ φ L (cid:1) + − k P k Q j (cid:0) P [ k − , k + ( ∇ x γ P ≤ k − Q > k − ( e − i γ )) P ≤ k + ˜ φ L (cid:1) + − k P k Q j (cid:0) P [ k − , k + ( ∇ x γ P > k − ( e − i γ )) P ≤ k + ˜ φ L (cid:1) . (8.34) ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 127
Then we estimate the first term on the right of (8.34) by(8.35) 2 j (cid:13)(cid:13)(cid:13) ∇ x − k P k Q j (cid:0) P [ k − , k + ( ∇ x γ P ≤ k − Q ≤ k − ( e − i γ )) P ≤ k + ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) L t L x . j (cid:13)(cid:13)(cid:13) P [ k − , k + Q ≤ k + C ∇ x γ (cid:13)(cid:13)(cid:13) L ∞ t L x (cid:13)(cid:13)(cid:13) P ≤ k + Q ≤ k ˜ φ L (cid:13)(cid:13)(cid:13) L t L ∞ x + j (cid:13)(cid:13)(cid:13) P [ k − , k + ∇ x γ (cid:13)(cid:13)(cid:13) L t L + x (cid:13)(cid:13)(cid:13) P ≤ k + Q > k ˜ φ L (cid:13)(cid:13)(cid:13) L ∞ t L − x . ( j − k ) (cid:13)(cid:13)(cid:13) P k + O (1) ∇ x ˜ A L (cid:13)(cid:13)(cid:13) L ∞ t L x (cid:13)(cid:13)(cid:13) ˜ φ L (cid:13)(cid:13)(cid:13) S . Further, we get for the second term on the right of (8.34) that2 j (cid:13)(cid:13)(cid:13) ∇ x − k P k Q j (cid:0) P [ k − , k + ( ∇ x γ P ≤ k − Q > k − ( e − i γ )) P ≤ k + ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) L t L x . j − k (cid:13)(cid:13)(cid:13) P k + O (1) ∇ x γ (cid:13)(cid:13)(cid:13) L t L x (cid:13)(cid:13)(cid:13) ∂ t γ (cid:13)(cid:13)(cid:13) L t L x (cid:13)(cid:13)(cid:13) P ≤ k + ˜ φ L (cid:13)(cid:13)(cid:13) L ∞ t L x . j − k k (cid:13)(cid:13)(cid:13) P k + O (1) ∇ x γ (cid:13)(cid:13)(cid:13) L t L x (cid:13)(cid:13)(cid:13) ( ˜ A L , ˜ φ L ) (cid:13)(cid:13)(cid:13) S (cid:13)(cid:13)(cid:13) ˜ φ L (cid:13)(cid:13)(cid:13) S . (8.36)The third term on the right hand side of (8.34)2 − k P k Q j (cid:0) P [ k − , k + ( ∇ x γ P > k − ( e − i γ )) P ≤ k + ˜ φ L (cid:1) is handled similarly, which concludes the treatment of the contribution of the second term on theright hand side of (8.32), namely P k Q j (cid:0) P [ k − , k + ( e − i γ ) ˜ φ L (cid:1) . To treat the third term on the right hand side of (8.32), i.e. the high-high interaction term P k Q j (cid:0) P > k + ( e − i γ ) ˜ φ L (cid:1) , we write it schematically as P k Q j (cid:0) P > k + ( e − i γ ) ˜ φ L (cid:1) = X k > k + k = k + O (1) P k Q j (cid:0) P k ( e − i γ ) P k ˜ φ L (cid:1) = X k > k + k = k + O (1) − k P k Q j (cid:0) P k ( ∇ x γ e − i γ ) P k ˜ φ L (cid:1) and so we can estimate this by2 j (cid:13)(cid:13)(cid:13) ∇ x P k Q j (cid:0) P > k + ( e − i γ ) ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) L t L x . X k > k + k = k + O (1) ( j + k ) k − k k − k (cid:13)(cid:13)(cid:13) ∇ x γ (cid:13)(cid:13)(cid:13) L t L x (cid:13)(cid:13)(cid:13) P k ∇ x ˜ φ L (cid:13)(cid:13)(cid:13) L ∞ t L x . (8.37)Combining the bounds (8.30) – (8.37) and square-summing over k , the estimate (cid:18)X k ∈ Z (cid:13)(cid:13)(cid:13) ∇ t , x P k Q ≤ k + C φ L (cid:13)(cid:13)(cid:13) X , ∞ (cid:19) . (cid:13)(cid:13)(cid:13) ( ˜ A L , ˜ φ L ) (cid:13)(cid:13)(cid:13) S with implied constant also depending on (cid:13)(cid:13)(cid:13) ( ˜ A L , ˜ φ L ) (cid:13)(cid:13)(cid:13) S easily follows. We omit the estimate for (cid:18)X k ∈ Z (cid:13)(cid:13)(cid:13) ∇ t , x P k Q > k + C φ L (cid:13)(cid:13)(cid:13) X , ∞ (cid:19) as it is similar. This proves the first bound (8.23) in the statement of Observation 1.
28 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION
Next, we turn to the proof of the second bound (8.24) and consider P k Q [ k + C , k + C ] ( e − i γ ˜ φ L ) = P k Q [ k + C , k + C ] (cid:0) P ≤ k − Q ≤ k − ( e − i γ ) ˜ φ L (cid:1) + P k Q [ k + C , k + C ] (cid:0) P ≤ k − Q > k − ( e − i γ ) ˜ φ L (cid:1) + P k Q [ k + C , k + C ] (cid:0) P [ k − , k + ( e − i γ ) ˜ φ L (cid:1) + P k Q [ k + C , k + C ] (cid:0) P > k + ( e − i γ ) ˜ φ L (cid:1) . (8.38)Each of these terms is straightforward to estimate. For the first term on the right, we obtain (cid:13)(cid:13)(cid:13) P k Q [ k + C , k + C ] (cid:0) P ≤ k − Q ≤ k − ( e − i γ ) ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) L t L + x . (cid:13)(cid:13)(cid:13) P k + O (1) Q k + O (1) ˜ φ L (cid:13)(cid:13)(cid:13) L t L + x . ( ν − k (cid:13)(cid:13)(cid:13) P k ˜ φ L (cid:13)(cid:13)(cid:13) S . Also, we get (cid:13)(cid:13)(cid:13) P k Q [ k + C , k + C ] ( P ≤ k − Q > k − ( e − i γ ) ˜ φ L ) (cid:13)(cid:13)(cid:13) L t L + x . − k (cid:13)(cid:13)(cid:13) ∂ t γ (cid:13)(cid:13)(cid:13) L t L x (cid:13)(cid:13)(cid:13) P k + O (1) ˜ φ L (cid:13)(cid:13)(cid:13) L ∞ t L + x . ( ν − k (cid:13)(cid:13)(cid:13) ∂ t γ (cid:13)(cid:13)(cid:13) L t L x (cid:13)(cid:13)(cid:13) P k ˜ φ L (cid:13)(cid:13)(cid:13) S , and (cid:13)(cid:13)(cid:13) P k Q [ k + C , k + C ] (cid:0) P [ k − , k + ( e − i γ ) ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) L t L + x . − k (cid:13)(cid:13)(cid:13) ∇ x γ (cid:13)(cid:13)(cid:13) L t L x (cid:13)(cid:13)(cid:13) P ≤ k + O (1) ˜ φ L (cid:13)(cid:13)(cid:13) L ∞ t L + x . ( ν − k (cid:13)(cid:13)(cid:13) ∇ x γ (cid:13)(cid:13)(cid:13) L t L x X l ≤ k + O (1) ν ( l − k ) (cid:13)(cid:13)(cid:13) P l ˜ φ L (cid:13)(cid:13)(cid:13) S . The last term on the right hand side of (8.38) can be handled similarly. These estimates then yieldthe second inequality (8.24) of Observation 1.We also observe that the estimates on γ established earlier yield the required bounds (8.25) and(8.26) for A L = ˜ A L + ∇ t , x γ .Now we turn to the last bounds (8.27) and (8.28) in the statement of Observation 1. We onlyprove (8.27), the proof of (8.28) being similar. We write P k Q ≤ k + C φ L = P k Q ≤ k + C (cid:0) P ≤ k − ( e − i γ ) ˜ φ L (cid:1) + P k Q ≤ k + C (cid:0) P [ k − , k + ( e − i γ ) ˜ φ L (cid:1) + P k Q ≤ k + C (cid:0) P > k + ( e − i γ ) ˜ φ L (cid:1) . (8.39)The first term is directly bounded by(8.40) (cid:13)(cid:13)(cid:13) ∇ x P k Q ≤ k + C (cid:0) P ≤ k − ( e − i γ ) ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) L ∞ t L x . (cid:13)(cid:13)(cid:13) ∇ x P k + O (1) ˜ φ L (cid:13)(cid:13)(cid:13) L ∞ t L x . The second term on the right of (8.39) is a bit more complicated. We write it schematically as P k Q ≤ k + C (cid:0) P [ k − , k + ( e − i γ ) ˜ φ L (cid:1) = P k Q ≤ k + C (cid:0) − k P [ k − , k + ( ∇ x γ e − i γ ) P ≤ k + O (1) ˜ φ L (cid:1) = P k Q ≤ k + C (cid:0) − k P [ k − , k + ( ∇ x γ P ≤ k − Q ≤ k − ( e − i γ )) P ≤ k − ˜ φ L (cid:1) + P k Q ≤ k + C (cid:0) − k P [ k − , k + ( ∇ x γ P ≤ k − Q > k − ( e − i γ )) P ≤ k − ˜ φ L (cid:1) + P k Q ≤ k + C (cid:0) − k P [ k − , k + ( ∇ x γ P > k − ( e − i γ )) P ≤ k − ˜ φ L (cid:1) + P k Q ≤ k + C (cid:0) − k P [ k − , k + ( ∇ x γ P ≤ k − ( e − i γ )) P > k − ˜ φ L (cid:1) . (8.41) ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 129
Then we get for the first term of the last list of four terms (cid:13)(cid:13)(cid:13) ∇ x P k Q ≤ k + C (cid:0) − k P [ k − , k + ( ∇ x γ P ≤ k − Q ≤ k − ( e − i γ )) P ≤ k − ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) L ∞ t L x . (cid:13)(cid:13)(cid:13) P [ k − , k + Q ≤ k + C + ∇ x γ (cid:13)(cid:13)(cid:13) L ∞ t L x (cid:13)(cid:13)(cid:13) P ≤ k − ˜ φ L (cid:13)(cid:13)(cid:13) L ∞ t L ∞ x . (cid:13)(cid:13)(cid:13) P k + O (1) ˜ A L (cid:13)(cid:13)(cid:13) L ∞ t L x X l ≤ k − l (cid:13)(cid:13)(cid:13) P l ˜ φ L (cid:13)(cid:13)(cid:13) L ∞ t L x . (cid:13)(cid:13)(cid:13) P k + O (1) ∇ x ˜ A L (cid:13)(cid:13)(cid:13) S (cid:13)(cid:13)(cid:13) ˜ φ L (cid:13)(cid:13)(cid:13) S , (8.42)where we have taken advantage of our previous considerations on the structure of γ . For the secondterm on the above list (8.41), we get(8.43) (cid:13)(cid:13)(cid:13) ∇ x P k Q ≤ k + C (cid:0) − k P [ k − , k + ( ∇ x γ P ≤ k − Q > k − ( e − i γ )) P ≤ k − ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) L ∞ t L x . (cid:13)(cid:13)(cid:13) P ≤ k + O (1) ∇ x γ (cid:13)(cid:13)(cid:13) L ∞ t L + x (cid:13)(cid:13)(cid:13) P ≤ k − Q > k − ( e − i γ ) (cid:13)(cid:13)(cid:13) L ∞ t L + x (cid:13)(cid:13)(cid:13) P ≤ k − ˜ φ L (cid:13)(cid:13)(cid:13) L ∞ t L ∞− x . (cid:13)(cid:13)(cid:13) ˜ A L (cid:13)(cid:13)(cid:13) S X l ≤ k − − σ ( k − l ) (cid:13)(cid:13)(cid:13) P l ˜ φ L (cid:13)(cid:13)(cid:13) S . The term P k Q ≤ k + C (cid:0) − k P [ k − , k + ( ∇ x γ P > k − ( e − i γ )) P ≤ k − ˜ φ L (cid:1) is handled similarly. Finally, we have (cid:13)(cid:13)(cid:13) ∇ x P k Q ≤ k + C (cid:0) − k P [ k − , k + ( ∇ x γ P ≤ k − ( e − i γ )) P > k − ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) L ∞ t L x . (cid:13)(cid:13)(cid:13) P k + O (1) ∇ x γ (cid:13)(cid:13)(cid:13) L ∞ t L x (cid:13)(cid:13)(cid:13) P k + O (1) ˜ φ L (cid:13)(cid:13)(cid:13) L ∞ t L x . (8.44)The bounds (8.40) – (8.44) su ffi ce to perform the square summation over k in the last inequality ofObservation 1. The last term on the right hand side of (8.39) P k Q ≤ k + C (cid:0) P > k + ( e − i γ ) ˜ φ L (cid:1) is treated similarly and hence omitted here. (cid:3) Observation 2:
We have the bound X k ≤ k − k (cid:13)(cid:13)(cid:13) P k Q ±≤ k + C φ L P k Q ±≤ k + C ∇ t , x φ L (cid:13)(cid:13)(cid:13) L t L x . (cid:13)(cid:13)(cid:13) ( ˜ A L , ˜ φ L ) (cid:13)(cid:13)(cid:13) S . Moreover, for any L -space-time integrable weight function m ( a ) , a ∈ R + , we have X k ≤ k − k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z R + m ( a ) P k Q ±≤ k + C φ L ( · − a ) P k Q ±≤ k + C ∇ t , x φ L ( · − a ) da (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L t L x . (cid:13)(cid:13)(cid:13) ( ˜ A L , ˜ φ L ) (cid:13)(cid:13)(cid:13) S . Similar bounds can be obtained upon replacing one or more factors by A L . We make the crucialobservation that these bounds are essentially invariant under mild Lorentz transformations. Thus,we infer similar bounds for A and φ .Proof of Observation 2. Here one places the low frequency input P k Q ±≤ k + C φ L into L t L ∞ x and the high frequency input P k Q ±≤ k + C ∇ t , x φ L into L ∞ t L x by using Observation 1. (cid:3)
30 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION
Using Observation 1 and Observation 2, we can now move toward controlling the norm k ( A , φ ) k S .From above, we know a priori that we control (cid:18)X k ∈ Z (cid:13)(cid:13)(cid:13) P k Q [ k + C , k + C ] c ∇ t , x φ L (cid:13)(cid:13)(cid:13) X , ∞ (cid:19) as well as norms of the form X k ≤ k − k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z R + m ( a ) P k Q ±≤ k + C φ L ( · − a ) P k Q ±≤ k + C ∇ t , x φ L ( · − a ) da (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L t L x ! / . The latter has the crucial divisibility property, i.e. for any δ > R into intervals I , I , . . . , I N , with N depending on the size of the norm as well as δ such that we havefor each n = , . . . , N that X k ≤ k − k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) χ I n ( t ) Z R + m ( a ) P k Q ±≤ k + C φ L ( · − a ) P k Q ±≤ k + C ∇ t , x φ L ( · − a ) da (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L t L x ! / ≤ δ. Of course, we get a similar statement for weakened versions of the former norm such as (cid:18)X k ∈ Z (cid:13)(cid:13)(cid:13) P k Q [ k − C , k + C ] ∇ t , x φ L (cid:13)(cid:13)(cid:13) X , ∞ (cid:19) . In order to infer the desired S norm bound on ( A , φ ), we shall partition the time axis R into finitelymany intervals I , . . . , I N , whose number depends on (cid:13)(cid:13)(cid:13)(cid:0) ˜ A L , ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) S and such that on each of these I n , we can infer via a direct bootstrap argument a bound on k ( A , φ ) k S ( I n × R ) . This will then su ffi ce to obtain the desired bound on k ( A , φ ) k S ( R × R ) . We do this in two steps, whichwe outline below. Step 1:
Given δ > and δ > , using the known a priori bound on (cid:13)(cid:13)(cid:13)(cid:0) ˜ A L , ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) S and choosing theintervals I n suitably as above (whose number will depend on δ , δ , as well as the assumed boundon (cid:13)(cid:13)(cid:13)(cid:0) ˜ A L , ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) S ) , we infer from the equation for A, upon writingA | I n = A f ree , ( I n ) + A nonlin , ( I n ) , that there exists a decompositionA nonlin , ( I n ) = A nonlin , ( I n ) + A nonlin , ( I n ) , where we have schematicallyA nonlin = X k (cid:3) − P k Q k + O (1) ( P k + O (1) Q < k − C φ ∇ x P k + O (1) Q < k − C φ ) , while we also have the bound (cid:13)(cid:13)(cid:13) A nonlin (cid:13)(cid:13)(cid:13) ℓ S ( I n × R ) ≤ δ + δ (cid:16) k ( A , φ ) k S ( I n × R ) + k ( A , φ ) k S ( I n × R ) (cid:17) for all n = , , . . . , N. Moreover, it holds that (cid:13)(cid:13)(cid:13) A nonlin (cid:13)(cid:13)(cid:13) S ( I n × R ) ≤ δ for all n = , , . . . , N. ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 131
The idea behind this bound is to insert it into the equation for φ and pick δ , δ small dependingon E crit . Proof of Step 1.
We proceed as in the proof of Lemma 7.6 and write the source term in the equationfor A in the schematic form (cid:3) A i = ∆ − ∇ r N ir (cid:0) φ, φ (cid:1) + A | φ | . Localizing this to frequency k =
0, we write the right hand side in the form P (cid:16) ∆ − ∇ r N ir (cid:0) φ, φ (cid:1) + A | φ | (cid:17) = P (cid:18) X k , k ∆ − ∇ r N ir (cid:0) P k φ, P k φ (cid:1) + X k , k , k P k AP k φ P k φ (cid:19) . We first deal with the quadratic null form term and reduce this to moderate frequencies by observingthat for C = C ( δ ) large enough, we obtain (for a suitable absolute constant σ independent of allother constants) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X | k | > C , k P (cid:16) ∆ − ∇ r N ir ( P k φ, P k φ ) (cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N ≤ δ X k > C − σ | k | (cid:13)(cid:13)(cid:13) P k φ (cid:13)(cid:13)(cid:13) S . Generalizing to arbitrary output frequencies, one easily gets from here the bound X k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X | k − k | > C , k P k (cid:16) ∆ − ∇ r N ir (cid:0) P k φ, P k φ (cid:1)(cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N k . δ (cid:13)(cid:13)(cid:13) φ (cid:13)(cid:13)(cid:13) S . Next, we pick C = C ( E crit ) such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P (cid:16) X | k | , | k | < C ∆ − ∇ r N ir (cid:0) P k Q > C φ, P k φ ) (cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N ≤ δ X | k | , | k | < C (cid:13)(cid:13)(cid:13) P k φ (cid:13)(cid:13)(cid:13) S k (cid:13)(cid:13)(cid:13) P k φ (cid:13)(cid:13)(cid:13) S k , and generalizing to general output frequencies, we then reduce to X k P k (cid:18) X | k , − k | < C ∆ − ∇ r N ir (cid:0) P k Q ≤ k + C φ, P k Q ≤ k + C φ (cid:1)(cid:19) . Depending on our choice of C , we may assume the Lorentz transform L to be chosen su ffi cientlyclose to the identity, i.e. | d | su ffi ciently small, such that according to Observation 1 we have (cid:18) X k (cid:13)(cid:13)(cid:13) P k Q [ k − C , k + C ] ∇ t , x φ (cid:13)(cid:13)(cid:13) X , ∞ (cid:19) . (cid:13)(cid:13)(cid:13) ( ˜ A L , ˜ φ L ) (cid:13)(cid:13)(cid:13) S . As observed before, this norm has the divisibility property, so that restricting to suitable time inter-vals I n , n = , . . . , N , which form a partition of the time axis R , we may assume (cid:18)X k (cid:13)(cid:13)(cid:13) P k Q [ k − C , k + C ] ∇ t , x φ (cid:13)(cid:13)(cid:13) X , ∞ ( I n × R ) (cid:19) ≤ δ for all n = , . . . , N . But then we easily infer the bound (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P (cid:18) X | k | , | k | < C ∆ − ∇ r N ir (cid:0) P k Q [ k − C , k + C ] φ, P k φ (cid:1)(cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N ( I n × R ) . (cid:13)(cid:13)(cid:13) P k Q [ k − C , k + C ] φ (cid:13)(cid:13)(cid:13) L t L x ( I n × R ) (cid:13)(cid:13)(cid:13) P k φ (cid:13)(cid:13)(cid:13) L t L ∞ x ≤ δ (cid:13)(cid:13)(cid:13) P k φ (cid:13)(cid:13)(cid:13) S k ,
32 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION and this su ffi ces again, after generalizing this to arbitrary output frequency. In fact, we get X k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P k (cid:18) X | k , − k | < C ∆ − ∇ r N ir (cid:0) P k Q [ k − C , k + C ] φ, P k φ (cid:1)(cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N k ( I n × R ) . (cid:18) X k (cid:13)(cid:13)(cid:13) P k Q [ k − C , k + C ] ∇ t , x φ (cid:13)(cid:13)(cid:13) X , ∞ ( I n × R ) (cid:19) (cid:13)(cid:13)(cid:13) φ (cid:13)(cid:13)(cid:13) S ( I n × R ) ≤ δ (cid:13)(cid:13)(cid:13) φ (cid:13)(cid:13)(cid:13) S ( I n × R ) . We have now reduced to the expression X k P k (cid:18) X | k , − k | < C ∆ − ∇ r N ir (cid:0) P k Q ≤ k − C φ, P k Q ≤ k − C φ (cid:1)(cid:19) . The last reduction here consists in removing extremely small angular separation between the inputs P k Q ≤ k − C φ, P k Q ≤ k − C φ. Thus, there exists C = C ( δ ) such that we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P (cid:18) X | k , | < C ∆ − ∇ r N ir (cid:0) P k Q ≤ k − C φ, P k Q ≤ k − C φ (cid:1)(cid:19) ′ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N ≤ δ X | k , | < C (cid:13)(cid:13)(cid:13) P k φ (cid:13)(cid:13)(cid:13) S k (cid:13)(cid:13)(cid:13) P k φ (cid:13)(cid:13)(cid:13) S k , where the prime indicates that the inputs are reduced to have closely aligned Fourier supports ofangular separation C − . Finally, we write X | k , | < C P (cid:18) ∆ − ∇ r N ir (cid:0) P k Q ≤ k − C φ, P k Q ≤ k − C φ (cid:1)(cid:19) = X | k , | < C P (cid:18) ∆ − ∇ r N ir (cid:0) P k Q ≤ k − C φ, P k Q ≤ k − C φ (cid:1)(cid:19) ′ + X | k , | < C P (cid:18) ∆ − ∇ r N ir (cid:0) P k Q ≤ k − C φ, P k Q ≤ k − C φ (cid:1)(cid:19) ′′ , where the second term is of the form A nonlin as required for Step 1. In fact, the angular separationof the inputs and small modulation forces the output to have modulation ∼
1. Moreover, replacingthe output frequency by k and square-summing over k results in a small norm due to the fact that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X | k , | < C P (cid:18) ∆ − ∇ r N ir (cid:0) P k Q ≤ k − C φ, P k Q ≤ k − C φ (cid:1)(cid:19) ′′ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N ( I n × R ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X | k , | < C P (cid:18) ∆ − ∇ r N ir (cid:0) P k Q ≤ k − C φ, P k Q ≤ k − C φ ) (cid:19) ′′ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L t L x ( I n × R ) and we can then take advantage of Observation 2 to obtain (cid:18) X k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X | k , − k | < C P k (cid:18) ∆ − ∇ r N ir (cid:0) P k Q ≤ k − C φ, P k Q ≤ k − C φ (cid:1)(cid:19) ′′ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N k ( I n × R ) (cid:19) ≤ δ by choosing the intervals I n suitably. The cubic term P k , , P k AP k φ P k φ is handled similarly. (cid:3) ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 133
Step 2:
Choosing the time intervals I n suitably as in Step 1, we obtain the equation X k ∈ Z (cid:3) A free < k P k φ = F , where we have k F k N ( I n × R ) ≤ δ + δ (cid:0) k ( A , φ ) k S ( I n × R ) + (cid:13)(cid:13)(cid:13) ( A , φ ) (cid:13)(cid:13)(cid:13) S ( I n × R ) (cid:1) , where A f ree is the free wave evolution of the data for A at the beginning endpoint of I n .Proof of Step 2. This to a large extent mimics the argument for the proof of Proposition 7.5. In fact,we recall from there that we can write F = P k ∈ Z F k with F k = − iP k (cid:0) A f ree ≥ k ,ν ∂ ν φ (cid:1) − [ P k , (cid:3) A free < k ] φ − P k (cid:0) ( (cid:3) A − (cid:3) A free ) φ (cid:1) + P k (cid:0) (cid:3) A free < k φ + i (cid:0) A f ree ≥ k ,ν ∂ ν φ (cid:1) − (cid:3) A free φ (cid:1) . As usual, we treat each term separately.
First term . Similar to the proof of Proposition 7.5, we reduce it to − iP k (cid:0) P k + O (1) A f ree ,ν ∂ ν P k + O (1) φ (cid:1) up to terms satisfying the conclusion of Step 2. Then using divisibility for the norm X k ∈ Z − k (cid:13)(cid:13)(cid:13) P k A f ree ,ν (cid:13)(cid:13)(cid:13) L t L ∞ x as well as the inequality (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k ∈ Z − iP k (cid:16) P k + O (1) A f ree ,ν ∂ ν P k + O (1) φ (cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N ( I n × R ) . (cid:18) X k ∈ Z − k (cid:13)(cid:13)(cid:13) χ I n P k A f ree ,ν (cid:13)(cid:13)(cid:13) L t L ∞ x (cid:19) k φ k S ( I n × R ) , we get the conclusion of Step 2 by choosing the intervals I n suitably and by subdividing the intervalsobtained from Step 1, if necessary. Second term . This is handled like the first term, since it can be written in the form X k ˜ P k (cid:0) i ∇ x A f ree < k ,ν ∂ ν φ (cid:1) + ˜ P k (cid:0) ∇ x (( A f ree < k ) ) φ (cid:1) . Third term . As usual this term is the most di ffi cult one, since it contains X k ∈ Z iP < k A nonlin ν ∂ ν P k φ. We essentially follow the reductions performed in the proof of Proposition 7.5, whence we shall becorrespondingly brief.
Reduction to H ∗ N lowhi . Using the same notation as in that proof and restricting to frequency k = (cid:13)(cid:13)(cid:13) N lowhi (cid:0) P < A nonlin , ( I n ) , P φ (cid:1) − H ∗ N lowhi (cid:0) P < A nonlin , ( I n ) , P φ (cid:1)(cid:13)(cid:13)(cid:13) N ( I n × R ) . (cid:13)(cid:13)(cid:13) A nonlin , ( I n ) (cid:13)(cid:13)(cid:13) S ( I n × R ) (cid:13)(cid:13)(cid:13) P φ (cid:13)(cid:13)(cid:13) S ( I n × R ) . Hence, replacing the output frequency by general k ∈ Z and square-summing gives the bound . k ( A , φ ) k S ( I n × R ) (cid:16) δ + δ (cid:0) k ( A , φ ) k S ( I n × R ) + k ( A , φ ) k S ( I n × R ) (cid:1)(cid:17) ,
34 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION which is of the desired form. This then reduces the estimate of N lowhi (cid:0) P < A nonlin , ( I n ) , P φ (cid:1) − H ∗ N lowhi (cid:0) P < A nonlin , ( I n ) , P φ (cid:1) to the contribution of P < A nonlin , ( I n ) , whose explicit form we recall from Step 1. This means wehave to estimate the expression (cid:0) − H ∗ (cid:1)(cid:18) X k < (cid:3) − P k Q k + O (1) (cid:0) P k + O (1) Q ≤ k − C φ ∇ x P k + O (1) Q ≤ k − C φ (cid:1) ∇ t , x P φ (cid:19) . The idea here is to use the a priori bounds from Observation 1 and Observation 2 to arrive at therequired estimate. For this, we split the above expression into the following (cid:0) − H ∗ (cid:1)(cid:18) X k < (cid:3) − P k Q k + O (1) (cid:0) P k + O (1) Q ≤ k − C φ ∇ x P k + O (1) Q ≤ k − C φ (cid:1) ∇ t , x P φ (cid:19) = (cid:0) − H ∗ (cid:1)(cid:18) X k < (cid:3) − P k Q k + O (1) (cid:0) P k + O (1) Q ≤ k − C φ ∇ x P k + O (1) Q ≤ k − C φ (cid:1) Q [ C , C ] ∇ t , x P φ (cid:19) + (cid:0) − H ∗ (cid:1)(cid:18) X k < (cid:3) − P k Q k + O (1) (cid:0) P k + O (1) Q ≤ k − C φ ∇ x P k + O (1) Q ≤ k − C φ (cid:1) Q [ k − C , C ] ∇ t , x P φ (cid:19) + (cid:0) − H ∗ (cid:1)(cid:18) X k < (cid:3) − P k Q k + O (1) (cid:0) P k + O (1) Q ≤ k − C φ ∇ x P k + O (1) Q ≤ k − C φ (cid:1) Q ≤ k − C ∇ t , x P φ (cid:19) + (cid:0) − H ∗ (cid:1)(cid:18) X k < (cid:3) − P k Q k + O (1) (cid:0) P k + O (1) Q ≤ k − C φ ∇ x P k + O (1) Q ≤ k − C φ (cid:1) Q > C ∇ t , x P φ (cid:19) ≡ I + II + III + IV . We now estimate each of the terms on the right in turn.
Estimate of term I . We distinguish between very small k and k = O (1). In the latter case, weschematically estimate the term in the following fashion. We shall suppress the distinction betweenspace-time translates of φ and φ , as our norms are invariant under these, and also keep in mind thatthe operator (cid:3) − P k Q k + O (1) is given by (space-time) convolution with a kernel of L -mass ∼ − k . Then we get in case k = O (1), (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:0) − H ∗ (cid:1)(cid:18) X k < (cid:3) − P k Q k + O (1) (cid:0) P k + O (1) Q ≤ k − C φ ∇ x P k + O (1) Q ≤ k − C φ (cid:1) Q [ C , C ] ∇ t , x P φ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N ( I n × R ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:0) − H ∗ (cid:1)(cid:18) X k < (cid:3) − P k Q k + O (1) (cid:0) P k + O (1) Q ≤ k − C φ ∇ x P k + O (1) Q ≤ k − C φ (cid:1) Q [ C , C ] ∇ t , x P φ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L t L x ( I n × R ) . − k (cid:13)(cid:13)(cid:13) P k + O (1) Q ≤ k − C φ (cid:13)(cid:13)(cid:13) L t L ∞ x (cid:13)(cid:13)(cid:13) ∇ x P k + O (1) Q ≤ k − C φ Q [ C , C ] ∇ t , x P φ (cid:13)(cid:13)(cid:13) L t L x . Here the second factor is essentially invariant under mild Lorentz transformations, and so we get(up to changing the meaning of the constants slightly) (cid:13)(cid:13)(cid:13) ∇ x P k + O (1) Q ≤ k − C φ Q [ C , C ] ∇ t , x P φ (cid:13)(cid:13)(cid:13) L t L x . (cid:13)(cid:13)(cid:13) ∇ x P k + O (1) Q ≤ k − C φ L Q [ C , C ] ∇ t , x P ≤ O (1) φ L (cid:13)(cid:13)(cid:13) L t L x . We estimate the last norm using Observation 1, resulting in the bound (cid:13)(cid:13)(cid:13) ∇ x P k + O (1) Q ≤ k − C φ Q [ C , C ] ∇ t , x P φ (cid:13)(cid:13)(cid:13) L t L x . (cid:13)(cid:13)(cid:13) ∇ x P k + O (1) Q ≤ k − C φ L (cid:13)(cid:13)(cid:13) L ∞ t L − x (cid:13)(cid:13)(cid:13) Q [ C , C ] ∇ t , x P ≤ O (1) φ L (cid:13)(cid:13)(cid:13) L t L + x ≤ C (cid:16)(cid:13)(cid:13)(cid:13)(cid:0) ˜ A L , ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) S (cid:17) . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 135
Then by divisibility of the norm (cid:18) X l ∈ Z − l (cid:16) X k − l = O (1) (cid:13)(cid:13)(cid:13) ∇ x P k + O (1) Q ≤ k − C φ Q [ l − C , l + C ] ∇ t , x P l φ (cid:13)(cid:13)(cid:13) L t L x (cid:17) (cid:19) , we arrive upon suitable choice of the intervals I n at the conclusion that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X l ∈ Z (cid:0) − H ∗ (cid:1)(cid:18) X | k − l | = O (1) (cid:3) − P k Q k + O (1) (cid:0) P k + O (1) Q ≤ k − C φ · ∇ x P k + O (1) Q ≤ k − C φ (cid:1) Q [ l + C , l + C ] ∇ t , x P l φ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N ( I n × R ) ≤ δ k φ k S ( I n × R ) . This completes the contribution of the term I when k = O (1). On the other hand, when k ≪ −
1, thesmallness gain comes directly from k . Indeed, we can then estimate (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:0) − H ∗ (cid:1)(cid:18) X k ≪− (cid:3) − P k Q k + O (1) (cid:0) P k + O (1) Q ≤ k − C φ ∇ x P k + O (1) Q ≤ k − C φ (cid:1) Q [ C , C ] ∇ t , x P φ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N ( I n × R ) . X k ≪− − k (cid:13)(cid:13)(cid:13) P k + O (1) Q ≤ k − C φ (cid:13)(cid:13)(cid:13) L ∞ t L ∞ x ( I n × R ) (cid:13)(cid:13)(cid:13) ∇ x P k + O (1) Q ≤ k − C φ (cid:13)(cid:13)(cid:13) L t L ∞ x ( I n × R ) (cid:13)(cid:13)(cid:13) Q [ C , C ] ∇ t , x P φ (cid:13)(cid:13)(cid:13) L t L x ( I n × R ) . X k ≪− k (cid:13)(cid:13)(cid:13) P φ (cid:13)(cid:13)(cid:13) S ( I n × R ) k φ k S ( I n × R ) ≤ δ (cid:13)(cid:13)(cid:13) P φ (cid:13)(cid:13)(cid:13) S ( I n × R ) . Replacing P φ by P l φ , l ∈ Z , and square-summing over l results in the desired bound. This com-pletes the estimate for term I . Estimate of term II . Here we use the bound (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:0) − H ∗ (cid:1)(cid:18) X k < (cid:3) − P k Q k + O (1) (cid:0) P k + O (1) Q ≤ k − C φ ∇ x P k + O (1) Q ≤ k − C φ (cid:1) Q [ k − C , C ] ∇ t , x P φ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N ( I n × R ) . X k < l ∈ [ k − C , C ] − k (cid:13)(cid:13)(cid:13) P k + O (1) Q ≤ k − C φ (cid:13)(cid:13)(cid:13) L t L ∞ x ( I n × R ) (cid:13)(cid:13)(cid:13) ∇ x P k + O (1) Q ≤ k − C φ (cid:13)(cid:13)(cid:13) L ∞ t L ∞ x ( I n × R ) (cid:13)(cid:13)(cid:13) Q l ∇ t , x P φ (cid:13)(cid:13)(cid:13) L t L x ( I n × R ) . Now if we further restrict the above term to | k − l | ≫
1, we easily bound it by ≤ δ (cid:13)(cid:13)(cid:13) φ (cid:13)(cid:13)(cid:13) S ( I n × R ) (cid:13)(cid:13)(cid:13) P φ (cid:13)(cid:13)(cid:13) X , ∞ ( I n × R ) ≤ δ k φ k S ( I n × R ) , which is as desired. On the other hand, when restricting the modulation of Q [ k − C , C ] ∇ t , x P φ to l = k + O (1), we use the fact that for k ≪ − (cid:13)(cid:13)(cid:13) ∇ x P k + O (1) Q ≤ k − C φ Q k + O (1) ∇ t , x P φ (cid:13)(cid:13)(cid:13) L t L x . (cid:13)(cid:13)(cid:13) ∇ x P k + O (1) Q ≤ k − C φ L Q k + O (1) ∇ t , x P φ L (cid:13)(cid:13)(cid:13) L t L x . (cid:13)(cid:13)(cid:13) P k + O (1) Q ≤ k − C ∇ x φ L (cid:13)(cid:13)(cid:13) L ∞ t L ∞ x (cid:13)(cid:13)(cid:13) Q k + O (1) ∇ t , x P φ L (cid:13)(cid:13)(cid:13) L t L x . Then changing the frequency 0 to general m ∈ Z and using Observation 1, we infer X k < m − C − k (cid:13)(cid:13)(cid:13) ∇ x P k + O (1) Q ≤ k − C φ Q k + O (1) ∇ t , x P m φ (cid:13)(cid:13)(cid:13) L t L x ≤ C (cid:16)(cid:13)(cid:13)(cid:13)(cid:0) ˜ A L , ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) S (cid:17) .
36 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION
Also, the square-sum norm on the left has the divisibility property, whence by restricting to suitabletime intervals I n , we may arrange it to be ≪ δ . Finally, we infer the bound X m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:0) − H ∗ (cid:1)(cid:18) X k < m − C (cid:3) − P k Q k + O (1) (cid:0) P k + O (1) Q ≤ k − C φ ∇ x P k + O (1) Q ≤ k − C φ (cid:1) Q k + O (1) ∇ t , x P m φ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N ( I n × R ) . (cid:18) X k − k (cid:13)(cid:13)(cid:13) P k + O (1) Q ≤ k − C φ (cid:13)(cid:13)(cid:13) L t L ∞ x (cid:19) (cid:18) X k < m − C − k (cid:13)(cid:13)(cid:13) ∇ x P k + O (1) Q ≤ k − C φ Q k + O (1) ∇ t , x P m φ (cid:13)(cid:13)(cid:13) L t L x (cid:19) ≤ δ k φ k S ( I n × R ) . Estimate of term III . This follows the same pattern as for term I , by placing the product ∇ x P k + O (1) Q ≤ k − C φ Q ≤ k − C ∇ t , x P φ into L t L x and using Observation 2. Estimate of term IV . Here one places (cid:3) − P k Q k + O (1) (cid:0) P k + O (1) Q ≤ k − C φ ∇ x P k + O (1) Q ≤ k − C φ (cid:1) into L t L ∞ x and Q > C ∇ t , x P φ into L t L x , keeping in mind that C ≫ C = C ( E crit ) is very large. Reduction to H ∗ N lowhi (cid:0) H P < A nonlin , ( I n ) , P φ (cid:1) . To begin with, recall the notation from the proof ofProposition 7.5 for the definition of the symbol H applied to bilinear expressions. To reduce to thisterm, we need to estimate the di ff erence (cid:13)(cid:13)(cid:13) H ∗ N lowhi (cid:0) P < A nonlin , ( I n ) , P φ (cid:1) − H ∗ N lowhi (cid:0) H P < A nonlin , ( I n ) , P φ (cid:1)(cid:13)(cid:13)(cid:13) N ( I n × R ) . Here we recall that H k M ( φ, ψ ) = X j ≤ k + C Q j M (cid:0) Q ≤ j − C φ, Q ≤ j − C ψ (cid:1) as well as H M (cid:0) φ, ψ (cid:1) = X k ≤ k , − C , k ≤ min { k , k }− C H k M (cid:0) P k φ, P k ψ (cid:1) . Then write for the spatial components of (cid:0) I − H (cid:1) P < A nonlin , ( I n ) , (cid:0) I − H (cid:1) P < A nonlin , ( I n ) = X k < k > max { k , k }− C (cid:3) − P k P x (cid:0) P k φ ∇ x P k φ (cid:1) + X k ≤ k , − Cj > k + C (cid:3) − P k Q j P x (cid:0) P k φ ∇ x P k φ (cid:1) + X k ≤ k , − Cj ≤ k + C (cid:3) − P k Q j P x (cid:0) P k Q > j − C φ ∇ x P k φ (cid:1) + X k ≤ k , − Cj ≤ k + C (cid:3) − P k Q j P x (cid:0) P k Q ≤ j − C φ ∇ x P k Q > j − C φ (cid:1) . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 137
For the first term on the right, employing notation introduced in [22] and also used in the proof ofProposition 7.5, we get upon further restricting to (cid:12)(cid:12)(cid:12) max { k , } − min { k , } (cid:12)(cid:12)(cid:12) ≫ , the smallness gain (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k < , k ≥ max { k , k }− C (cid:3) − P k P x (cid:0) P k φ ∇ x P k φ (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z ≪ δ (cid:13)(cid:13)(cid:13) φ (cid:13)(cid:13)(cid:13) S and the corresponding contribution to H ∗ N lowhi (cid:0) ( I − H ) P < A nonlin , ( I n ) , P φ (cid:1) can then be boundedwith respect to (cid:13)(cid:13)(cid:13) · (cid:13)(cid:13)(cid:13) N ( I n × R ) by ≤ δ (cid:13)(cid:13)(cid:13) φ (cid:13)(cid:13)(cid:13) S (cid:13)(cid:13)(cid:13) P φ (cid:13)(cid:13)(cid:13) S , which upon replacing 0 by general frequencies and square summing gives the desired bound. Sim-ilarly, for the remaining terms on the right above, one may reduce to k , = k + O (1), see estimate(134) in [22]. Finally, in each of these terms, we may reduce the output to modulation ∼ k , sinceelse one gains smallness due to the null form structure for (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H ∗ N lowhi (cid:0) P < A nonlin , ( I n ) , P φ (cid:1) − H ∗ N lowhi (cid:0) H P < A nonlin , ( I n ) , P φ (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N ( I n × R ) . Thus we have now reduced to estimating (and gaining a smallness factor) for the schematic expres-sion X k < , k , = k + O (1) (cid:3) − P k Q k + O (1) P x (cid:0) P k φ ∇ x P k φ (cid:1) ∂ ν Q ≤ k − C P φ. Here we can suppress the operator (cid:3) − P k Q k + O (1) , which is given by convolution with a space-timekernel of L -norm ∼ − k , and then schematically estimate the preceding via (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k < k , = k + O (1) (cid:3) − P k Q k + O (1) P x (cid:0) P k φ ∇ x P k φ (cid:1) ∂ ν Q ≤ k − C P φ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N ( I n × R ) . X k = k + O (1) < O (1) − k (cid:13)(cid:13)(cid:13) P k φ (cid:13)(cid:13)(cid:13) L t L ∞ x ( I n × R ) (cid:13)(cid:13)(cid:13) ∇ x P k φ Q ≤ k − C P φ (cid:13)(cid:13)(cid:13) L t L x ( I n × R ) . Here we exploit Lorentz invariance of the norm of the right factor to obtain X k < − k (cid:13)(cid:13)(cid:13) ∇ x P k φ Q ≤ k − C P φ (cid:13)(cid:13)(cid:13) L t L x . X k < − k (cid:13)(cid:13)(cid:13) ∇ t , x ( P k φ ) L Q ≤ k − C ( P φ ) L (cid:13)(cid:13)(cid:13) L t L x . (cid:13)(cid:13)(cid:13) ( ˜ A L , ˜ φ L ) (cid:13)(cid:13)(cid:13) S . In fact, distinguishing as usual between di ff erent frequency / modulation configurations for either ofthe factors, one estimates the L t L x -norm of the input by placing the first input into L t L ∞ x and thesecond into L ∞ t L x , both of which are controlled by Observation 1. Using divisibility of the L t L x norm, it now follows that upon proper choice of the intervals I n , whose number of course only
38 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION depends on (cid:13)(cid:13)(cid:13)(cid:0) ˜ A L , ˜ φ L (cid:1)(cid:13)(cid:13)(cid:13) S , we get the estimate (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k < k , = k + O (1) (cid:3) − P k Q k + O (1) P x (cid:0) P k φ ∇ x P k φ (cid:1) ∂ ν Q ≤ k − C P φ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N ( I n × R ) . (cid:18) X k < − k (cid:13)(cid:13)(cid:13) P k φ (cid:13)(cid:13)(cid:13) L t L ∞ x ( I n × R ) (cid:19) (cid:18) X k < − k (cid:13)(cid:13)(cid:13) ∇ x P k φ Q ≤ k − C P φ (cid:13)(cid:13)(cid:13) L t L x ( I n × R ) (cid:19) ≪ δ (cid:13)(cid:13)(cid:13) φ (cid:13)(cid:13)(cid:13) S ( I n × R ) . Of course, one gets the same bound upon replacing the frequency 0 by general l ∈ Z and squaresumming.As usual, similar reductions can be applied to the elliptic interaction term P < A ∂ t P φ . Dealing with H ∗ N lowhi (cid:0) H P < A nonlin , ( I n ) , P φ (cid:1) . Here we exploit the null structure arising fromcombining the elliptic as well as hyperbolic terms, just as in the proof of Proposition 7.5, or asin [22]. Correspondingly, we have to analyze three null forms, each in turn. The first null form . We can write it as X j ≤ k < , k , > k + O (1) (cid:3) − P k Q j (cid:0) Q ≤ j − C P k φ∂ α Q ≤ j − C P k φ (cid:1) ∂ α Q ≤ j − C P φ. From (148) in [22], it follows that we may restrict to j = k + O (1), as otherwise the desired smallnessjust follows from the o ff -diagonal decay of the estimate (even without restriction to smaller timeintervals). Furthermore, if k , k <
0, then we gain exponentially in the di ff erence k − k , while if k , k ≥
0, we gain exponentially in k . So we may further restrict to X k < , k , = k + O (1) (cid:3) − P k Q k + O (1) (cid:0) Q ≤ k − C P k φ∂ α Q ≤ k − C P k φ (cid:1) ∂ α Q ≤ k − C P φ and from here the argument proceeds exactly as before by suppressing the operator (cid:3) − P k Q k + O (1) and schematically estimating (cid:13)(cid:13)(cid:13) ∂ α Q ≤ k − C P k φ∂ α Q ≤ k − C P φ (cid:13)(cid:13)(cid:13) L t L x using Observation 2, while placing Q ≤ k − C P k φ into L t L ∞ x . The second and third null forms . These are treated identically and hence omitted here. (cid:3)
This completes Step 2. Together with Step 1, the linear theory for the operator P k (cid:3) A free < k P k anda standard bootstrap argument, this yields the bounds claimed in Proposition 8.7 for the localizednorms k ( A , φ ) k S ( I n × R ) . From there one can glue the localized components together to get the globalbounds. (cid:3) Rigidity I: Infinite time interval and reduction to the self-similar case for finite time in-tervals.
As in [10, Theorem 5.1], our goal is now to establish the following rigidity result.
Proposition 8.9.
Let ( A ∞ , Φ ∞ ) be as before with lifespan I = ( − T , T ) . Then it is not possibleto have T < ∞ or T < ∞ . Moreover, if λ ( t ) ≥ λ > for all t ∈ R , then we necessarily have ( A ∞ , Φ ∞ ) = (0 , , whence there cannot be a minimal energy blowup solution under the givenassumptions. ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 139
We begin the proof of Proposition 8.9 in the case when T = ∞ and λ ( t ) ≥ λ > , ∞ ). Tothis end we follow the method of proof in [20], which in turn follows the strategy in [10], but alsoadds a crucial Vitali type covering argument that is inspired by the covering argument in [42]. Usingthe assumption λ ( t ) ≥ λ > , ∞ ) and the compactness property expressed in Theorem 7.23,we obtain that for any ε >
0, there exists R ( ε ) > t ≥ Z(cid:8) | x + x ( t ) λ ( t ) |≥ R ( ε ) (cid:9)(cid:18)X α (cid:12)(cid:12)(cid:12) ∇ t , x A ∞ α ( t , x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ∇ t , x Φ ∞ ( t , x ) (cid:12)(cid:12)(cid:12) + | Φ ∞ ( t , x ) | | x | (cid:19) dx ≤ ε. Then we have in perfect analogy with [20, Lemma 10.9] and [10, Lemma 5.4] the following
Lemma 8.10.
There exist ε > and C > such that if ε ∈ (0 , ε ) , there exists R ( ε ) so that ifR > R ( ε ) , then there exists t = t ( R , ε ) with ≤ t ≤ CR and the property that for all < t < t we have (cid:12)(cid:12)(cid:12)(cid:12) x ( t ) λ ( t ) (cid:12)(cid:12)(cid:12)(cid:12) < R − R ( ε ) , (cid:12)(cid:12)(cid:12)(cid:12) x ( t ) λ ( t ) (cid:12)(cid:12)(cid:12)(cid:12) = R − R ( ε ) . Proof.
We adjust the proof of [20, Lemma 10.9] by using the weighted momentum monotonicityidentity (8.6) from Proposition 8.2. To begin with, we show that there exists α > Z I Z R (cid:18)X k F ∞ k ( t , x ) + (cid:12)(cid:12)(cid:12) D Φ ∞ ( t , x ) (cid:12)(cid:12)(cid:12) (cid:19) dx dt ≥ α > I ⊂ [0 , ∞ ) of unit length. We argue by contradiction. Suppose not, then there existsa sequence of intervals J n = [ t n , t n +
1] with t n → ∞ such that(8.47) Z J n Z R (cid:18)X k F ∞ k ( t , x ) + (cid:12)(cid:12)(cid:12) D Φ ∞ ( t , x ) (cid:12)(cid:12)(cid:12) (cid:19) dx dt ≤ n . For a sequence of times { s n } n with s n ∈ J n , the set ((cid:18) λ ( s n ) ∇ t , x A ∞ x (cid:18) s n , · − x ( s n ) λ ( s n ) (cid:19) , λ ( s n ) ∇ t , x Φ ∞ (cid:18) s n , · − x ( s n ) λ ( s n ) (cid:19)(cid:19)) n is pre-compact in (cid:0) L x ( R ) (cid:1) by Theorem 7.23. Then by Corollary 6.3, there exists a non-emptyopen interval I ∗ around t = λ ( s n ) (cid:18) ∇ t , x A ∞ x , ∇ t , x Φ ∞ (cid:19)(cid:18) s n + t λ ( s n ) − , · − x ( s n ) λ ( s n ) (cid:19) converges to a limiting function (cid:0) ∇ t , x A ∗ x , ∇ t , x Φ ∗ (cid:1) ∈ C (cid:0) I ∗ , ( L x ( R )) (cid:1) as n → ∞ in the given topology. ( A ∗ , Φ ∗ ) is a weak solution to MKG-CG on I ∗ × R in the L t ˙ H x -sense and satisfies the Coulomb condition.We now distinguish two cases: Either there exists a sequence of times { s n } n with s n ∈ J n suchthat { λ ( s n ) } n remains bounded or { λ ( s n ) } n does not remain bounded for any sequence of times { s n } n with s n ∈ J n . In the first case, noting that λ ( t ) ≥ λ >
0, we may replace I ∗ by a smaller non-emptytime interval I † and assume that s n + λ ( s n ) − I † ⊂ J n for all n ≥
1. From (8.47) we infer that Z I † Z R (cid:12)(cid:12)(cid:12)(cid:0) ∂ t Φ ∗ + i A ∗ Φ ∗ (cid:1) ( t , x ) (cid:12)(cid:12)(cid:12) dx dt = ,
40 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION whence ∂ t Φ ∗ + i A ∗ Φ ∗ ≡ I † × R . But then we have in the weak sense that X k = (cid:0) ∂ k + i A ∗ k (cid:1) Φ ∗ ≡ I † × R . This implies Φ ∗ | I † × R ≡ ∂ t Φ ∗ | I † × R ≡
0. We conclude that ( A ∗ , Φ ∗ ) must bea “trivial” solution in that the spatial components of A ∗ are finite energy free waves, while thetemporal component vanishes, and we have Φ ∗ ≡
0. But this solution has finite S -bounds, whichis a contradiction upon applying Proposition 7.20.Next, we consider the case that { λ ( s n ) } n does not remain bounded for any sequence of times { s n } n with s n ∈ J n . Then we essentially replicate the preceding argument, but need to also add a Vitalitype covering trick. We write for each n ≥ J n = [ s ∈ J n (cid:2) s − λ ( s ) − , s + λ ( s ) − (cid:3) ∩ J n . Applying Vitali’s covering lemma, we may pick a disjoint subcollection of intervals { I s } s ∈ A n with I s : = [ s − λ ( s ) − , s + λ ( s ) − ] ∩ J n for some subset A n ⊂ J n such that X s ∈ A n | I s | ≥ . It follows that we may pick a sequence of times { s n } n with s n ∈ J n such that we have Z I sn Z R (cid:18)X k F ∞ k ( t , x ) + (cid:12)(cid:12)(cid:12) D Φ ∞ ( t , x ) (cid:12)(cid:12)(cid:12) (cid:19) dx dt = o ( λ ( s n ) − ) . In particular, we obtain Z − Z R χ J n (cid:18)X k (cid:0) F ∞ k (cid:1) + (cid:12)(cid:12)(cid:12) D Φ ∞ (cid:12)(cid:12)(cid:12) (cid:19)!(cid:0) s n + t λ ( s n ) − , x (cid:1) dx dt = o (1) . But then, passing to a subsequence, we may again extract a limiting function from1 λ ( s n ) (cid:18) ∇ t , x A ∞ x , ∇ t , x Φ ∞ (cid:19)(cid:18) s n + t λ ( s n ) − , · − x ( s n ) λ ( s n ) (cid:19) , which yields a time independent solution and leads to a contradiction as before.This shows that (8.46) is indeed valid for suitable α >
0. We note that λ ( t ) ≥ λ > x (0) =
0. If the assertion of the lemma was false, then we would have (cid:12)(cid:12)(cid:12)(cid:12) x ( t ) λ ( t ) (cid:12)(cid:12)(cid:12)(cid:12) < R − R ( ε )for all 0 ≤ t < CR with C > ffi ciently large later on. We now use the weightedmomentum monotonicity identity (8.6) to obtain a contradiction. In view of (8.45), we concludethat the corresponding remainder term (8.7) satisfies r ( R ) . ε. Now choose ε > I of unit length, Z I Z R X k F ∞ k ( t , x ) + (cid:12)(cid:12)(cid:12) D Φ ∞ ( t , x ) (cid:12)(cid:12)(cid:12) dx + O ( r ( R )) ! dt ≥ α , ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 141 provided CR is su ffi ciently large. In particular, this implies Z CR Z R X k F ∞ k ( t , x ) + (cid:12)(cid:12)(cid:12) D Φ ∞ ( t , x ) (cid:12)(cid:12)(cid:12) dx + O ( r ( R )) ! dt ≥ α CR − . On the other hand, integrating (8.6) in time from 0 to CR , we find Z CR Z R X k F ∞ k ( t , x ) + (cid:12)(cid:12)(cid:12) D Φ ∞ ( t , x ) (cid:12)(cid:12)(cid:12) dx + O ( r ( R )) ! dt . RE crit with a universal implied constant. The two preceding bounds contradict each other for C large. (cid:3) To finish o ff the proof of Proposition 8.9 in the case when T = ∞ and λ ( t ) ≥ λ > , ∞ ),we now use Proposition 8.8 to conclude a contradiction to the preceding Lemma 8.10. Lemma 8.11.
There exist ε > , R ( ε ) > , C > such that if R > R ( ε ) , t = t ( R , ε ) are as inLemma 8.10, then for < ε < ε , t ( R , ε ) ≥ C R ε . Proof.
The proof proceeds exactly as in [10, Lemma 5.5] using the weighted energy identity (8.5)and that the minimal blowup solution ( A ∞ , Φ ∞ ) has vanishing momentum by Proposition 8.8. (cid:3) It remains to prove Proposition 8.9 when T < ∞ . As in [10] and [20], we first reduce this caseto a self-similar blowup scenario. After rescaling we may assume that T =
1. We recall fromLemma 8.3 that there exists a constant C ( K ) > < C ( K )1 − t ≤ λ ( t )for all 0 ≤ t <
1. Moreover, we know from Lemma 8.4 that after spatial translation(8.48) supp (cid:16) F ∞ αβ ( t , · ) , Φ ∞ ( t , · ) (cid:17) ⊂ B (0 , − t )for all 0 ≤ t < α, β ∈ { , , . . . , } . Next, we prove an upper bound on λ ( t ). Lemma 8.12.
Let ( A ∞ , Φ ∞ ) be as above with T = . Then there exists C ( K ) > such that λ ( t ) ≤ C ( K )1 − tfor all ≤ t < .Proof. We follow the argument in Lemma 10.11 in [20]. Suppose the claim was false. Thenconsider for 0 ≤ t < z ( t ) = Z R X k x k (cid:18)X j F ∞ j F ∞ k j + Re (cid:0) D Φ ∞ D k Φ ∞ (cid:1)(cid:19) dx + Z R Re (cid:0) Φ ∞ D Φ ∞ (cid:1) dx . From the weighted momentum monotonicity identity (8.6) in Proposition 8.2 we obtain that z ′ ( t ) = − Z R (cid:18)X k (cid:0) F ∞ k (cid:1) + (cid:12)(cid:12)(cid:12) D Φ ∞ (cid:12)(cid:12)(cid:12) (cid:19) dx . Since we have by (8.48) and Hardy’s inequality that z ( t ) → t →
1, we can write z ( t ) = Z t Z R (cid:18)X k F ∞ k ( s , x ) + (cid:12)(cid:12)(cid:12) D Φ ∞ ( s , x ) (cid:12)(cid:12)(cid:12) (cid:19) dx ds .
42 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION
Now we distinguish between two possibilities: Either there exists α > ≤ t < Z t Z R (cid:18)X k F ∞ k ( s , x ) + (cid:12)(cid:12)(cid:12) D Φ ∞ ( s , x ) (cid:12)(cid:12)(cid:12) (cid:19) dx ds ≥ α (1 − t ) , or else there exists a sequence { t n } n ⊂ [0 ,
1) with t n → J n : = [ t n , | J n | Z J n Z R (cid:18)X k F ∞ k ( s , x ) + (cid:12)(cid:12)(cid:12) D Φ ∞ ( s , x ) (cid:12)(cid:12)(cid:12) (cid:19) dx ds → . In the former case, we argue exactly as in [10, Lemma 5.6] to obtain the conclusion of the lemma.In particular, here we invoke Proposition 8.5. In the latter case, a contradiction ensues as follows.Using the same Vitali covering argument as in the proof of Lemma 8.10, we can select a sequenceof intervals J ′ n = [ s n − λ ( s n ) − , s n + λ ( s n ) − ] with s n ∈ J n such that1 | J ′ n | Z J ′ n Z R (cid:18)X k F ∞ k ( s , x ) + (cid:12)(cid:12)(cid:12) D Φ ∞ ( s , x ) (cid:12)(cid:12)(cid:12) (cid:19) dx ds → . But then, using compactness, we again extract a trivial limiting solution, and obtain a contradictionas in the proof of Lemma 8.10. (cid:3)
We are now in a position to reduce to the exactly self-similar case.
Corollary 8.13.
Let ( A ∞ , Φ ∞ ) be as above with T = . Then the set (cid:26)(cid:16) (1 − t ) (cid:0) ∇ t , x A ∞ x (cid:1) ( t , (1 − t ) x ) , (1 − t ) (cid:0) ∇ t , x Φ ∞ (cid:1) ( t , (1 − t ) x ) (cid:17) : 0 ≤ t < (cid:27) is pre-compact in (cid:0) L x ( R ) (cid:1) .Proof. Here we can proceed similarly to the proof of Proposition 5.7 in [10]. Our point of departureis Theorem 7.23. From Lemma 8.3 and Lemma 8.12 we know that C ( K ) ≤ (1 − t ) λ ( t ) ≤ C ( K )for all 0 ≤ t <
1. Using the sharp support properties (8.48) and that E crit >
0, we also conclude that | x ( t ) | ≤ C for all 0 ≤ t < C >
0. Then the claim follows from the compactnessassertion in Theorem 7.23. (cid:3)
Rigidity II: The self-similar case.
In this subsection we rule out the existence of a minimalblowup solution (cid:0) A ∞ , Φ ∞ (cid:1) as in Corollary 8.13. To this end we use self-similar variables andderive a suitable Lyapunov functional. For ease of notation we drop the superscript ∞ and denotethe minimal blowup solution from Corollary 8.13 just by ( A , Φ ). Following [10] and [20], weintroduce the self-similar variables y = x − t , s = − log(1 − t ) , ≤ t < e Φ ( s , y , = e − s Φ (1 − e − s , e − s y ) , e A α ( s , y , = e − s A α (1 − e − s , e − s y ) , ≤ α ≤ . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 143
We also define the associated covariant derivatives of e Φ and the curvature 2-form associated with e A in self-similar variables by ] D α Φ ( s , y , = e − s D α Φ (1 − e − s , e − s y ) , ≤ α ≤ , g F αβ ( s , y , = e − s F αβ (1 − e − s , e − s y ) , ≤ α, β ≤ . (8.49)Observe that (cid:0) e A , e Φ (cid:1) ( s , y ,
0) are defined for 0 ≤ s < ∞ . Moreover, in view of (8.48), e Φ ( s , · ,
0) andthe curvature components g F αβ ( s , · ,
0) have support in { y ∈ R : | y | ≤ } . For small δ >
0, we alsodefine y = x + δ − t , s = − log(1 + δ − t ) , ≤ t < e Φ ( s , y , δ ) = e − s Φ (1 + δ − e − s , e − s y ) , e A α ( s , y , δ ) = e − s A α (1 + δ − e − s , e − s y ) , ≤ α ≤ . Analogously to (8.49), we introduce ] D α Φ ( s , y , δ ) for 0 ≤ α ≤ g F αβ ( s , y , δ ) for 0 ≤ α, β ≤ (cid:0) e A , e Φ (cid:1) ( s , y , δ ) is defined for − log(1 + δ ) ≤ s < log( δ ).In self-similar variables the Maxwell-Klein-Gordon system is given by(8.50) ∂ k g F k = Im (cid:0)e Φ ] D Φ (cid:1) , − (cid:0) ∂ s + + y · ∇ y (cid:1)g F j + ∂ k g F jk = Im (cid:0)e Φ g D j Φ (cid:1) , (cid:0) ∂ s + i e A + + y · ∇ y (cid:1) ] D Φ = (cid:0) ∂ k + i e A k (cid:1) g D k Φ , where ∂ k denotes partial di ff erentiation with respect to the y variable. We begin by stating thefollowing properties of (cid:0) e A , e Φ (cid:1) . Lemma 8.14. (i) For fixed δ > , we have for all ≤ s < log( δ ) that supp (cid:0)e Φ ( s , · , δ ) (cid:1) ⊂ (cid:8) y ∈ R : | y | ≤ − δ (cid:9) , supp (cid:0)g F αβ ( s , · , δ ) (cid:1) ⊂ (cid:8) y ∈ R : | y | ≤ − δ (cid:9) . (8.51) (ii) Uniformly for all δ > and all ≤ s < log( δ ) , it holds that (8.52) Z B X α (cid:12)(cid:12)(cid:12) ] D α Φ ( s , y , δ ) (cid:12)(cid:12)(cid:12) dy . E crit and (8.53) Z B | e Φ ( s , y , δ ) | (1 − | y | ) dy . E crit . Proof. (i) For 0 ≤ s < log( δ ) we infer from the support properties (8.48) thatsupp (cid:0)e Φ ( s , · , δ ) (cid:1) ⊂ (cid:26) y ∈ R : | y | ≤ − t + δ − t = e − s − δ e − s ≤ − δ (cid:27) and similarly for the support of g F αβ ( s , · , δ ).(ii) The estimate (8.52) follows immediately from a change of variables. Noting that e Φ ( s , · , δ ) ∈
44 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION H ( B ) for all δ > ≤ s < log( δ ), we then use the Hardy-type inequality (0.5) from [3]together with the diamagnetic inequality to conclude that Z B | e Φ | (1 − | y | ) dy ≤ Z B | e Φ | (1 − | y | ) dy . Z B (cid:12)(cid:12)(cid:12) ∇ y | e Φ | (cid:12)(cid:12)(cid:12) dy . Z B X k (cid:12)(cid:12)(cid:12) g D k Φ (cid:12)(cid:12)(cid:12) dy . E crit . (cid:3) For small δ > (cid:0) e A , e Φ (cid:1) ( s , y , δ ) with associated covariant derivatives ] D α Φ ( s , y , δ ) and curva-ture components g F αβ ( s , y , δ ) as above, we now introduce a Lyapunov functional e E ( s ) = Z B (cid:18) X j g F j + X j , k g F jk − X j , k y k g F j g F jk (cid:19) dy (1 − | y | ) + Z B (cid:18) X α (cid:12)(cid:12)(cid:12) ] D α Φ (cid:12)(cid:12)(cid:12) − X k y k Re (cid:0) g D k Φ ] D Φ (cid:1) − Re (cid:0)e Φ ] D Φ (cid:1) − | e Φ | − | y | (cid:19) dy (1 − | y | ) and define the non-negative quantity e Ξ ( s ) = Z B X k (cid:18)g F k − (cid:16) X j y j | y | g F j (cid:17) y k | y | + X j y j g F jk (cid:19) dy (1 − | y | ) + Z B | y | (cid:18) X j y j g F j (cid:19) dy (1 − | y | ) + Z B (cid:12)(cid:12)(cid:12)(cid:12) ] D Φ − X k y k g D k Φ − e Φ (cid:12)(cid:12)(cid:12)(cid:12) dy (1 − | y | ) . We emphasize that both e E and e Ξ are gauge invariant quantities. They are well-defined for all δ > e E . Proposition 8.15.
Let (cid:0) e A , e Φ (cid:1) ( s , y , δ ) for δ > be as above. Then we have for ≤ s < s < log (cid:16) δ (cid:17) that (8.54) e E ( s ) − e E ( s ) = Z s s e Ξ ( s ) ds . Moreover, it holds that (8.55) lim s → log( δ ) e E ( s ) ≤ E crit . The crucial monotonicity identity (8.54) can be derived in a gauge invariant manner. However,the computations simplify significantly by imposing the Cronstrom-type gauge condition(8.56) X k = x k A k ( t , x ) = ≤ t < x ∈ R . This does not change the energy regularity of (cid:0) A , Φ (cid:1) . In self-similarvariables the gauge condition (8.56) reads(8.57) X k = y k e A k ( s , y , δ ) = ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 145 for all δ >
0, 0 ≤ s < log( δ ) and y ∈ R . Under the gauge condition (8.57) the functional e E can bewritten as e E ( s ) = Z B (cid:18) X j (cid:0) ∂ s e A j − ∂ j e A (cid:1) + X j , k (cid:0) ∂ j e A k − ∂ k e A j (cid:1) − X j (cid:0) (1 + y · ∇ y ) e A j (cid:1) (cid:19) dy (1 − | y | ) + Z B (cid:18) (cid:12)(cid:12)(cid:12) ( ∂ s + i e A ) e Φ (cid:12)(cid:12)(cid:12) + X k (cid:12)(cid:12)(cid:12) ( ∂ k + i e A k ) e Φ (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) (1 + y · ∇ y ) e Φ (cid:12)(cid:12)(cid:12) − | e Φ | − | y | (cid:19) dy (1 − | y | ) and the quantity e Ξ reads e Ξ ( s ) = Z B X k (cid:18) ∂ k e A − (cid:16) y | y | · ∇ y e A (cid:17) y k | y | − ∂ s e A k (cid:19) dy (1 − | y | ) + Z B | y | (cid:16) y · ∇ y e A (cid:17) dy (1 − | y | ) + Z B (cid:12)(cid:12)(cid:12) ( ∂ s + i e A ) e Φ (cid:12)(cid:12)(cid:12) dy (1 − | y | ) , where ∂ k denotes partial di ff erentiation with respect to the y variable. It is not obvious that theabove expressions for e E and e Ξ in the Cronstrom-type gauge (8.57) are even well-defined for all δ >
0. However, this follows from the gauge invariant support properties of e Φ and g F αβ , and thefollowing easily verified identities (assuming the gauge condition (8.57)) ∂ s e A j − ∂ j e A = g F j − X k y k g F k j ,∂ j e A k − ∂ k e A j = g F jk , y · ∇ y e A = X j y j g F j , (cid:0) + y · ∇ y (cid:1) e A j = X k y k g F k j , (cid:0) ∂ s + i e A (cid:1)e Φ = ] D Φ − X k y k g D k Φ − e Φ , (cid:0) ∂ k + i e A k (cid:1)e Φ = g D k Φ , y · ∇ y e Φ = X k y k g D k Φ . (8.58) Proof of Proposition 8.15.
In order to justify the computations in the derivation of the monotonicityidentity (8.54), we shall have to assume smoothness of (cid:0) A , Φ (cid:1) . However, smoothing the compo-nents as in Definition 5.3 destroys the crucial support properties of Φ and the curvature components F αβ , and thus of e Φ and g F αβ . In that sense certain expressions below involving weights of the form(1 − | y | ) − or (1 − | y | ) − become singular at | y | =
1. To deal with this, we need to introducean additional smooth cuto ff χ (cid:0) −| y | ε (cid:1) that smoothly localizes away from the boundary, but such thatlim ε → χ (cid:0) y ε (cid:1) = χ (0 , ( | y | ), where χ (0 , is the sharp characteristic cuto ff to the the interval (0 , χ (cid:18) − | y | ε (cid:19) − | y | ) − , which will lead to additional error terms localized near the boundary. But then letting the frequencycuto ff converge toward | ξ | = + ∞ in the regularization for fixed but su ffi ciently small ε >
0, it will be
46 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION easy to convince oneself that the additional errors vanish in the limit due to the support properties(8.48) of the underlying ( A , Φ ). We shall formally omit this additional cuto ff .Further, in order to simplify the computations below, we impose the Cronstrom-type gauge con-dition (8.56) on ( A , Φ ). This leads to another technical complication in that the C ∞ smoothnessof the regularized ( A , Φ ) in Coulomb gauge will be lost. This can again be dealt with via smoothtruncation of the functional, this time away from the origin by including the cuto ff χ (cid:0) | y | ε (cid:1) . Since allthe integrations by parts to be performed below involve an operator y · ∇ y , the error terms are seento be controllable in terms of the energy on smaller and smaller balls, and hence negligible in thelimit as ε →
0. Again, we shall gloss over this technicality in the formulas below.We now begin with the derivation of the monotonicity identity (8.54), where we assume that (cid:0) e A , e Φ (cid:1) ( s , y , δ ) satisfy the Cronstrom-type gauge condition(8.59) X k = y k e A k ( s , y , δ ) = ≤ s < log( δ ), y ∈ R and are smooth solutions to the Maxwell-Klein-Gordon system (8.50)in self-similar variables. In order to make the notation less heavy in this derivation, we write ( ˜ A , ˜ φ )instead of (cid:0) e A , e Φ (cid:1) , and g D α φ, g F αβ instead of ] D α Φ , g F αβ . We will repeatedly apply the following easilyverified identities without further referencing,( ∂ s + i ˜ A + + y · ∇ y ) ˜ φ = g D φ, ( ∂ k + i ˜ A k )( ∂ s + i ˜ A ) = ( ∂ s + i ˜ A )( ∂ k + i ˜ A k ) + i ( ∂ k ˜ A − ∂ s ˜ A k ) , where ∂ k denotes partial di ff erentiation with respect to the y variable. We also recall the identities(8.58). Moreover, we use the notation ρ ( y ) = − | y | ) and observe that ∂ k ρ ( y ) = y k ρ ( y ) , (1 + y · ∇ y ) ρ ( y ) = ρ ( y ) . The equation for ˜ φ can be written in expanded form as( ∂ s + i ˜ A + + y · ∇ y )( ∂ s + i ˜ A + + y · ∇ y ) ˜ φ = X k ( ∂ k + i ˜ A k ) ˜ φ, or alternatively as( ∂ s + i ˜ A ) ˜ φ + (3 + y · ∇ y )( ∂ s + i ˜ A ) ˜ φ + (2 + y · ∇ y )(1 + y · ∇ y ) ˜ φ − i ( y · ∇ y ˜ A ) ˜ φ = X k ( ∂ k + i ˜ A k ) ˜ φ. We start analyzing the derivative with respect to s of the following energy functional dds Z | ( ∂ s + i ˜ A ) ˜ φ | ρ ( y ) dy = Z Re (cid:16) ∂ s ( ∂ s + i ˜ A ) ˜ φ ( ∂ s + i ˜ A ) ˜ φ (cid:17) ρ ( y ) dy = Z Re (cid:16) ( ∂ s + i ˜ A ) ˜ φ ( ∂ s + i ˜ A ) ˜ φ (cid:17) ρ ( y ) dy . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 147
Inserting the equation for ˜ φ , we obtain Z Re (cid:16) ( ∂ s + i ˜ A ) ˜ φ ( ∂ s + i ˜ A ) ˜ φ (cid:17) ρ ( y ) dy = X k Z Re (cid:16) ( ∂ k + i ˜ A k ) ˜ φ ( ∂ s + i ˜ A ) ˜ φ (cid:17) ρ ( y ) dy − Z Re (cid:16) (3 + y · ∇ y )( ∂ s + i ˜ A ) ˜ φ ( ∂ s + i ˜ A ) ˜ φ (cid:17) ρ ( y ) dy − Z Re (cid:16) (2 + y · ∇ y )(1 + y · ∇ y ) ˜ φ ( ∂ s + i ˜ A ) ˜ φ (cid:17) ρ ( y ) dy + Z Re (cid:16) i ( y · ∇ y ˜ A ) ˜ φ ( ∂ s + i ˜ A ) ˜ φ (cid:17) ρ ( y ) dy ≡ I + II + III + IV . Integrating by parts in the term I , we find I = − X k Z Re (cid:16) ( ∂ k + i ˜ A k ) ˜ φ ( ∂ k + i ˜ A k )( ∂ s + i ˜ A ) ˜ φ (cid:17) ρ ( y ) dy − X k Z Re (cid:16) ( ∂ k + i ˜ A k ) ˜ φ ( ∂ s + i ˜ A ) ˜ φ (cid:17) ∂ k ρ ( y ) dy = − X k Z Re (cid:16) ( ∂ k + i ˜ A k ) ˜ φ ( ∂ s + i ˜ A )( ∂ k + i ˜ A k ) ˜ φ (cid:17) ρ ( y ) dy − X k Z Re (cid:16) ( ∂ k + i ˜ A k ) ˜ φ i ( ∂ k ˜ A − ∂ s ˜ A k ) ˜ φ (cid:17) ρ ( y ) dy − X k Z Re (cid:16) ( ∂ k + i ˜ A k ) ˜ φ ( ∂ s + i ˜ A ) ˜ φ (cid:17) y k ρ ( y ) dy = − dds Z (cid:18)X k | ( ∂ k + i ˜ A k ) ˜ φ | (cid:19) ρ ( y ) dy − X k Z Im (cid:16) ˜ φ g D k φ (cid:17) ( ∂ s ˜ A k − ∂ k ˜ A ) ρ ( y ) dy − X k Z Re (cid:16) ∂ k ˜ φ ( ∂ s + i ˜ A ) ˜ φ (cid:17) y k ρ ( y ) dy , where in the last step we took advantage of the gauge condition (8.59). We expect the second tolast term to cancel against a corresponding term from a suitable energy functional for ˜ A . On theother hand, the last term is expected to cancel against other terms from the equation for ˜ φ . Next,integrating by parts in the term II yields II = − Z Re (cid:16) (3 + y · ∇ y )( ∂ s + i ˜ A ) ˜ φ ( ∂ s + i ˜ A ) ˜ φ (cid:17) ρ ( y ) dy = Z | ( ∂ s + i ˜ A ) ˜ φ | ρ ( y ) dy .
48 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION
Performing another round of integration by parts, now in the term
III , we find that
III = − Z Re (cid:16) (2 + y · ∇ y )(1 + y · ∇ y ) ˜ φ ( ∂ s + i ˜ A ) ˜ φ (cid:17) ρ ( y ) dy = Z Re (cid:16) (1 + y · ∇ y ) ˜ φ ( ∂ s + i ˜ A )(1 + y · ∇ y ) ˜ φ (cid:17) ρ ( y ) dy + Z Re (cid:16) (1 + y · ∇ y ) ˜ φ i ( y · ∇ y ˜ A ) ˜ φ (cid:17) ρ ( y ) dy + Z Re (cid:16) y · ∇ y ˜ φ ( ∂ s + i ˜ A ) ˜ φ (cid:17) ρ ( y ) dy + Z Re (cid:16) ˜ φ ( ∂ s + i ˜ A ) ˜ φ (cid:17) ρ ( y ) dy , and thus, III = dds Z | (1 + y · ∇ y ) ˜ φ | ρ ( y ) dy + Z Re (cid:16) (1 + y · ∇ y ) ˜ φ i ( y · ∇ y ˜ A ) ˜ φ (cid:17) ρ ( y ) dy + X k Z Re (cid:16) ∂ k ˜ φ ( ∂ s + i ˜ A ) ˜ φ (cid:17) y k ρ ( y ) dy + dds Z | ˜ φ | ρ ( y ) dy . Now we note that the second term on the right hand side of
III and the term IV nicely combine togive Z Re (cid:16) (1 + y · ∇ y ) ˜ φ i ( y · ∇ y ˜ A ) ˜ φ (cid:17) ρ ( y ) dy + Z Re (cid:16) i ( y · ∇ y ˜ A ) ˜ φ ( ∂ s + i ˜ A ) ˜ φ (cid:17) ρ ( y ) dy = Z Re (cid:16) i ( y · ∇ y ˜ A ) ˜ φ ( ∂ s + i ˜ A + + y · ∇ y ) ˜ φ (cid:17) ρ ( y ) dy = − Z ( y · ∇ y ˜ A ) Im (cid:16) ˜ φ g D φ (cid:17) ρ ( y ) dy . Moreover, we observe that the last term on the right hand side of I cancels out with the third termon the right hand side of III . We summarize the preceding computations as follows dds
Z (cid:18) | ( ∂ s + i ˜ A ) ˜ φ | + X k | ( ∂ k + i ˜ A k ) ˜ φ | − | (1 + y · ∇ y ) ˜ φ | − | ˜ φ | ρ ( y ) (cid:19) ρ ( y ) dy = Z | ( ∂ s + i ˜ A ) ˜ φ | ρ ( y ) dy − X k Z Im (cid:16) ˜ φ g D k φ (cid:17) ( ∂ s ˜ A k − ∂ k ˜ A ) ρ ( y ) dy − Z ( y · ∇ y ˜ A ) Im (cid:16) ˜ φ g D φ (cid:17) ρ ( y ) dy . (8.60)We expect the last two terms on the right hand side to cancel against corresponding terms gen-erated by di ff erentiating a suitable energy functional for ˜ A , while the first term furnishes the keymonotonicity. ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 149
At this point we have to pass to the corresponding equation for ˜ A . It is given in expanded formfor j = , . . . , ∂ s ( ∂ s ˜ A j − ∂ j ˜ A ) + (3 + y · ∇ y )( ∂ s ˜ A j ) + (2 + y · ∇ y )(1 + y · ∇ y ) ˜ A j − (2 + y · ∇ y )( ∂ j ˜ A ) − X k ∂ k ( ∂ k ˜ A j − ∂ j ˜ A k ) = Im (cid:16) ˜ φ g D j φ (cid:17) . We begin with a tentative ansatz for the correct energy functional for ˜ A to leading order, which wedi ff erentiate with respect to s , dds Z (cid:18) X j ( ∂ s ˜ A j − ∂ j ˜ A ) + X j , k ( ∂ j ˜ A k − ∂ k ˜ A j ) (cid:19) ρ ( y ) dy = Z X j ∂ s ( ∂ s ˜ A j − ∂ j ˜ A ) ( ∂ s ˜ A j − ∂ j ˜ A ) ρ ( y ) dy + Z X j , k ∂ s ( ∂ j ˜ A k − ∂ k ˜ A j ) ( ∂ j ˜ A k − ∂ k ˜ A j ) ρ ( y ) dy ≡ ① + ② . Inserting the equation for ˜ A in the term ① , we obtain ① = − X j Z (3 + y · ∇ y )( ∂ s ˜ A j ) ( ∂ s ˜ A j − ∂ j ˜ A ) ρ ( y ) dy − X j Z (2 + y · ∇ y )(1 + y · ∇ y ) ˜ A j ( ∂ s ˜ A j − ∂ j ˜ A ) ρ ( y ) dy + X j Z (2 + y · ∇ y )( ∂ j ˜ A ) ( ∂ s ˜ A j − ∂ j ˜ A ) ρ ( y ) dy + X j , k Z ∂ k ( ∂ k ˜ A j − ∂ j ˜ A k ) ( ∂ s ˜ A j − ∂ j ˜ A ) ρ ( y ) dy + X j Z Im (cid:16) ˜ φ g D j φ (cid:17) ( ∂ s ˜ A j − ∂ j ˜ A ) ρ ( y ) dy = e I + e II + f III + f IV + e V ,
50 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION where we already see that the term e V cancels against the second term on the right hand side of(8.60). The term ② can be rewritten as ② = X j , k Z (cid:16) ∂ j ( ∂ s ˜ A k − ∂ k ˜ A ) − ∂ k ( ∂ s ˜ A j − ∂ j ˜ A ) (cid:17) ( ∂ j ˜ A k − ∂ k ˜ A j ) ρ ( y ) dy = X j , k Z ∂ j ( ∂ s ˜ A k − ∂ k ˜ A ) ( ∂ j ˜ A k − ∂ k ˜ A j ) ρ ( y ) dy = − X j , k Z ( ∂ s ˜ A k − ∂ k ˜ A ) ∂ j ( ∂ j ˜ A k − ∂ k ˜ A j ) ρ ( y ) dy − X j , k Z ( ∂ s ˜ A k − ∂ k ˜ A ) ( ∂ j ˜ A k − ∂ k ˜ A j ) y j ρ ( y ) dy = − X j , k Z ( ∂ s ˜ A k − ∂ k ˜ A ) ∂ j ( ∂ j ˜ A k − ∂ k ˜ A j ) ρ ( y ) dy − X k Z ( ∂ s ˜ A k − ∂ k ˜ A ) (1 + y · ∇ y ) ˜ A k ρ ( y ) dy , where in the second to last step we integrated by parts and in the last step we used that X j y j ( ∂ j ˜ A k − ∂ k ˜ A j ) = (1 + y · ∇ y ) ˜ A k due to the gauge condition (8.59). We see that the term f IV on the right hand side of ① cancelsagainst the first term on the right hand side of ② . Next, we integrate by parts in the term e I to find e I = − X j Z (3 + y · ∇ y )( ∂ s ˜ A j ) ( ∂ s ˜ A j ) ρ ( y ) dy + X j Z (3 + y · ∇ y )( ∂ s ˜ A j ) ( ∂ j ˜ A ) ρ ( y ) dy = X j Z ( ∂ s ˜ A j ) ρ ( y ) dy − X j Z ( ∂ s ˜ A j ) (5 + y · ∇ y )( ∂ j ˜ A ) ρ ( y ) dy − X j Z ( ∂ s ˜ A j ) ( ∂ j ˜ A ) y · ∇ y ρ ( y ) dy . Integrating by parts also in the term e II yields e II = X j Z (1 + y · ∇ y ) ˜ A j (1 + y · ∇ y )( ∂ s ˜ A j − ∂ j ˜ A ) ρ ( y ) dy + X j Z (1 + y · ∇ y ) ˜ A j ( ∂ s ˜ A j − ∂ j ˜ A ) ρ ( y ) dy = dds Z (cid:18)X j | (1 + y · ∇ y ) ˜ A j | (cid:19) ρ ( y ) dy − X j Z (1 + y · ∇ y ) ˜ A j (1 + y · ∇ y )( ∂ j ˜ A ) ρ ( y ) dy + X j Z (1 + y · ∇ y ) ˜ A j ( ∂ s ˜ A j − ∂ j ˜ A ) ρ ( y ) dy ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 151 and we observe that the third term on the right hand side of e II cancels against the second term onthe right hand side of ② . Another round of integration by parts, now in the term f III , leads to f III = X j Z (2 + y · ∇ y )( ∂ j ˜ A ) ( ∂ s ˜ A j ) ρ ( y ) dy + X j Z
12 ( ∂ j ˜ A ) y · ∇ y ρ ( y ) dy . Combining the above expressions, we are thus reduced to ① + ② = X j Z ( ∂ s ˜ A j ) ρ ( y ) dy + dds Z (cid:18)X j | (1 + y · ∇ y ) ˜ A j | (cid:19) ρ ( y ) dy + X j Z
12 ( ∂ j ˜ A ) y · ∇ y ρ ( y ) dy − X j Z ( ∂ s ˜ A j ) ( ∂ j ˜ A ) y · ∇ y ρ ( y ) dy − X j Z ( ∂ s ˜ A j ) (3 + y · ∇ y )( ∂ j ˜ A ) ρ ( y ) dy − X j Z (1 + y · ∇ y ) ˜ A j (1 + y · ∇ y )( ∂ j ˜ A ) ρ ( y ) dy + X j Z Im (cid:16) ˜ φ g D j φ (cid:17) ( ∂ s ˜ A j − ∂ j ˜ A ) ρ ( y ) dy . We reformulate this as ① + ② = X j Z ( ∂ s ˜ A j ) ρ ( y ) dy + dds Z (cid:18)X j | (1 + y · ∇ y ) ˜ A j | (cid:19) ρ ( y ) dy + X j Z
12 ( ∂ j ˜ A ) y · ∇ y ρ ( y ) dy − X j Z ( ∂ s ˜ A j ) ( ∂ j ˜ A ) ρ ( y ) dy − X j Z ( ∂ s + + y · ∇ y ) ˜ A j (1 + y · ∇ y )( ∂ j ˜ A ) ρ ( y ) dy + X j Z Im (cid:16) ˜ φ g D j φ (cid:17) ( ∂ s ˜ A j − ∂ j ˜ A ) ρ ( y ) dy . (8.61)Next, we further analyze the second to last term on the right hand side of the above identity. Inte-gration by parts gives − X j Z ( ∂ s + + y · ∇ y ) ˜ A j (1 + y · ∇ y )( ∂ j ˜ A ) ρ ( y ) dy = − X j Z ( ∂ s + + y · ∇ y ) ˜ A j ∂ j ( y · ∇ y ˜ A ) ρ ( y ) dy = + X j Z ∂ j ( ∂ s + + y · ∇ y ) ˜ A j ( y · ∇ y ˜ A ) ρ ( y ) dy + X j Z ( ∂ s + y + y · ∇ y ) ˜ A j ( y · ∇ y ˜ A ) y j ρ ( y ) dy = + X j Z ∂ j ( ∂ s + + y · ∇ y ) ˜ A j ( y · ∇ y ˜ A ) ρ ( y ) dy , (8.62)
52 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION where in the last step we used that due to the gauge condition (8.59), X j y j ( ∂ s + + y · ∇ y ) ˜ A j = . Moreover, one easily verifies that(8.63) X j ∂ j ( ∂ s + + y · ∇ y ) ˜ A j = X j ∂ j ˜ A + X j ∂ j g F j = X j ∂ j ˜ A + Im (cid:16) φ g D φ (cid:17) , where in the last equality we linked with the equation for ˜ A . Inserting (8.63) back into (8.62) andintegrating by parts several times more, we conclude that − X j Z ( ∂ s + + y · ∇ y ) ˜ A j (1 + y · ∇ y )( ∂ j ˜ A ) ρ ( y ) dy = X j Z ( ∂ j ˜ A ) ρ ( y ) dy + X j Z
12 ( ∂ j ˜ A ) y · ∇ y ρ ( y ) dy − Z ( y · ∇ y ˜ A ) ρ ( y ) dy + Z Im (cid:16) φ g D φ (cid:17) ( y · ∇ y ˜ A ) ρ ( y ) dy . (8.64)Finally, inserting (8.64) back into (8.61) and combining terms, we may summarize the precedingcomputations as follows dds Z (cid:18) X j ( ∂ s ˜ A j − ∂ j ˜ A ) + X j , k ( ∂ j ˜ A k − ∂ k ˜ A j ) − X j | (1 + y · ∇ y ) ˜ A j | (cid:19) ρ ( y ) dy = Z X j ( ∂ s ˜ A j − ∂ j ˜ A ) ρ ( y ) dy − Z ( y · ∇ y ˜ A ) ρ ( y ) dy + Z Im (cid:16) φ g D φ (cid:17) ( y · ∇ y ˜ A ) ρ ( y ) dy + X j Z Im (cid:16) ˜ φ g D j φ (cid:17) ( ∂ s ˜ A j − ∂ j ˜ A ) ρ ( y ) dy . (8.65)We observe that the last two terms on the right hand side cancel against the last two terms on theright hand side of (8.60). However, it is not yet obvious that the first two terms on the right handside of the above identity (8.65) yield the desired monotonicity. To this end we decompose the 4-vector ( ∂ j ˜ A ) j = into its radial and angular part. The gauge condition (8.59) then allows to rewritethis as Z X j ( ∂ s ˜ A j − ∂ j ˜ A ) ρ ( y ) dy − Z ( y · ∇ y ˜ A ) ρ ( y ) dy = Z X j (cid:18) ∂ s ˜ A j − ∂ j ˜ A + (cid:16) y | y | · ∇ y ˜ A (cid:17) y j | y | (cid:19) ρ ( y ) dy + Z (cid:16) y | y | · ∇ y ˜ A (cid:17) ρ ( y ) dy − Z (cid:0) y · ∇ y ˜ A (cid:1) ρ ( y ) dy = Z X j (cid:18) ∂ s ˜ A j − ∂ j ˜ A + (cid:16) y | y | · ∇ y ˜ A (cid:17) y j | y | (cid:19) ρ ( y ) dy + Z | y | (cid:0) y · ∇ y ˜ A (cid:1) ρ ( y ) dy . (8.66)Combining (8.60), (8.65), and (8.66) finishes the proof of the monotonicity identity (8.54). ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 153
It remains to prove (8.55). Using the gauge invariant formulation of the Lyapunov functional e E ,we proceed exactly as in the proof of Proposition 6.2 (iii) in [10] to show that for all δ > s → log( δ ) Z B (cid:18) X j g F j + g F jk + X α (cid:12)(cid:12)(cid:12) ] D α Φ (cid:12)(cid:12)(cid:12) (cid:19) dy (1 − | y | ) ≤ E crit , whilelim s → log( δ ) Z B (cid:18)X j , k y k g F j g F jk + X k y k Re (cid:0) g D k Φ ] D Φ (cid:1) + Re (cid:0)e Φ ] D Φ (cid:1) + | e Φ | − | y | (cid:19) dy (1 − | y | ) = . (cid:3) Next, we prove upper and lower bounds for the Lyapunov functional e E ( s ) uniformly in δ > ≤ s < log( δ ). Lemma 8.16.
For all δ > and all ≤ s < log( δ ) , we have (8.67) − CE crit ≤ e E ( s ) ≤ E crit for some absolute constant C > .Proof. The upper bound is immediate from (8.55) and the monotonicity property (8.54) of thefunctional e E . In order to prove the lower bound, we work with the gauge invariant formulation ofthe Lyapunov functional e E and first observe that for | y | ≤
1, the quantities12 X α (cid:12)(cid:12)(cid:12) ] D α Φ (cid:12)(cid:12)(cid:12) − X k y k Re (cid:0) g D k Φ ] D Φ (cid:1) and 12 X j g F j + X j , k g F jk − X j , k y k g F j g F jk are non-negative. This is straightforward to see for the first expression, while for the second one weuse that X j , k y k g F j g F jk = | y | X j , k (cid:16) y k | y | g F j − y j | y | g F k (cid:17)g F jk ≤ | y | X j , k (cid:16) y k | y | g F j − y j | y | g F k (cid:17) + | y | X j , k g F j , k . From the general identity X j , k (cid:0) ω k r j − ω j r k (cid:1) = (cid:0) r − ( r · ω ) (cid:1) ≤ r for r , ω ∈ R with | ω | =
1, we then conclude that X j , k y k g F j g F jk ≤ | y | X j g F j + | y | X j , k g F jk . It therefore su ffi ces to obtain an upper bound on Z B (cid:18) Re (cid:0)e Φ ] D Φ (cid:1) + | e Φ | − | y | (cid:19) dy (1 − | y | )
54 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION uniformly for all δ > ≤ s < log( δ ). From H ¨older’s inequality, (8.52) and (8.53) we easilyinfer that Z B (cid:18) Re (cid:0)e Φ ] D Φ (cid:1) + | e Φ | − | y | (cid:19) dy (1 − | y | ) . Z B | ] D Φ | dy + Z B | e Φ | (1 − | y | ) dy . E crit . (cid:3) As a corollary of Proposition 8.15 and Lemma 8.16, we obtain the following decay property as δ → Corollary 8.17.
For each δ > , there exists s δ ∈ (cid:0) log( δ ) , log( δ ) (cid:1) such that (8.68) Z s δ + log( δ ) s δ e Ξ ( s ) ds ≤ CE crit log( δ ) . Proof.
From (8.54) and Lemma 8.16 we have that Z log( δ )0 e Ξ ( s ) ds ≤ CE crit . Then the claim is immediate. (cid:3)
Our goal is now to extract a limiting solution (cid:0) A ∗ , Φ ∗ (cid:1) and to eventually show that Φ ∗ mustvanish. This will yield a contradiction to the minimal blowup solution (cid:0) A , Φ (cid:1) having infinite S norm.Let t δ = + δ − e − s δ , where s δ is as in Corollary 8.17. By Corollary 8.13 we can pick a sequence δ l → l → ∞ such that (cid:16) (1 − t δ l ) ∇ t , x A x (cid:0) t δ l , (1 − t δ l ) x (cid:1) , (1 − t δ l ) ∇ t , x Φ (cid:0) t δ l , (1 − t δ l ) x (cid:1)(cid:17) → (cid:16) ∇ t , x A ∗ x ( x ) , ∇ t , x Φ ∗ ( x ) (cid:17) strongly in (cid:0) L x ( R ) (cid:1) as δ l →
0. We may also arrange that(8.69) (cid:16) (1 + δ l − t δ l ) ∇ t , x A x (cid:0) t δ l , (1 + δ l − t δ l ) x (cid:1) , (1 + δ l − t δ l ) ∇ t , x Φ (cid:0) t δ l , (1 + δ l − t δ l ) x (cid:1)(cid:17) → (cid:16) ∇ t , x A ∗ x ( x ) , ∇ t , x Φ ∗ ( x ) (cid:17) in (cid:0) L x ( R ) (cid:1) as δ l →
0. We now consider the MKG-CG evolutions in the sense of Definition 5.3of the energy class Coulomb data given by the left hand side of (8.69). Denote these evolutions by (cid:0) A l ∗ , Φ l ∗ (cid:1) . By the perturbative results from Corollary 6.3, these evolutions exist on some fixed timeinterval [0 , T ∗ ], where we may assume that 0 < T ∗ <
1. Moreover, we have on [0 , T ∗ ] that A l ∗ = (1 + δ l − t δ l ) A (cid:0) t δ l + (1 + δ l − t δ l ) t , (1 + δ l − t δ l ) x (cid:1) , Φ l ∗ ( t , x ) = (1 + δ l − t δ l ) Φ (cid:0) t δ l + (1 + δ l − t δ l ) t , (1 + δ l − t δ l ) x (cid:1) , and (cid:0) ∇ t , x A l ∗ x ( t , · ) , ∇ t , x Φ l ∗ ( t , · ) (cid:1) → (cid:0) ∇ t , x A ∗ x ( t , · ) , ∇ t , x Φ ∗ ( t , · ) (cid:1) in (cid:0) L x ( R ) (cid:1) as l → ∞ uniformly for all 0 ≤ t ≤ T ∗ , where (cid:0) A ∗ , Φ ∗ (cid:1) is a weak solution to MKG-CGon [0 , T ∗ ] × R . Note that on account of these identities we havesupp (cid:0) Φ l ∗ ( t , · ) (cid:1) ⊂ (cid:26) x ∈ R : | x | ≤ − t δ l + δ l − t δ l − t < − t (cid:27) and similarly supp (cid:0) ( ∂ α A l ∗ β − ∂ β A l ∗ α )( t , · ) (cid:1) ⊂ (cid:8) x ∈ R : | x | < − t (cid:9) . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 155
We now switch to the self-similar variables s = − log(1 − t ) , y = x − t , ≤ t ≤ T ∗ , and define g A l ∗ α ( s , y ) = e − s A l ∗ α (1 − e − s , e − s y ) , f Φ l ∗ ( s , y ) = e − s Φ l ∗ (1 − e − s , e − s y ) , and similarly for (cid:0) f A ∗ , f Φ ∗ (cid:1) . We conclude exactly as in [10] after Remark 6.8 there that g A l ∗ α ( s , y ) = e A α ( s δ l + s , y , δ l ) , f Φ l ∗ ( s , y ) = e Φ ( s δ l + s , y , δ l ) , (8.70)and(8.71) (cid:0) ∇ s , y g A l ∗ x , ∇ s , y f Φ l ∗ (cid:1) ( s , · ) → (cid:0) ∇ s , y f A ∗ x , ∇ s , y f Φ ∗ (cid:1) ( s , · )in (cid:0) L y ( R ) (cid:1) as l → ∞ uniformly for all 0 ≤ s ≤ − log(1 − T ∗ ) = : S . Then (cid:0) f A ∗ , f Φ ∗ (cid:1) is a weaksolution to the Maxwell-Klein-Gordon system in self-similar variables (8.50). Denoting by ] D α Φ ∗ and g F ∗ αβ the covariant derivatives and curvature components in self-similar variables associated with (cid:0) f A ∗ , f Φ ∗ (cid:1) , we conclude that supp (cid:8)f Φ ∗ ( s , · ) (cid:9) ⊂ { y ∈ R : | y | ≤ } , supp (cid:8)g F ∗ αβ ( s , · ) (cid:9) ⊂ { y ∈ R : | y | ≤ } . (8.72) Lemma 8.18.
Let (cid:0) f A ∗ , f Φ ∗ (cid:1) be as above. Then it holds that (8.73) X j y j g F ∗ j ≡ , ] D Φ ∗ − X k y k ] D k Φ ∗ − f Φ ∗ ≡ .
56 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION
Proof.
For large l we obtain from (8.70), (8.71), and Corollary 8.17 that Z S Z B | y | (cid:18) X j y j g F ∗ j ( s , y ) (cid:19) dy (1 − | y | ) ds + Z S Z B (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) ] D Φ ∗ − X k y k ] D k Φ ∗ − f Φ ∗ (cid:17) ( s , y ) (cid:12)(cid:12)(cid:12)(cid:12) dy (1 − | y | ) ds ≤ lim inf l →∞ ( Z S Z B | y | (cid:18) X j y j g F j ( s δ l + s , y , δ l ) (cid:19) dy (1 − | y | ) ds + Z S Z B (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) ] D Φ − X k y k g D k Φ − e Φ (cid:17) ( s δ l + s , y , δ l ) (cid:12)(cid:12)(cid:12)(cid:12) dy (1 − | y | ) ds ) ≤ lim inf l →∞ ( Z s δ l + Ss δ l Z B | y | (cid:18) X j y j g F j ( s , y , δ l ) (cid:19) dy (1 − | y | ) ds + Z s δ l + Ss δ l Z B (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) ] D Φ − X k y k g D k Φ − e Φ (cid:17) ( s , y , δ l ) (cid:12)(cid:12)(cid:12)(cid:12) dy (1 − | y | ) ds ) ≤ lim inf l →∞ CE crit log( δ l ) = . (cid:3) Proposition 8.19.
Let (cid:0) f A ∗ , f Φ ∗ (cid:1) be as above. Then we have f Φ ∗ ≡ . Going back to the ( t , x ) coordinates, the preceding proposition implies that A ∗ k is a free wavefor k = , . . . ,
4, while A ∗ ≡
0. This contradicts Proposition 6.1 and hence completes the proof ofProposition 8.9.
Proof of Proposition 8.19.
In order to simplify the computations below, we assume that (cid:0) f A ∗ , e Φ ∗ (cid:1) satisfy the Cronstrom-type gauge condition (in self-similar variables)(8.74) X k = y k f A ∗ k ( s , y ) = ≤ s ≤ S and y ∈ R . Then the properties (8.73) of the limiting solution (cid:0) f A ∗ , f Φ ∗ (cid:1) can bewritten as y · ∇ y f A ∗ ≡ , (cid:0) ∂ s + i f A ∗ (cid:1)e Φ ∗ ≡ f Φ ∗ simplifies to(2 + y · ∇ y )(1 + y · ∇ y ) f Φ ∗ = X k (cid:0) ∂ k + i f A ∗ k (cid:1) f Φ ∗ . Integrating this equation against f Φ ∗ , we find(8.75) Z R (cid:16) (2 + y · ∇ y )(1 + y · ∇ y ) f Φ ∗ (cid:17)f Φ ∗ dy = − X k Z R (cid:12)(cid:12)(cid:12)(cid:0) ∂ k + i f A ∗ k (cid:1)f Φ ∗ (cid:12)(cid:12)(cid:12) dy . ONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION 157
A simple integration by parts shows that the left hand side of (8.75) is given by Z R (cid:16) (2 + y · ∇ y )(1 + y · ∇ y ) f Φ ∗ (cid:17)f Φ ∗ dy = Z R (cid:12)(cid:12)(cid:12)f Φ ∗ (cid:12)(cid:12)(cid:12) dy − Z R (cid:12)(cid:12)(cid:12) y · ∇ y f Φ ∗ (cid:12)(cid:12)(cid:12) dy . Decomposing the 4-vector (cid:0) ∂ k f Φ ∗ (cid:1) k = into its radial and angular part, we observe that the gaugecondition (8.74) allows to rewrite the right hand side of (8.75) as − X k Z (cid:12)(cid:12)(cid:12) ( ∂ k + i f A ∗ k ) f Φ ∗ (cid:12)(cid:12)(cid:12) dy = − Z R (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) y | y | · ∇ y f Φ ∗ (cid:12)(cid:12)(cid:12)(cid:12) + X k (cid:12)(cid:12)(cid:12)(cid:12) ∂ k f Φ ∗ − (cid:16) y | y | · ∇ y f Φ ∗ (cid:17) y k | y | + i f A ∗ k f Φ ∗ (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) dy ≤ − Z R (cid:12)(cid:12)(cid:12)(cid:12) y | y | · ∇ y f Φ ∗ (cid:12)(cid:12)(cid:12)(cid:12) dy . Thus, we find that 4 Z R (cid:12)(cid:12)(cid:12)f Φ ∗ (cid:12)(cid:12)(cid:12) dy ≤ − Z R (cid:12)(cid:12)(cid:12)(cid:12) y | y | · ∇ y f Φ ∗ (cid:12)(cid:12)(cid:12)(cid:12) dy − Z R (cid:12)(cid:12)(cid:12) y · ∇ y f Φ ∗ (cid:12)(cid:12)(cid:12) dy ! , and in view of the support properties (8.72) of f Φ ∗ , we must have f Φ ∗ ≡ (cid:3) To conclude the rigidity argument, we need to reduce to the additional assumption λ ( t ) ≥ λ > t ∈ R made in the statement of Proposition 8.9. However, this follows as in Lemma 10.18in [20].Finally, we summarize the proof of the global existence assertion in Theorem 1.2 and addressthe proof of the scattering assertion. Proof of Theorem 1.2.
From the concentration compactness step in Section 7 and the rigidity argu-ment in this section, we infer the existence of a non-decreasing function K : (0 , ∞ ) → (0 , ∞ ) withthe following property: Let ( A x , φ )[0] be admissible Coulomb class data of energy E . Then thereexists a unique global admissible solution ( A , φ ) to MKG-CG with initial data ( A x , φ )[0] satisfyingthe a priori bound (cid:13)(cid:13)(cid:13) ( A x , φ ) (cid:13)(cid:13)(cid:13) S ( R × R ) ≤ K ( E ) . It remains to prove that the dynamical variables ( A x , φ ) of the global solution ( A , φ ) to MKG-CGscatter to finite energy free waves. To this end it su ffi ces to show that k (cid:3) A j k N ( R × R ) < ∞ for j = , . . . , k (cid:3) φ k N ( R × R ) < ∞ . Here the only concern is to bound the low-high interactions in the magnetic interaction term − iA f reej ∂ j φ in the equation for φ , where A f reej is the free wave evolution of the initial data A j [0]. Inthis case, the bound k ( A x , φ ) k S ( R × R ) < ∞ does not su ffi ce and we have to invest our strong assump-tions about the spatial decay of the initial data. More precisely, from [22] we have the followingestimate for dyadic frequencies k ≤ k − C , (cid:13)(cid:13)(cid:13) P k A f reej P k ∂ j φ (cid:13)(cid:13)(cid:13) N . (cid:13)(cid:13)(cid:13) P k A x [0] (cid:13)(cid:13)(cid:13) ˙ H x × L x (cid:13)(cid:13)(cid:13) P k φ (cid:13)(cid:13)(cid:13) S .
58 CONCENTRATION COMPACTNESS FOR THE CRITICAL MKG EQUATION
Thus, we may bound the low-high interactions in the magnetic interaction term − iA f reej ∂ j φ by (cid:13)(cid:13)(cid:13)(cid:13)X k ∈ Z P ≤ k − C A f reej P k ∂ j φ (cid:13)(cid:13)(cid:13)(cid:13) N . (cid:13)(cid:13)(cid:13) A x [0] (cid:13)(cid:13)(cid:13) ℓ ( ˙ H x × L x ) k φ k S . To see that (cid:13)(cid:13)(cid:13) A x [0] (cid:13)(cid:13)(cid:13) ℓ ( ˙ H x × L x ) is finite, we observe that in the Coulomb gauge we have for j = , . . . , A j = − ∆ − ∂ l F jl . Hence, we obtain for j = , . . . , X k ∈ Z k P k A j (0) k ˙ H x . X k ∈ Z X l = k P k ∆ − ∂ l F jl (0) k ˙ H x . X k ∈ Z X l = k P k F jl (0) k L x < ∞ , since the spatial curvature components F jl (0) are of Schwartz class by assumption. Similarly, weconclude that k ∂ t A x (0) k ℓ L x < ∞ , which finishes the proof. (cid:3) R eferences
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