Small data global existence and decay for two dimensional wave maps
aa r X i v : . [ m a t h . A P ] O c t Small data global existence and decay for twodimensional wave maps
Willie Wai Yeung Wong * Based on commit ab2ee4c of 2019-10-02 11:04
Abstract
We prove the small-data global existence for the wave-map equa-tion on R , using a variant of the vector field method. The main innova-tions lie in the introduction of two new linear estimates. First is the controlof the dispersive decay of the solution φ itself (as opposed to its derivatives),via a logarithmic weighted Hardy inequality. This control has not been pre-viously established using purely physical space methods in two spatial di-mensions. Second is a point-wise decay estimate for a twisted derivative of φ associated to the Morawetz K multiplier, that cannot be reduced to point-wise decay estimates associated to the standard commutator vector fields.As the linear theory is largely similar between dimensions, and in view ofthe novelty of the second innovation even in higher dimensions, we includea discussion of the method for R ,d in general. Both linear estimates areused crucially in our study: the control of the wave-map equation in thesmall data regime necessarily requires understanding the dispersive behav-ior of the bare solution φ itself, by virtue of the equation it satisfies. Thepoint-wise decay for the twisted derivative allows us to avoid certain top-order logarithmic energy growths; this is indispensable for extending ourargument from the case of compactly supported initial data (to which ourmethods are most naturally adapted) to initial data that are strongly local-ized but not necessarily of compact support, via an iterative construction.
1. Introduction
We consider the equation(1.1) (cid:3) η φ = φ · η (d φ, d φ )which models the wave-map equation, in the small data regime. Here η is theMinkowski metric on R ,d , represented by the matrix diag( − , , . . . ,
1) in rectan-gular coordinates, and so in standard coordinates we can rewrite (1.1) as − ∂ tt φ + d X i =1 ∂ ii φ = φ h − ( ∂ t φ ) + d X i =1 ( ∂ i φ ) i . * Michigan State University, East Lansing, USA; [email protected] φ = ( φ , . . . , φ N ) : R ,d → R N given by thecomponent-wise equations (cid:3) η φ C = N X A,B =1 Γ CAB ( φ ) η (d φ A , d φ B )where the Christo ff el symbols Γ captures the geometry of the target manifold.For solutions which are perturbations of a constant solution, we can perform aTaylor expansion of the functions Γ and arrive at (1.1), which captures the lead-ing order contributions. In the small-data regime where decay can be proven,one can rather straightforwardly upgrade results concerning (1.1) to the fullwave-map system. For more about the wave-map equation in general, see [15].Using the dispersive decay of the linear wave equation, one can easily show, us-ing the vector field method of Klainerman, that when the domain has spatialdimension d ≥ ffi ciently small in certain weighted energy norm. In-deed, multiplying (1.1) by ∂ t φ and integrating by parts on the domain [0 , T ] × R d ,we see the energy inequality(1.2) 12 Z { T }× R d ( ∂ t φ ) + (cid:12)(cid:12)(cid:12) ∇ φ (cid:12)(cid:12)(cid:12) d x ≤ Z { }× R d ( ∂ t φ ) + (cid:12)(cid:12)(cid:12) ∇ φ (cid:12)(cid:12)(cid:12) d x + T Z Z R d (cid:12)(cid:12)(cid:12) φ ( ∂φ ) (cid:12)(cid:12)(cid:12) d x d t. By Gronwall, we see then(1.3) Z { T }× R d ( ∂ t φ ) + (cid:12)(cid:12)(cid:12) ∇ φ (cid:12)(cid:12)(cid:12) d x ≤ (cid:20) Z { }× R d ( ∂ t φ ) + (cid:12)(cid:12)(cid:12) ∇ φ (cid:12)(cid:12)(cid:12) d x (cid:21) · exp (cid:20) C T Z sup R d (cid:12)(cid:12)(cid:12) φ∂φ (cid:12)(cid:12)(cid:12) d t (cid:21) . Using that the expected L ∞ decay for the linear wave equation is at the rate (1 + t ) (1 − d ) / , we see that when d ≥ L ∞ decay of the solution.In dimension d = 2 we see, however, that the expected L ∞ decay for the linearwave equation is only at the rate (1+ t ) − / , and so sup R (cid:12)(cid:12)(cid:12) φ∂φ (cid:12)(cid:12)(cid:12) is not integrable intime. This di ffi culty can in principle be overcome by the fact that the nonlinearity η (d φ, d φ ) satisfies the null condition [10, 4, 1, 2], which would imply that for allintents and purposes the termsup R (cid:12)(cid:12)(cid:12) φ∂φ (cid:12)(cid:12)(cid:12) . (1 + t ) − / . t − / decay upgrades the nonlinearity to be integrable in time.There is however, one other di ffi culty in dimension d = 2, and this relates toobtaining the decay estimate for φ itself. Classical Klainerman-Sobolev inequal-ities, when applied to the standard energy estimates, only control (cid:12)(cid:12)(cid:12) ∂φ (cid:12)(cid:12)(cid:12) in L ∞ .One can naïvely estimate (cid:12)(cid:12)(cid:12) φ ( t ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) φ (0) (cid:12)(cid:12)(cid:12) + R t (cid:12)(cid:12)(cid:12) ∂φ ( s ) (cid:12)(cid:12)(cid:12) d s , but at a loss of one factorof t decay (see also discussion in Remark 2, §6.5 of [6]). A better method for di-mensions d ≥ ∂ t φ and integrating by parts, one mul-tiplies by ( t + r ) ∂ t φ + 2 tr∂ r φ + ( d − tφ ). When d ≥
3, the resulting energy iscoercive on (cid:13)(cid:13)(cid:13) φ (cid:13)(cid:13)(cid:13) L ( R d ) , and direct applications of the classical vector field methodyields also pointwise decay for the solution φ itself. This argument is worked outin detail in [7]. When d = 2, however, the coercivity is lost, and while we havegood control on η (d φ, d φ ) by virtue of the Klainerman-Sobolev inequalities andthe null condition, we have no control on the φ factor in the nonlinearity in (1.1).The goal of the present manuscript is to present a modified vector field methodthat allows us to recover, when d = 2, decay estimates on φ itself up to a logarith-mic loss. Through this control we are able to prove: Let φ , φ ∈ C ∞ ( B (0 , k ≥
4. There exists an ǫ > (cid:13)(cid:13)(cid:13) φ (cid:13)(cid:13)(cid:13) H k +1 + (cid:13)(cid:13)(cid:13) φ (cid:13)(cid:13)(cid:13) H k < ǫ , there exists a global solution to (1.1) with φ (0 , x ) = φ ( x )and ∂ t φ (0 , x ) = φ ( x ). Furthermore the solution decays to zero as t → + ∞ . (cid:4) We will state and prove a more precise version of this theorem, including theboundedness of weighted energies as well as the pointwise peeling estimates, inSection 6. The above argument uses crucially that the initial data has boundedsupport. For data that has unbounded support (but su ffi cient decay near infinityin a Sobolev sense), this can be overcome via an approximation procedure takingadvantage of the scaling invariance of the wave-map equation. We discuss thisin Section 8. Crucial to this argument is a nonlinear stability result for large-data“dispersive” solutions to the wave-map equation, in the spirit of [16], which maybe of independent interest and which we discuss in Section 7.In closing, we note that the d ≥ d = 2. We include a discussion of the higher dimensionalresults to highlight the di ff erences, as well as to showcase the minor notationaladvantage in our system which allows us to reduce the set of commutators usedfor analyzing the nonlinear problems. 3 cknowledgements— The author would like to thank Pin Yu and Shiwu Yangfor useful comments. Part of this research was conducted when the author wasa visiting scholar at The Institute of Mathematical Sciences at The Chinese Uni-versity of Hong Kong. The author would also like to express his gratitude toZhouping Xin, Po Lam Yung, and The IMS for their hospitality during his visit.
2. Hyperboloidal Global Sobolev Inequalities
We begin by o ff ering a streamlined proof of a family of weighted global Sobolevinequalities adapted to hyperboloids in Minkowski space. Inequalities of thistype was first introduced by Klainerman [11, 8] for the study of the decay ofthe Klein-Gordon equation, and has been adapted by LeFloch and Ma [14] fortreatment of coupled Klein-Gordon–Wave systems. The main novelty in thissection of our work is the geometric formulation of our main inequalities as weighted Sobolev inequalities adapted to the Lorentz boosts, in the sense that stan-dard Sobolev inequalities on Euclidean domains are adapted to coordinate par-tial derivatives. In applications this means that the only commutator vector fieldswe will use are Lorentz boosts , and in particular we can omit commutations withspatial rotations which are used in arguments based on Klainerman’s originalwork.We restrict our attention to the interior of the future light-cone in Minkowskispace, that is, the set { t > | x |} ⊂ R ,d . (We use x , . . . , x d for the standard coordi-nates on R d , and use x or t for the time coordinate.) Let(2.1) τ def = q t − | x | and denote by Σ τ its level sets. The Σ τ are hyperboloids that asymptote to theforward light-cone centered at the origin.Denote by L i , i ∈ { , . . . , d } the Lorentz boost vector fields(2.2) L i def = t∂ x i + x i ∂ t . By definition L i are tangent to the hypersurfaces Σ τ . The set { L i } is linearly inde-pendent and span the tangent space of Σ τ . We will also mention the rotationalvector fields(2.3) Ω ij def = x i ∂ x j − x j ∂ x i . The Ω ij are also tangent to Σ τ , and they admit the decomposition(2.4) Ω ij = x i t L j − x j t L i . L i , L j ] = Ω ij [ Ω ij , Ω jk ] = Ω ik [ L i , Ω ij ] = L j We can define a system of radial coordinates on { t > | x |} \ { x = 0 } . Let(2.6) ( τ, ρ, θ ) ∈ R + × R + × S d − (where we identify S d − canonically as the unit sphere in R d ) be the coordinateof the point(2.7) t = τ cosh( ρ ) ,x = τ sinh( ρ ) · θ. The Minkowski metric takes the warped product form(2.8) η = − d t + d X i =1 (d x i ) = − d τ + τ d ρ + τ sinh( ρ ) d θ were by d θ we refer to the standard metric on S d − . Then clearly the inducedRiemannian metrics on Σ τ , which we denote by h τ , have the coordinate expres-sions relative to the ( ρ, θ ) coordinates(2.9) h τ = τ (d ρ + sinh( ρ ) d θ )( h τ ) − = 1 τ (cid:16) ∂ ρ ⊗ ∂ ρ + 1sinh( ρ ) ∂ θ ⊗ ∂ θ (cid:17) where ∂ θ ⊗ ∂ θ is the inverse standard metric on S d − .One can check the identity(2.10) d X i =1 L i ⊗ L i = ∂ ρ ⊗ ∂ ρ + cosh( ρ ) sinh( ρ ) ∂ θ ⊗ ∂ θ which implies that(2.11) ( τ − h τ ) − + X i 2. Let f be a function on the cylinder R + × S d − with the product metric. Thensup ρ> (cid:12)(cid:12)(cid:12) f ( ρ, θ ) (cid:12)(cid:12)(cid:12) . X k ≤ s ( d ) ∞ Z Z S d − (cid:12)(cid:12)(cid:12) ∇ k f (cid:12)(cid:12)(cid:12) d θ d ρ. (cid:4) , define the index set I m = { , . . . , d } m , and set I ≤ m def = [ ≤ m ′ ≤ m I m ′ ; I def = [ ≤ m ′ I m ′ . For α ∈ I , its order (denoted | α | ), is the (unique) non-negative integer m suchthat α ∈ I m . For α = ( α , . . . , α m ) ∈ I m , we denote by L α the m th order di ff erentialoperator(2.17) L α φ def = L α m L α m − · · · L α φ. By convention, for the unique element α ∈ I we define the corresponding L α φ def = φ .Putting everything together we have the following global Sobolev inequality adaptedto Lorentz boosts. Let ℓ ∈ R be fixed. We have the following uniform estimate (the implicit constantdepending on d and ℓ ) for functions f defined on the set { t > | x |} ⊂ R ,d : (cid:12)(cid:12)(cid:12) f ( τ, ρ, θ ) (cid:12)(cid:12)(cid:12) . τ − d cosh( ρ ) − d − ℓ X α ∈ I ≤ s ( d ) Z Σ τ cosh( ρ ) ℓ | L α f | τ d sinh( ρ ) d − d θ d ρ | {z } dvol Σ τ . (cid:4) Proof First we prove the estimate for ρ < : using that ( Σ τ , h τ ) is conformal to( H d , h ) with h τ = τ h , we see that they have the same Levi-Civita connection. Sothe first part of Proposition 2.16 impliessup ρ< | f | . X k ≤ s ( d ) Z Σ τ ∩{ ρ< } (cid:12)(cid:12)(cid:12) ∇ k f (cid:12)(cid:12)(cid:12) τ − h τ dvol τ − h τ . Next, observe that for a fixed ( ρ, θ ), the higher derivative (cid:12)(cid:12)(cid:12) ∇ k L i (cid:12)(cid:12)(cid:12) τ − h τ is indepen-dent of τ , and hence has universal bounds when ρ < 2. Observing that L i ∇ a f = ∇ a L i ( f ) − ∇ ∇ a L i f We cannot use the standard multi-index notation, whose definition uses the fact that partialderivatives commute. Similarly, we cannot use the natural generalization of the multi-index no-tation as usually used in discussion of the vector field method, since there the set of first orderoperators used generate a Lie algebra (are closed under commutations). As we saw in (2.5), theLorentz boost vector fields themselves are not closed under commutation. 7e see after applying Lemma 2.12 repeatedly by induction that X k ≤ s ( d ) (cid:12)(cid:12)(cid:12) ∇ k f (cid:12)(cid:12)(cid:12) τ − h τ . X α ∈ I ≤ s ( d ) | L α f | on the domain Σ τ ∩ { ρ < } , for some universal constant depending only on thedimension d . Using that cosh( ρ ) is bounded both above and below on the domain { ρ < } , we see our claim follows, with the τ − d decay originating from τ − d dvol Σ τ =dvol τ − h τ .Next we prove the estimate for ρ > : for this we apply the second part ofProposition 2.16 to the function f cosh( ρ ) ℓ/ sinh( ρ ) ( d − / . We note that when ρ > 1, both cosh( ρ ) and sinh( ρ ) are uniformly comparable to e ρ , and so we have( ∂ ρ ) j cosh( ρ ) ℓ/ sinh( ρ ) ( d − / ≈ cosh( ρ ) ℓ/ sinh( ρ ) ( d − / . Hence immediately weget(2.19) sup ρ> h | f | cosh( ρ ) ℓ sinh( ρ ) d − i . X k ≤ s ( d ) ∞ Z cosh( ρ ) ℓ (cid:12)(cid:12)(cid:12) ∇ k f (cid:12)(cid:12)(cid:12) sinh( ρ ) d − d θ d ρ ;we remark that the Levi-Civita connection and the norm are both taken withrespect to the product metric on R + × S d − . Note that the inverse product metrichas the decompsition ∂ ρ ⊗ ∂ ρ + X i 3. Decay of linear waves: ∂ t -energy In this section we apply Theorem 2.18 to get point-wise decay for the derivatives ∂φ of solutions to the linear wave equation (cid:3) φ = 0 on R ,d . The main contri-bution of this paper is the explicit geometric decomposition (3.5) of the energydensity of the linear wave equation along a hyperboloid. This makes explicitthe fact that we see improved decay in “good directions” already at the level of8rst derivatives, and is compatible with our formulation of the global Sobolevinequality in Theorem 2.18.Let φ : R ,d → R , define the standard energy quantity(3.1) (cid:16) E [ φ ] (cid:17) def = Z { }× R d (cid:12)(cid:12)(cid:12) ∂ t φ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ∇ φ (cid:12)(cid:12)(cid:12) d x. Let φ be a solution to (cid:3) φ = 0. Then we have the estimate in the region { t > | x |} (cid:12)(cid:12)(cid:12) L i φ (cid:12)(cid:12)(cid:12) . τ − d cosh( ρ ) − d | {z } t − d X α ∈ I ≤ s ( d ) E [ L α φ ] , (cid:12)(cid:12)(cid:12) ∂ t φ (cid:12)(cid:12)(cid:12) . τ − d cosh( ρ ) − d | {z } t − d ( t + r ) − / ( t − r ) − / X α ∈ I ≤ s ( d ) E [ L α φ ] . (cid:4) Note that the coe ffi cients of the vector field L i in rectangular coordinates havesize ≈ t , so the first inequality really states that certain “good derivatives” (theones in the span of L i with size ≈ /t coe ffi cients) decay like t − d/ , which capturesexactly the expected improve decay over the naive t − ( d − / . We also see from thesecond inequality that in the “generic direction” ∂ t φ decays only uniformly like t − ( d − / as expected; however, we also see improved interior decay by ( t − r ) − / .These peeling properties do not obviously hold using the classical vector fieldmethod, using the ∂ t -energy, for the lowest order derivatives of the solution. (cid:4) We do not need compact support for the estimates derived in this section. Seealso Footnote 2 below.An interesting feature of our decay estimates, when compared to the classicalargument of Klainerman, is that we do not need to commute with the full set ofgenerators of Poincaré group, nor do we need to commute with the scaling vectorfield S = t∂ t + P x i ∂ x i . Here we only commute with the Lorentz boosts and notspatial rotations. (cid:4) Proof Let Q ab as usual denote the stress energy tensor ∂ a φ∂ b φ − η ab η (d φ, d φ ).Consider the energy current J a = Q ab ( ∂ t ) b . It is standard that the space-time divergence ∇ a J a = (cid:3) φ∂ t φ , which in our casevanishes. 9ntegrating in the region between Σ τ and { } × R d , by the divergence theorem wehave Z Σ τ Q ( ∂ τ , ∂ t ) dvol Σ τ ≤ E [ φ ] . We can compute that ∂ t = cosh( ρ ) ∂ τ − τ − sinh( ρ ) ∂ ρ which implies Q ( ∂ t , ∂ τ ) = cosh( ρ ) Q ( ∂ τ , ∂ τ ) − τ − sinh( ρ ) Q ( ∂ ρ , ∂ τ ) . By orthogonality we have Q ( ∂ ρ , ∂ τ ) = ∂ ρ φ∂ τ φ ;and a standard computation gives Q ( ∂ τ , ∂ τ ) = 12 ( ∂ τ φ ) + 12 τ ( ∂ ρ φ ) + 12 τ sinh( ρ ) (cid:12)(cid:12)(cid:12) ∂ θ φ (cid:12)(cid:12)(cid:12) . Combining the two and completing the square we get the following identity2 Q ( ∂ τ , ∂ t ) = cosh( ρ ) τ sinh( ρ ) (cid:12)(cid:12)(cid:12) ∂ θ φ (cid:12)(cid:12)(cid:12) + 1 τ cosh( ρ ) ( ∂ ρ φ ) + 1cosh( ρ ) ( ∂ t φ ) . By (2.10) we finally have(3.5) Q ( ∂ τ , ∂ t ) = 12 τ cosh( ρ ) d X i =1 ( L i φ ) + 12 cosh( ρ ) ( ∂ t φ ) . Now, as L i commutes with the wave operator we have that if φ solves (cid:3) φ = 0, sodoes L α φ . This implies(3.6) X α ∈ I ≤ s ( d ) Z Σ τ τ cosh( ρ ) d X i =1 ( L i L α φ ) + 1cosh( ρ ) ( ∂ t L α φ ) dvol Σ τ ≤ X α ∈ I ≤ s ( d ) E [ L α φ ] . Note that we don’t necessarily have a conservation law; there is formally another boundaryintegral along null infinity that captures the energy radiated away. By the construction of theenergy current, this quantity is signed and its omission in the formula below is what gives the ≤ sign instead of the = sign. L i φ now follows immediately from Theorem 2.18.For ∂ t φ , we need to control the integral of ( L α ∂ t φ ) ; the energy estimate onlycontrols ( ∂ t L α φ ) . We do so by computing the commutator:[ L i , ∂ t ] = − ∂ x i , [ L i , ∂ x i ] = − δ ij ∂ t . By induction this implies (cid:12)(cid:12)(cid:12) L α ∂ t φ (cid:12)(cid:12)(cid:12) . X | β | ≤| α | (cid:12)(cid:12)(cid:12) ∂ t L β φ (cid:12)(cid:12)(cid:12) + X | β | ≤| α |− d X i =1 (cid:12)(cid:12)(cid:12) ∂ x i L β φ (cid:12)(cid:12)(cid:12) . Noting that ∂ x i = 1 t ( L i − x i ∂ t )we can bound X | β | ≤| α |− d X i =1 (cid:12)(cid:12)(cid:12) ∂ x i L β φ (cid:12)(cid:12)(cid:12) ≤ X | β | ≤| α |− (cid:12)(cid:12)(cid:12) ∂ t L β φ (cid:12)(cid:12)(cid:12) + 1 τ cosh ρ X | β | ≤| α | (cid:12)(cid:12)(cid:12) L β φ (cid:12)(cid:12)(cid:12) . And therefore we have the estimate(3.7) X α ∈ I ≤ s ( d ) Z Σ τ τ cosh( ρ ) d X i =1 ( L i L α φ ) + 1cosh( ρ ) ( L α ∂ t φ ) dvol Σ τ . X α ∈ I ≤ s ( d ) Z Σ τ τ cosh( ρ ) d X i =1 ( L i L α φ ) + 1cosh( ρ ) ( ∂ t L α φ ) dvol Σ τ . Applying Theorem 2.18 we also get the decay estimate for ∂ t φ . (cid:3) 4. Decay of linear waves: K -energy The Morawetz energy can also be used with the hyperboloidal foliation; this givesimproved decay properties. As in the previous section, the main contribution isthe identity (4.6). For convenience, we shall assume that the data has compactsupport in this section. Let φ be a solution to (cid:3) φ = 0. Suppose φ (2 , x ) and ∂ t φ (2 , x ) are both supportedon the ball of radius 1 centered at the origin. Then in the region t > max(2 , | x | )we have (cid:12)(cid:12)(cid:12) L i φ (cid:12)(cid:12)(cid:12) . τ − d cosh( ρ ) − d (cid:16)(cid:13)(cid:13)(cid:13) ∂ t φ (2 , —) (cid:13)(cid:13)(cid:13) H s ( d ) + (cid:13)(cid:13)(cid:13) φ (2 , —) (cid:13)(cid:13)(cid:13) H s ( d )+1 (cid:17) . (cid:4) .2 Remark We see that this gives a gain of an additional factor of τ − decay for derivatives inthe “good directions”. In terms of uniform-in- x decay, this means a gain of t − / (as well as a gain of ( t − r ) − / which is not uniform on constant t hyperplanes). (cid:4) Proof Define the modified current(4.3) K a = Q ab K b + d − t∂ a ( φ ) − d − φ ∂ a t where K is the vector field(4.4) K def = ( t + | x | ) ∂ t + 2 tr∂ r = τ cosh( ρ ) ∂ τ + τ sinh( ρ ) ∂ ρ . It is a simple exercise to check that the space-time divergence(4.5) ∇ a K a = (cid:3) φ [ K φ + ( d − tφ ] . By virtue of finite speed of propagation and the divergence theorem, we havethat Z Σ τ K a ( ∂ τ ) a dvol Σ τ = Z { }× R d K a ( ∂ t ) a d x. The integral on the right is entirely determined by the initial data, so we focuson evaluating the integral over Σ τ . The integrand is K a ( ∂ τ ) a = Q ( K , ∂ τ ) + d − τ cosh( ρ ) ∂ τ ( φ ) − d − φ cosh( ρ ) . The middle term, observe, can be re-written in terms of K and ∂ ρ . K a ( ∂ τ ) a = Q ( K , ∂ τ ) + d − τ K ( φ ) − d − 12 sinh( ρ ) ∂ ρ ( φ ) − d − φ cosh( ρ )= Q ( K , ∂ τ ) + d − τ K ( φ ) − d − ∂ ρ [sinh( ρ ) φ ] . We wish to integrate Z Σ τ K a ( ∂ τ ) a dvol Σ τ = ∞ Z Z S d − ( Q ( K , ∂ τ ) + d − τ K ( φ ) − d − ∂ ρ [sinh( ρ ) φ ] ) τ d sinh( ρ ) d − d θ d ρ. φ hascompact support on each Σ τ , we can integrate the final term in the braces byparts to get Z Σ τ K a ( ∂ τ ) a dvol Σ τ = ∞ Z Z S d − ( Q ( K , ∂ τ ) + d − τ K ( φ ) + ( d − ρ ) φ ) τ d sinh( ρ ) d − d θ d ρ. Next, we can write Q ( K , ∂ τ ) = τ cosh( ρ ) Q ( ∂ τ , ∂ τ ) + τ sinh( ρ ) Q ( ∂ τ , ∂ ρ )= 12 cosh( ρ ) " τ ( ∂ τ φ ) + ( ∂ ρ φ ) + 1sinh( ρ ) (cid:12)(cid:12)(cid:12) ∂ θ φ (cid:12)(cid:12)(cid:12) + τ sinh( ρ ) ∂ τ φ∂ ρ φ = 12 cosh( ρ ) (cid:26)h τ cosh( ρ ) ∂ τ φ + sinh( ρ ) ∂ ρ φ i + ( ∂ ρ φ ) + cosh( ρ ) sinh( ρ ) (cid:12)(cid:12)(cid:12) ∂ θ φ (cid:12)(cid:12)(cid:12) (cid:27) = 12 cosh( ρ ) (cid:20) τ ( K φ ) + d X i =1 ( L i φ ) (cid:21) . So finally, completing the square one more time we get(4.6) Z Σ τ K a ( ∂ τ ) a dvol Σ τ = Z Σ τ 12 cosh( ρ ) d X i =1 ( L i φ ) + 12 τ cosh( ρ ) h K φ + ( d − tφ i dvol Σ τ . Applying this estimate to L α φ for α ∈ I ≤ s ( d ) , we see that X α ∈ I ≤ s ( d ) Z Σ τ ρ ) d X i =1 (cid:12)(cid:12)(cid:12) L i L α φ (cid:12)(cid:12)(cid:12) dvol Σ τ is uniformly bounded by the initial data, which in turn is bounded by (cid:13)(cid:13)(cid:13) ∂ t φ (2 , —) (cid:13)(cid:13)(cid:13) H s ( d ) + (cid:13)(cid:13)(cid:13) φ (2 , —) (cid:13)(cid:13)(cid:13) H s ( d )+1 (we make use of the compact support again to control the various weights). Oneapplication of Theorem 2.18 gives the desired decay. (cid:3) r p -weighted energy estimates of Dafermos and Rodnianski [5].To make this comparison more explicit, we show that using the estimate (4.6) onecan also derive decay estimates for the twisted derivative K φ + ( d − tφ . Observefirst the commutator identity(4.7) [ L i , K + ( d − t ] = x i t ( K + ( d − t ) + τ t L i which implies that(4.8) Z τ cosh( ρ ) (cid:12)(cid:12)(cid:12) L i [ K + ( d − t ] φ (cid:12)(cid:12)(cid:12) dvol Σ τ . Z τ cosh( ρ ) (cid:18)(cid:12)(cid:12)(cid:12) [ K + ( d − t ] L i φ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) [ K + ( d − t ] φ (cid:12)(cid:12)(cid:12) (cid:19) + 1cosh( ρ ) ( L i φ ) dvol Σ τ . Therefore applying this to L α φ , we see that the energy identity shows that X α ∈ I ≤ s ( d ) Z Σ τ ρ ) d X i =1 (cid:12)(cid:12)(cid:12) L α [ K + ( d − t ] φ (cid:12)(cid:12)(cid:12) dvol Σ τ is uniformly bounded by the K -energies for up to s ( d ) derivatives of the solution,which after an application of Theorem 2.18 gives that(4.9) | K φ + ( d − tφ | . τ − d cosh( ρ ) − d . The point-wise bound (4.9) is, to the author’s knowledge, new. Observe thatsince K decomposes as K = τ ∂ t + 2 P di =1 x i L i , this estimate cannot be obtainedfrom Proposition 4.1 and Proposition 3.2. The twisted structure involving thelower order term ( d − tφ is important as this imparts additional cancellations.Indeed, the pointwise estimates for ∂ t φ , L i φ , and ( K + ( d − t ) φ are independentof each other in the sense that any two of the three is not su ffi cient to derive thethird. (cid:4) This point-wise bound (4.9) will play an important role in the discussion in thesequel. When applying the K -energy to nonlinear applications, one pays for theadditional decay gained by requiring stronger estimates on the inhomogeneities.In traditional applications of the vector field method this often translates to alogarithmic growth of the top-order energies, when certain terms most natu-rally estimated using ( K + ( d − t ) φ are instead estimated using the decompo-sition described in the previous remark. We will take full advantage of this esti-mate by performing the decomposition (6.8) and fully expressing the nonlinear-ities, where necessary, using this operator. This allows us to avoid the log-loss14f top-order energies, which is then itself crucial for proving our perturbationTheorem 7.4. (cid:4) 5. Decay of φ itself In this section we prove that the energy integrals considered in the previous twosections given in (3.5) and (4.6) have nice coercivity properties on the L -integralfor φ itself. In the context of the study of wave equations, our results highlightthe following two ideas:1. First, it shows that the Morawetz energy is not necessary for controllingthe decay of the solution φ itself. This has applications to situations wherethe Morwawetz energy is not available, due to the lack of conformal sym-metry of the underlying equation. A particular such application is to thestudy of wave equations on Kaluza-Klein backgrounds. See the discussionin Remark 5.8.2. Second, it shows that one can in fact obtain estimates on the solution φ itself using purely physical space techniques, in dimension d = 2. Suchestimates were previously unavailable. The downside to our argument isthat compact support of initial data seems essential in this case.The basic technical tool for obtaining the coercivity are the following Hardy in-equalities. d ≥ ) Let d ≥ 3, then Z Σ τ ρ ) φ dvol Σ τ ≤ d − Z Σ τ ρ ) d X i =1 ( L i φ ) dvol Σ τ . (cid:4) d = 2 ) When d = 2, then Z Σ τ ρ ) φ dvol Σ τ . Z Σ τ (1 + ρ )cosh( ρ ) d X i =1 ( L i φ ) dvol Σ τ . (cid:4) When d ≥ 3, noting that sinh( ρ ) d − / cosh( ρ ) grows asymptotically like sinh( ρ ) d − ,we see that our Hardy inequality can be viewed as the statement that the hy-perbolic space H d − of one lower dimension has a spectral gap (and hence ˚ H controls L ). When d = 2 this obviously degenerates, and the best we can have15s essentially the Hardy inequality on the half-line. Note that compared to theexponential weight in ρ , the polynomial ρ is e ff ectively a logarithm. (cid:4) The Hardy type inequalities of Lemma 5.1 is well-known in the study of weightedSobolev spaces. See, e.g. Theorem 1.3 in [3]; the d = 2 case corresponds to theforbidden δ = 0 case in that theorem. Our main contribution in this section isLemma 5.2. We include the proof of both for completeness. (cid:4) Proof (Lemma 5.1) For any f : R + → R that decays rapidly as ρ → ∞ , observethat ∞ Z ∂ ρ h sinh( ρ ) α f ( ρ ) i d ρ = 0by the fundamental theorem of calculus. So we have α ∞ Z f ( ρ ) cosh( ρ ) sinh( ρ ) α − d ρ ≤ (cid:12)(cid:12)(cid:12)(cid:12) ∞ Z f ( ρ ) f ′ ( ρ ) sinh( ρ ) α d ρ (cid:12)(cid:12)(cid:12)(cid:12) . Applying Cauchy-Schwarz to the right, and squaring both sides, give α ∞ Z f ( ρ ) cosh( ρ ) sinh( ρ ) α − d ρ ≤ ∞ Z [ f ′ ( ρ )] sinh( ρ ) α +1 cosh( ρ ) d ρ. Now set α = d − 2, so α + 1 = d − 1. We can integrating in the spherical directionsand use (2.10) to control ( ∂ ρ f ) . On the left we apply the simple observation thattanh( ρ ) ≤ 1. This leads to exactly the claimed inequality. (cid:3) Proof (Lemma 5.2) For any f : R + → R that decays rapidly as ρ → ∞ , we observethat ∞ Z ∂ ρ h ρf ( ρ ) i d ρ = 0which implies ∞ Z f ( ρ ) d ρ ≤ ∞ Z f ( ρ ) f ′ ( ρ ) ρ d( ρ ) ≤ (cid:20) ∞ Z ρ p ρ f ( ρ ) d ρ (cid:21) · (cid:20) ∞ Z ρ q ρ [ f ′ ( ρ )] d ρ (cid:21) 16y Cauchy-Schwarz. This implies, since ρ/ p ρ < ∞ Z f ( ρ ) d ρ ≤ ∞ Z [ f ′ ( ρ )] ρ q ρ d ρ. Now since lim ρ → tan( ρ ) /ρ = 1, there exists some constant C such that ρ ≤ C tan( ρ ) p ρ for all ρ > 0. This implies ∞ Z f ( ρ ) tanh( ρ ) d ρ ≤ C ∞ Z (1 + ρ ) tanh( ρ )[ f ′ ( ρ )] d ρ which, after integrating in the spherical directions and using (2.10) is exactly theclaimed inequality. (cid:3) Applying the Hardy inequalities to the energy estimates (3.6) and (4.6) we getthe following decay estimates for φ itself when dimension d ≥ 3. We omit theobvious proofs. d ≥ , ∂ t -energy) Let φ solve the linear wave equation on R ,d with finite energy initial data, in thesense that P α ∈ I ≤ s ( d ) E [ L α φ ] < ∞ . Then (cid:12)(cid:12)(cid:12) φ (cid:12)(cid:12)(cid:12) . t − d/ . (cid:4) d ≥ , K -energy) Let φ solve the linear wave equation on R ,d with compactly supported initialdata (e.g. satisfying the hypotheses of Proposition 4.1), then (cid:12)(cid:12)(cid:12) φ (cid:12)(cid:12)(cid:12) . t d − p ( t + r )( t − r ) . (cid:4) We note that the estimates are sharp. The finite ∂ t -energy condition is compat-ible with initial data φ (0 , x ) = 0 and ∂ t φ (0 , x ) = (1 + r ) − d/ − ǫ . One can checkusing the fundamental solution that the solution φ ( t, 0) for this data decays like t − d/ − ǫ .For the case with the K -energy, we note that the rate t − ( d − / is exactly the stan-dard L – L ∞ rate predicted by the fundamental solution (or alternatively station-ary phase arguments). This is correlated with the extra spatial weights of the K energy: Q ( K , ∂ t ) | t =0 = r Q ( ∂ t , ∂ t ). (cid:4) .8 Remark (Kaluza-Klein backgrounds) That estimates for φ itself is available using only the ∂ t energy has importantapplications. Previously the only control for φ itself in the vector field methodis via the K energy as described in [7]. The availability of the K energy how-ever depends on the conformal symmetries of the wave equation: indeed, thevector field K is also known as the “conformally inverted time translation” andcan be obtained by pushing forward the vector field ∂ t under the LorentzianKelvin transform. Equations not exhibiting (an approximate version of) this con-formal symmetry cannot be expected to have the same inequalities hold. Oneplace where the K energy is not available is the case of Klein-Gordon equations.However, as was shown originally by Klainerman [12], the t − d/ decay of φ itselfis available in using the ∂ t energy. In our formulation, this observation is due tothe fact that, for the Klein-Gordon equation (cid:3) φ − m φ = 0the analogue of the estimate (3.6) takes the form Z Σ τ τ cosh( ρ ) d X i =1 ( L i φ ) + 1cosh( ρ ) ( ∂ t φ ) + cosh( ρ ) m φ dvol Σ τ ≤ Z { }× R d ( ∂ t φ ) + (cid:12)(cid:12)(cid:12) ∇ φ (cid:12)(cid:12)(cid:12) + m φ d x. Then Theorem 2.18 implies that, after commuting with L α , (cid:12)(cid:12)(cid:12) φ (cid:12)(cid:12)(cid:12) . t − d/ directly. These type or arguments were also used in [14] for handling coupledsystems of wave and Klein-Gordon equations.We note here that similar arguments can also be made for linear waves on Kaluza-Klein backgrounds. More precisely, consider the space-time R ,d × S where S iscompact and equipped with some Riemannian metric g S . Take the spacetimemetric to be the product metric g = η + g S . Consider a solution φ to the linearwave equation (cid:3) g φ = 0 on this background. Due to the lack of conformal symme-try only the ∂ t -energy is available. For convenience denote by / ∇ the derivatives18angent to S , then one can check that the analogue of (3.6) in this case would be(5.9) Z Σ τ × S τ cosh( ρ ) d X i =1 ( L i φ ) + 1cosh( ρ ) ( ∂ t φ ) + cosh( ρ ) (cid:12)(cid:12)(cid:12) / ∇ φ (cid:12)(cid:12)(cid:12) dvol Σ τ × S ≤ Z { }× R d × S ( ∂ t φ ) + (cid:12)(cid:12)(cid:12) ∇ φ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) / ∇ φ (cid:12)(cid:12)(cid:12) d x d σ. And the Global Sobolev inequalities then implies the following decay rates in theforward lightcone { t > r } : (cid:12)(cid:12)(cid:12) φ (cid:12)(cid:12)(cid:12) . t − d/ (only when d ≥ (cid:12)(cid:12)(cid:12) L i φ (cid:12)(cid:12)(cid:12) . t − d/ (5.10b) (cid:12)(cid:12)(cid:12) ∂ t φ (cid:12)(cid:12)(cid:12) . t − d/ ( t + r ) − / ( t − r ) − / (5.10c) (cid:12)(cid:12)(cid:12) / ∇ φ (cid:12)(cid:12)(cid:12) . t − d/ (5.10d)which are exactly what one would expect from taking spectral projections on the S component and treating φ as a sum of solutions to the wave equation and aninfinite family of Klein-Gordon equations.We emphasize that classical vector field method, integrating along the hyper-planes { t } × R d × S , can only recover (cid:12)(cid:12)(cid:12) ∂φ (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) L i ∂φ (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) / ∇ ∂φ (cid:12)(cid:12)(cid:12) . t (1 − d ) / ( t − r ) − / and in particular cannot see any of the improved decay for / ∇ φ and its deriva-tives. (cid:4) We conclude this section with the decay estimates for φ when the dimension d =2. For technical reasons our proof only works when φ has compactly supportedinitial data, and hence we only state the version available from the K energy. d = 2 , K -energy) Let φ solve the linear wave equation on R , with compactly supported initialdata (e.g. satisfying the hypotheses of Proposition 4.1). Then (cid:12)(cid:12)(cid:12) φ (cid:12)(cid:12)(cid:12) . ln[( t + r )( t − r )] p ( t + r )( t − r ) . (cid:4) Proof Without loss of generality we shall assume that φ satisfies the conditionsof Proposition 4.1, this implies the quantity in (4.6) is bounded by the initialdata. 19bserve as the data, prescribed at time t = 2, is supported in the unit ball, thismeans by finite speed of propagation, on the support of φ , when t ≥ t ≥ r + 1. Within the forward lightcone from the origin this inequalitycan be rewritten as(5.12) τ (cosh ρ − sinh ρ ) ≥ ⇐⇒ τe − ρ ≥ ⇐⇒ ln τ ≥ ρ. Returning to (5.2) we see that, when τ ≥ Z Σ τ ρ ) φ dvol Σ τ . Z Σ τ (1 + ρ )cosh( ρ ) X ( L i φ ) dvol Σ τ ≤ (1 + ln τ ) Z Σ τ ρ ) X ( L i φ ) dvol Σ τ ≤ (1 + ln τ ) Z Σ τ K a ( ∂ τ ) a dvol Σ τ . Here, in the second inequality, we can use (5.12) thanks to the compact-dataassumption.Applying this estimate to φ and L α φ as before, we see by Theorem 2.18 that (cid:12)(cid:12)(cid:12) φ (cid:12)(cid:12)(cid:12) . τ − ln( τ ) as claimed. (cid:3) 6. Two-dimensional wave-maps Let us now apply the method developed in the previous sections to study theglobal existence and decay of solutions to (1.1) under the assumption that its ini-tial data has compact support. First we state the precise version of our theorem. Consider the future-evolution governed by (1.1) on R , , with initial data pre-scribed at time t = 2 such that φ (2 , x ) and ∂ t φ (2 , x ) are supported in the ball ofradius 1. Let k ≥ ǫ > (cid:13)(cid:13)(cid:13) φ (2 , —) (cid:13)(cid:13)(cid:13) H k +1 + (cid:13)(cid:13)(cid:13) ∂ t φ (2 , —) (cid:13)(cid:13)(cid:13) H k < ǫ, then there exists a future-global solution φ : [2 , ∞ ) × R d → R . Furthermore, thissolution obeys the following pointwise decay estimates: (cid:12)(cid:12)(cid:12) ∂ t L α φ (cid:12)(cid:12)(cid:12) . τ , | α | ≤ k − (cid:12)(cid:12)(cid:12) L α φ (cid:12)(cid:12)(cid:12) . τ , ≤ | α | ≤ k − (cid:12)(cid:12)(cid:12) φ (cid:12)(cid:12)(cid:12) . ln ττ . τ = q t − | x | as defined previously. (Note that by finite speed of propaga-tion, within the support of φ we have the lower bound τ ≥ √ (cid:4) As the equation is semilinear, local existence theory using energy method is stan-dard. In particular, for su ffi ciently small initial data it is clear that the solutionexists at least up to, and including { τ = 2 } (here we also use finite speed of propa-gation and boundedness of initial support). We concentrate on obtaining a priorienergy bounds when τ ≥ 2. Define E τ [ φ ] def = Z Σ τ τ cosh( ρ ) X i =1 ( L i φ ) + 1cosh( ρ ) ( ∂ t φ ) dvol Σ τ ;(6.2) F τ [ φ ] def = Z Σ τ ρ ) X i =1 ( L i φ ) + 1 τ cosh( ρ ) [ K φ + tφ ] dvol Σ τ . (6.3)We have the following energy identities for τ > τ > E τ [ φ ] ≤ E τ [ φ ] + 2 τ Z τ (cid:12)(cid:12)(cid:12) (cid:3) φ · ( ∂ t φ ) (cid:12)(cid:12)(cid:12) dvol Σ τ d τ (6.4) F τ [ φ ] ≤ F τ [ φ ] + 2 τ Z τ Z Σ τ (cid:12)(cid:12)(cid:12) (cid:3) φ · ( K φ + tφ ) (cid:12)(cid:12)(cid:12) dvol Σ τ d τ (6.5)where we used that the space-time volume-element is exactly dvol Σ τ d τ as seenin the metric decomposition (2.8).The basic strategy is standard: we wish to estimate E [ L α φ ] and F [ L α φ ] for all | α | less than some fixed constant. This will require estimating integrals of the forms(6.6a) Z Σ τ L α φ · η (d L α φ, d L α φ ) · ∂ t L α φ dvol Σ τ and(6.6b) Z Σ τ L α φ · η (d L α φ, d L α φ ) · ( K + t ) L α φ dvol Σ τ for | α | + | α | + | α | = | α | . In writing down the integrals above we implicitly usedthat vector fields acts on scalars by Lie di ff erentiation, and η is invariant under21orentz boosts L i , and that exterior di ff erentiation commutes with Lie di ff erenti-ation, so that if φ solves (1.1), then L α φ solves an equation of the form(6.7) (cid:3) L α φ = X c α ,α ,α ; α L α φη (d L α φ, d L α φ )where | α | + | α | + | α | = | α | , and c α ,α ,α ; α are combinatorial constants.Therefore we are led to consider integrals of the form Z Σ τ ζη (d ψ, d φ ) ∂ t ξ dvol Σ τ and Z Σ τ ζη (d ψ, d φ )( K ξ + tξ ) dvol Σ τ where the functions ζ, ψ, φ, ξ stand in place of L α derivatives of the original un-known. To get good control, we begin by decomposing the form η (d ψ, d φ ) = ( h τ ) − (d ψ, d φ ) − ( ∂ τ ψ )( ∂ τ φ ) . Using that ∂ τ = 1 tτ K − rtτ ∂ ρ = τt ∂ t + rtτ ∂ ρ we have η (d ψ, d φ ) = ( h τ ) − (d ψ, d φ ) − h tτ K ψ − rtτ ∂ ρ ψ ih τt ∂ t φ + rtτ ∂ ρ φ i . Introducing the short hand (cid:12)(cid:12)(cid:12) Lφ (cid:12)(cid:12)(cid:12) def = h P i =1 ( L i φ ) i we can bound (cid:12)(cid:12)(cid:12) η (d ψ, d φ ) (cid:12)(cid:12)(cid:12) . τ (cid:12)(cid:12)(cid:12) Lψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Lφ (cid:12)(cid:12)(cid:12) + 1 t (cid:12)(cid:12)(cid:12) K ψ∂ t φ (cid:12)(cid:12)(cid:12) + rt (cid:12)(cid:12)(cid:12) ∂ ρ ψ∂ t φ (cid:12)(cid:12)(cid:12) + rt τ (cid:12)(cid:12)(cid:12) K ψ∂ ρ φ (cid:12)(cid:12)(cid:12) + r t τ ( ∂ ρ φ )( ∂ ρ ψ ) . This we can bound by(6.8) (cid:12)(cid:12)(cid:12) η (d ψ, d φ ) (cid:12)(cid:12)(cid:12) . τ (cid:12)(cid:12)(cid:12) Lψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Lφ (cid:12)(cid:12)(cid:12) + 1 t (cid:12)(cid:12)(cid:12) ∂ t φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Lψ (cid:12)(cid:12)(cid:12) + 1 t (cid:12)(cid:12)(cid:12) ( K + t ) ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ t φ (cid:12)(cid:12)(cid:12) + 1 t (cid:12)(cid:12)(cid:12) ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ t φ (cid:12)(cid:12)(cid:12) + 1 tτ (cid:12)(cid:12)(cid:12) ( K + t ) ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Lφ (cid:12)(cid:12)(cid:12) + 1 τ (cid:12)(cid:12)(cid:12) ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Lφ (cid:12)(cid:12)(cid:12) where we simplified using r/t ≤ 1. 22 .9 Remark We use (6.8) rather than the more common decomposition (cid:12)(cid:12)(cid:12) η (d ψ, d φ ) (cid:12)(cid:12)(cid:12) . (cid:12)(cid:12)(cid:12) ∂ t ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ t φ (cid:12)(cid:12)(cid:12) + 1 t (cid:12)(cid:12)(cid:12) Lψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Lφ (cid:12)(cid:12)(cid:12) to take full advantage of both the L and L ∞ controls we have on ( K + t ) ψ , whichgives us additional decay. (cid:4) With this estimate, we can first control the nonlinearity when the quantity α in(6.6a) and (6.6b) has order | α | ≤ | α | − | α | ≤ | α | − ) We have the estimates (the L ∞ norms are taken along Σ τ ) Z Σ τ ζη (d ψ, d φ ) ∂ t ξ dvol Σ τ . ln ττ k ζ k L ∞ h(cid:13)(cid:13)(cid:13) Lφ (cid:13)(cid:13)(cid:13) L ∞ + (cid:13)(cid:13)(cid:13) ∂ t φ (cid:13)(cid:13)(cid:13) L ∞ i F τ [ ψ ] E τ [ ξ ] , Z Σ τ ζη (d ψ, d φ )( K ξ + tξ ) dvol Σ τ . ln τ k ζ k L ∞ h(cid:13)(cid:13)(cid:13) Lφ (cid:13)(cid:13)(cid:13) L ∞ + (cid:13)(cid:13)(cid:13) ∂ t φ (cid:13)(cid:13)(cid:13) L ∞ i F τ [ ψ ] F τ [ ξ ] . (cid:4) Proof The proof of the two cases are similar, we focus on the harder case whichis the second inequality. The basic idea is to put all the terms involving ξ and ψ in weighted L (controlled by E [ ξ ] , F [ ξ ] , F [ ψ ]), and the remainder ( ζ and φ ) in L ∞ . At parts of the argument we will also use the fact that by our finite speed ofpropagation property, cosh( ρ ) ≤ e ρ ≤ τ and hence tτ ≤ F [ ξ ] controls the square integral of ( K ξ + tξ ) / √ tτ . So it su ffi cesto show that √ tτη (d ψ, d φ ) is square integrable on Σ τ . Observe now that F [ ψ ] controls the square integral of (cid:12)(cid:12)(cid:12) Lψ (cid:12)(cid:12)(cid:12) / p cosh( ρ ), ( K ψ + tψ ) / √ tτ and (cid:12)(cid:12)(cid:12) ψ (cid:12)(cid:12)(cid:12) / [ln( τ ) p cosh( ρ )](the last through Lemma 5.2 and finite speed of propagation). So we can check23ach term that appears on the right of (6.8): √ tττ (cid:12)(cid:12)(cid:12) Lψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Lφ (cid:12)(cid:12)(cid:12) = cosh ρτ (cid:12)(cid:12)(cid:12) Lφ (cid:12)(cid:12)(cid:12) · p cosh ρ (cid:12)(cid:12)(cid:12) Lψ (cid:12)(cid:12)(cid:12) √ tτt (cid:12)(cid:12)(cid:12) Lψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ t φ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ∂ t φ (cid:12)(cid:12)(cid:12) · p cosh ρ (cid:12)(cid:12)(cid:12) Lψ (cid:12)(cid:12)(cid:12) √ tτt (cid:12)(cid:12)(cid:12) ( K + t ) ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ t φ (cid:12)(cid:12)(cid:12) = τt (cid:12)(cid:12)(cid:12) ∂ t φ (cid:12)(cid:12)(cid:12) · √ tτ (cid:12)(cid:12)(cid:12) ( K + t ) ψ (cid:12)(cid:12)(cid:12) √ tτt (cid:12)(cid:12)(cid:12) ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ t φ (cid:12)(cid:12)(cid:12) = ln( τ ) (cid:12)(cid:12)(cid:12) ∂ t φ (cid:12)(cid:12)(cid:12) · τ ) p cosh ρ (cid:12)(cid:12)(cid:12) ψ (cid:12)(cid:12)(cid:12) √ tτtτ (cid:12)(cid:12)(cid:12) ( K + t ) ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Lφ (cid:12)(cid:12)(cid:12) = 1 τ (cid:12)(cid:12)(cid:12) Lφ (cid:12)(cid:12)(cid:12) · √ tτ (cid:12)(cid:12)(cid:12) ( K + t ) ψ (cid:12)(cid:12)(cid:12) √ tττ (cid:12)(cid:12)(cid:12) ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Lφ (cid:12)(cid:12)(cid:12) = t ln ττ (cid:12)(cid:12)(cid:12) Lφ (cid:12)(cid:12)(cid:12) · τ ) p cosh ρ (cid:12)(cid:12)(cid:12) ψ (cid:12)(cid:12)(cid:12) Using that t/τ ≤ (cid:3) For taking care of the terms where | α | = | α | , we will place the ζ term in L , and( K + t ) ψ in L ∞ . The corresponding integral estimates are | α | = | α | ) We have the estimates (the L ∞ norms are taken along Σ τ ) Z Σ τ ( Lζ ) η (d ψ, d φ ) ∂ t ξ dvol Σ τ . τ h t (cid:13)(cid:13)(cid:13) K ψ + tψ (cid:13)(cid:13)(cid:13) L ∞ + (cid:13)(cid:13)(cid:13) Lψ (cid:13)(cid:13)(cid:13) L ∞ + (cid:13)(cid:13)(cid:13) ψ (cid:13)(cid:13)(cid:13) L ∞ i · h(cid:13)(cid:13)(cid:13) ∂ t φ (cid:13)(cid:13)(cid:13) L ∞ + (cid:13)(cid:13)(cid:13) Lφ (cid:13)(cid:13)(cid:13) L ∞ i F [ ζ ] E [ ξ ] , Z Σ τ ( Lζ ) η (d ψ, d φ )( K ξ + tξ ) dvol Σ τ . h t (cid:13)(cid:13)(cid:13) K ψ + tψ (cid:13)(cid:13)(cid:13) L ∞ + (cid:13)(cid:13)(cid:13) Lψ (cid:13)(cid:13)(cid:13) L ∞ + (cid:13)(cid:13)(cid:13) ψ (cid:13)(cid:13)(cid:13) L ∞ i · h(cid:13)(cid:13)(cid:13) ∂ t φ (cid:13)(cid:13)(cid:13) L ∞ + (cid:13)(cid:13)(cid:13) Lφ (cid:13)(cid:13)(cid:13) L ∞ ii F [ ζ ] F [ ξ ] . (cid:4) Proof Again we focus on the second inequality, the first is similar. In this sit-uation we must put Lζ in L , and both φ and ψ in L ∞ . The square integral of( Lζ ) / p cosh ρ is controlled by F τ [ ζ ] , so it su ffi ces to compute the L ∞ estimatesof tη (d ψ, d φ ). The results follow straightforwardly from (6.8), after using againthat cosh( ρ ) ≤ τ from finite speed of propagation. (cid:3) E τ ,τ ,k = sup τ ∈ [ τ ,τ ] X α ∈ I k E τ [ L α φ ] F τ ,τ ,k = sup τ ∈ [ τ ,τ ] X α ∈ I k F τ [ L α φ ] E τ ,τ , ≤ k = k X j =0 E τ ,τ ,j F τ ,τ , ≤ k = k X j =0 F τ ,τ ,j The global Sobolev inequality Theorem 2.18 states, when d = 2, that, for τ ∈ [ τ , τ ] and ψ = L α φ with | α | = k : (cid:12)(cid:12)(cid:12) ∂ t ψ (cid:12)(cid:12)(cid:12) . τ − E τ ,τ , ≤ k +2 (6.12a) (cid:12)(cid:12)(cid:12) Lψ (cid:12)(cid:12)(cid:12) . τ − F τ ,τ , ≤ k +2 (6.12b) (cid:12)(cid:12)(cid:12) K ψ + tψ (cid:12)(cid:12)(cid:12) . F τ ,τ , ≤ k +2 (6.12c) (cid:12)(cid:12)(cid:12) ψ (cid:12)(cid:12)(cid:12) . ln( τ ) τ − F τ ,τ , ≤ k +1 (6.12d)where in the last estimate (6.12d) we used the compact support assumption, andin the second to last estimate (6.12c) we used the commutator estimate (4.8).Applying the global Sobolev inequality results above to Lemma 6.10 and Lemma 6.11,25e obtain (here interpreting ( ζ, ψ, ϕ, ξ ) = ( L α φ, L β φ, L γ φ, L δ φ )) Z Σ τ ζη (d ψ, d ϕ ) ∂ t ξ dvol Σ τ . (ln τ ) τ F τ ,τ, ≤| α | +1 h E τ ,τ, ≤ | γ | +2 + F τ ,τ, ≤ | γ | +2 i F τ ,τ, | β | E τ ,τ, | δ | Z Σ τ ζη (d ψ, d ϕ )( K ξ + tξ ) dvol Σ τ . (ln τ ) τ F τ ,τ, ≤| α | +1 h E τ ,τ, ≤ | γ | +2 + F τ ,τ, ≤ | γ | +2 i F τ ,τ, | β | F τ ,τ, | δ | , Z Σ τ ( Lζ ) η (d ψ, d ϕ ) ∂ t ξ dvol Σ τ . ln( τ ) τ F τ ,τ, ≤ | β | +2 h E τ ,τ, ≤ | γ | +2 + F τ ,τ, ≤ | γ | +2 i F τ ,τ, | α | E τ ,τ, | δ | , Z Σ τ ( Lζ ) η (d ψ, d ϕ )( K ξ + tξ ) dvol Σ τ . ln( τ ) τ F τ ,τ, ≤ | β | +2 h E τ ,τ, ≤ | γ | +2 + F τ ,τ, ≤ | γ | +2 i F τ ,τ, | α | F τ ,τ, | δ | . Combining with (6.4) we arrive at(6.13) E τ ,τ , ≤ k − E τ ,τ , ≤ k . τ Z τ (ln τ ) τ (cid:16) F τ ,τ, ≤ k/ + E τ ,τ, ≤ k/ (cid:17) F τ ,τ, ≤ k/ F τ ,τ, ≤ k d τ. Similarly we get for the K -energy(6.14) F τ ,τ , ≤ k − F τ ,τ , ≤ k . τ Z τ (ln τ ) τ (cid:16) F τ ,τ, ≤ k/ + E τ ,τ, ≤ k/ (cid:17) F τ ,τ, ≤ k/ F τ ,τ, ≤ k d τ. The above estimates imply the following bootstrapping estimate: Fix k ≥ C depending on k such that the following holds: Let φ solve (1.1) on the spacetime region sand-wiched between Σ τ and Σ τ with 2 ≤ τ < τ be such that φ vanishes when e ρ ≥ τ .26uppose φ satisfies the following assumptions: we have the initial data bound(6.16) E τ ,τ , ≤ k + F τ ,τ , ≤ k ≤ ǫ ;and the bootstrap bounds(6.17) E τ ,τ , ≤ k + F τ ,τ , ≤ k ≤ δ . Then we have the improved bounds(6.18) (cid:4) E τ ,τ , ≤ k + F τ ,τ , ≤ k ≤ ǫ + Cδ . Proof The proposition follows immediately from the integrability of the func-tion (ln τ ) τ on [ τ , ∞ ). (cid:3) In particular, if δ is such that Cδ ≤ , and ǫ ≤ (ln 2) δ , then the conclusionof the above proposition implies(6.19) E τ ,τ , ≤ k + F τ ,τ , ≤ k ≤ δ . In this case, the (semi-)global existence part of our main Theorem 6.1 follows bya continuity argument, and the decay estimates follow from an application ofTheorem 2.18 to the energy bounds derived above. In the arguments above we were able to close the argument by commuting theequation with only Lorentz boost vector fields L i . If we were to wish to also con-trol the derivatives relative to the full algebra of the symmetries of Minkowskispace, generated by the vector fields(6.21) Z def = { ∂ t , ∂ x i , Ω ij , L i } , we see that the argument goes through largely unchanged, with the only signifi-cant modification coming in Lemma 6.11. As in this case we can no longer guar-antee that the derivative on ζ is an L derivative, we need to use the followingobservations:1. In the case of ∂ t ζ , whereas in the proof of Lemma 6.11 we used that thesquare integral of ( Lζ ) / p cosh ρ is controlled by F τ [ ζ ] , we have that thesquare integral of ∂ t ζ/ p cosh ρ is controlled by E τ [ ζ ] , and hence we replacethe F [ ζ ] on the right of the inequality by E [ ζ ].2. In the case of Ω ij ζ , the decomposition (2.4) and the boundedness of xt al-lows us to bound the square integral of ( Ω ij ζ ) / p cosh ρ also by F [ ζ ] .27. Finally, in the case of the ∂ x i , we decompose ∂ x i = 1 t L i − x i t ∂ t and therefore the square integral of ∂ x i ζ/ p cosh ρ is also bounded by E [ ζ ] .These changes can be accommodated in (6.13) and (6.14) by changing the factor F τ ,τ, ≤ k to ( F τ ,τ, ≤ k + E τ ,τ, ≤ k ), following which the bootstrap argument proceedsunchanged. (cid:4) 7. A perturbation result The argument in the previous section immediately implies a “stability of large-data dispersive solutions” result, in the spirit of and generalizing the results of[16]. For convenience, fix throughout the section a positive integer k ≥ 4. Implicitconstants are understood to depend on k .First, let us make precise what we mean by “dispersive solutions”. Observe thatthe global solutions of Theorem 6.1 are automatically “dispersive in the forwardlight-cone” per the following definition. Let Φ be a global solution of (1.1). We say that it is dispersive in the forward light-cone if there exists a constant M > { t > max(1 + | x | , } :• Uniform-in- τ L ∞ bounds for all α ∈ I ≤ k − . | ∂ t L α Φ | ≤ Mτ − (cid:12)(cid:12)(cid:12) L i L α Φ (cid:12)(cid:12)(cid:12) ≤ Mτ − (cid:12)(cid:12)(cid:12) ( K + t ) L α φ (cid:12)(cid:12)(cid:12) ≤ M | Φ | ≤ M ln( τ ) τ − • Space-time L bounds. For every α ∈ I ≤ k , Z { t> max(1+ | x | , } ln( τ ) τ cosh( ρ ) (cid:20) ( ∂ t L α Φ ) + ( LL α Φ ) + 1 τ ([ K + t ] L α Φ ) (cid:21) dvol ≤ M (cid:4) The background solutions considered in [16], which are formed from lookingat solutions to the wave-maps equation whose range restrict to a geodesic on28he target manifold, are automatically conjugate to a solution of the linear waveequation and therefore also satisfy the above bounds making them dispersive inthe forward light-cone. (cid:4) We will study here perturbations of a given solution Φ . That is, we look for φ such that Φ + φ is a solution to (1.1) and study the corresponding initial problem.The equation satisfied by φ is(7.3) (cid:3) φ = ( Φ + φ ) · η (d( Φ + φ ) , d( Φ + φ )) − Φ · η ( d Φ , d Φ ) . Let Φ be a global solution of (1.1) that is dispersive in the forward light-cone,with the associated constant M . Consider the future-evolution governed by (7.3)on R ,d , with initial data prescribed at time t = 2 such that φ (2 , x ) and ∂ t φ (2 , x )are supported in the ball of radius 1. Then there exists some constant λ > independent of M such that if (cid:13)(cid:13)(cid:13) φ (2 , —) (cid:13)(cid:13)(cid:13) H k +1 + (cid:13)(cid:13)(cid:13) ∂ t φ (2 , —) (cid:13)(cid:13)(cid:13) H k ≤ λ (1 + M ) )then there exists a future-global solution φ : [2 , ∞ ) × R d → R . (cid:4) We keep track of the M dependence for applications in the next section; for stabil-ity of large-data dispersive solutions the precise dependence is not needed. Thelarge exponential loss in M is due entirely to the linear -in- φ terms that appear onthe right of (7.3), and seems hard to avoid. In the next section this exponentialloss implies that our method only addresses very strongly localized non-compactinitial data, with tails that decay super-exponentially. (cid:4) Proof (Sketch) We proceed largely as in the previous section, and apply theestimates from Lemma 6.10 and Lemma 6.11, with the slots ( ζ, ψ, φ ) in the Lem-mas replaced by L α φ and L β Φ . If we make the bootstrap assumptions (6.16)and (6.17) as before, then in the place of (6.13) and (6.14) we obtain the energy29stimate(7.6) E τ ,τ , ≤ k + F τ ,τ , ≤ k − ǫ . τ Z τ (ln τ ) τ ( M + δ ) ( E τ ,τ, ≤ k + F τ ,τ, ≤ k ) d τ + X α ∈ I ≤ k τ Z τ (ln τ ) τ ( M + δ )( E τ ,τ, ≤ k/ + F τ ,τ, ≤ k/ ) · Z Σ τ | ∂ t L α Φ | + | LL α Φ | + 1 τ | [ K + t ] L α Φ | ) ρ dvol Σ τ / d τ. Applying Grönwall’s inequality we get (where λ is the implicit structural con-stant that appears in the previous inequality):(7.7) E τ ,τ , ≤ k + F τ ,τ , ≤ k ≤ ǫ exp λ ( M + δ ) τ Z τ (ln τ ) τ d τ + λ ( M + δ ) τ Z τ X α ∈ I ≤ k (cid:18) Z Σ τ | ∂ t L α Φ | + | LL α Φ | + 1 τ | [ K + t ] L α Φ | ) ρ dvol Σ τ (cid:19) / d τ . The first integral inside the exponential is obviously bounded. The second inte-gral is also bounded after noting that τ − ln( τ ) d τ is a finite measure and so wecan apply Hölder’s inequality to control L by L , whereupon our space-time L bound comes into play. If δ is initially chosen to be 1, then we can find λ > E τ ,τ , ≤ k + F τ ,τ , ≤ k ≤ ǫ exp( 12 λ (1 + M ) ) . So provided we choose ǫ < exp( − λ (1 + M ) ) we obtain the improved bootstrapbound E τ ,τ , ≤ k + F τ ,τ , ≤ k ≤ exp( − λ ) < δ from which global existence follows from a standard continuity argument. (cid:3) The phenomenon discussed here holds rather generally for large-classes of semi-linear equations. In particular, one can regard the above as related to the con-struction described in [17] for semilinear wave equations. There a version ofChristodoulou’s short pulse method is used to construct large data solutions to30uch equations. Their method can be interpreted as constructing first a future-global, dispersive approximate solution via the almost-radial initial data (whichcan, heuristically speaking, by studying an approximate evolution for the radialpart), and perturbing it using that both the approximate solution and residual er-ror terms have good dispersive decay. Our approach provides in a general setting,the argument for the perturbative step. (cid:4) The perturbed solution Φ + φ also satisfies uniform-in- τ L ∞ bounds and space-time L bounds by triangle inequality. Thus for λ > ffi ciently large the samebounds as Φ hold, but with M replaced by ( M + 1). (cid:4) The Definition 7.1 and Theorem 7.4 are stated using only Lorentz boost commu-tators. However, in view of Remark 6.20, the same results hold if we use also thefull set Z of (6.21). For the result in the next section, having available the full setof commutators is convenient (but not necessary). (cid:4) 8. Extension to non-compact data The discussion in the previous section only deals with compactly-supported ini-tial data, which is a common problem with studying nonlinear hyperbolic equa-tions using the hyperboloidal-based energy methods. In [13], LeFloch and Maintroduced the so-called Euclidean-Hyperboloidal method to overcome this dif-ficulty in the context of the stability problem of Minkowski space. Their methodrequires gluing asymptotically Euclidean slicings to the hyperboloids and is notobviously compatible with the weight loss of Lemma 5.2. For the wave-map prob-lem we circumvent this di ffi culty by taking advantage of the scaling invariance ofthe wave-map equation in d = 2, this allows us to show that in the case of stronglylocalized (but not necessarily of compact support) initial data, a small-data globalexistence theorem persists.We begin by observing that if φ is a solution to (1.1), then so is(8.1) S m φ ( t, x ) def = φ (2 m ( t − 2) + 2 , m x ) , the extra time-translation is to keep the constant time slice { t = 2 } invariant underthe transformation.Unfortunately, as the argument requires studying the initial data in H k norm(and not a scale invariant norm), the scaling argument does not automaticallyallow us to approximate non-compact initial data by compact ones. Specifically,31bserve that for any multi-index α Z { }× R d | ∂ α S m φ ( t, x ) | d x = 2 m ( | α |− Z { }× R d | ∂ α φ ( t, x ) | d x. This scaling property rules out naïvely approaching non-compact initial data viaa cut-o ff /rescaling procedure, even if one were to measure the initial data in aweighted Sobolev space (since to control the higher-order initial energy, we mustmeasure the initial data against non-singular weights).Our approach is to perform a spatial dyadic decomposition of the initial data andapply the Perturbation Theorem 7.4 to iteratively construct the solution. Moreprecisely, let χ be a smooth, positive function of compact support on R suchthat it vanishes outside the ball of radius 1 and is ≡ / m ≥ χ m ( x ) = χ (2 − m x ) − χ (2 − m x )so that χ m is supported in the annulus | x | ∈ [2 m − , m ].Consider now the initial value problem for (1.1), with initial data prescribed at t = 2(8.3) φ (2 , x ) = ˚ φ ( x ) ∂ t φ (2 , x ) = ˚ ϕ ( x ) . We decompose the initial data spatially. For m ≥ φ m def = χ m · ˚ φ, (8.4) ˚ ϕ m def = χ m · ˚ ϕ. (8.5)Let φ solve (1.1) with initial data ( ˚ φ , ˚ ϕ ). For m ≥ Φ m = P m − n =0 φ n , and φ m the solution to(8.6) (cid:3) φ m = ( Φ m + φ m ) η (d( Φ m + φ m ) , d( Φ m + φ m )) − Φ m η (d Φ m , d Φ m )with initial data ( ˚ φ m , ˚ ψ m ). Then provided we can show that each φ m exists glob-ally, the infinite sum φ = P ∞ m =0 φ m will be a solution to the original problem. Theconvergence of this sum is automatically uniform on any compact space-timesubset: in fact, due to finite-speed-of-propagation, φ m vanishes within the set {| t | + | x | ≤ m − } , and hence for every ( t, x ) the sum φ converges after finitely manyterms. 32n order to apply the uniform estimates of Theorem 7.4, we will rescale the prob-lem solved by φ m . Define the scaling operator(8.7) ˚ S m f ( x ) def = f (2 m x )Then we have that S m φ m solves the equation(8.8) (cid:3) S m φ m = ( S m Φ m + S m φ m ) η (d( S m Φ m + S m φ m ) , d( S m Φ m + S m φ m )) − ( S m Φ m ) η (d S m Φ m , d S m Φ m )with initial data ( ˚ S m ˚ φ m , m ˚ S m ˚ ϕ m ).(Notice that our scaling operators form a semi-group under composition: S m =( S ) ◦ m and ˚ S m = ( ˚ S ) ◦ m .)Therefore to arrive at a su ffi cient condition for global existence, it su ffi ces tostudy how the uniform-in- τ L ∞ bounds and space-time L bounds behave underrescaling. Observe first that Ω ij S m φ = S m Ω ij φ (8.9a) ∂S m φ = 2 m S m ∂φ (8.9b) L i S m φ = S m ( L i φ ) + (2 m +1 − S m ∂ t φ (8.9c)and(8.9d) ( K + t ) S m φ = 2 − m S m ( K + t ) φ + 4(1 − − m ) S m ( X x i t L i φ + τ t ∂ t φ )+ 2 − m (2 m +1 − S m ∂ t φ + 2(1 − − m ) S m φ. Notice that operators ∂ t , L i , and ( K + t ) are respectively homogeneous of degrees − 1, 0, and 1 under space-time scaling of R , . So such estimates as above areexpected. However, since our transformation S m is the conjugation of the scalingoperation with a time-translation, and as L i and ( K + t ) are not invariant undertime-translation, we pick up lower order correction terms. (cid:4) Also, on the set { t ≥ max(1 + | x | , } one sees(8.11) S m τ = 2 m q ( t − − m ) − | x | < m τ. Restricted to the set { S m t > max(1 + | S m x | , } we have S m t > max(1 + | S m x | , 2) = ⇒ t > max( | x | + 2 − − m , t − √ t − τ > − − m implies(8.12) τ + 4(1 − − m − ) > − − m − ) t. Plugging (8.12) into the expression for S m τ we see that, on the support of S m Φ m ,we have ( S m τ ) = 2 m (cid:16) τ − − − m ) t + 4(1 − − m ) (cid:17) > m − τ − − m − − − m ! > m − (cid:16) τ − (cid:17) . Finally, noting that when restricted to the future { t ≥ } , the bound (8.12) implies(when m ≥ τ > − − m − ) > S m τ ≥ m/ √ τ establishing that τ and S m τ are comparable up to a factor depending on m .Now, fixing k ≥ 4, the above computations imply the following lemma. Here let Z ℓ be strings of length ℓ drawn from (6.21), and implicitly we are only workingwithin the forward light cone { t > max(1 + | x | , } . There exists a constant C which depends on k such that if S m − Φ m satisfies the L ∞ bounds for all 0 ≤ ℓ ≤ k − (cid:12)(cid:12)(cid:12) ∂ t Z ℓ Φ (cid:12)(cid:12)(cid:12) ≤ Mτ − (cid:12)(cid:12)(cid:12) LZ ℓ Φ (cid:12)(cid:12)(cid:12) ≤ Mτ − (cid:12)(cid:12)(cid:12) ( K + t ) Z ℓ φ (cid:12)(cid:12)(cid:12) ≤ M | Φ | ≤ M ln( τ ) τ − and the L bounds for all 0 ≤ ℓ ≤ k Z { t> max(1+ | x | , } ln( τ ) τ cosh( ρ ) (cid:20) ( ∂ t Z ℓ Φ ) + ( LZ ℓ Φ ) + 1 τ ([ K + t ] Z ℓ Φ ) (cid:21) dvol ≤ M , then S m Φ m satisfies the same bounds with M replaced by CM . (cid:4) .15 Remark We are forced to deal with an enlarged set of commutators (more than just L i ),due to (8.9c). This means to estimate higher L derivatives of S m Φ m we must alsoestimate higher ∂ t derivative of S m − Φ m . Therefore it is not necessary that wefully embrace the full Poincaré algebra Z ; we merely need to add control of the ∂ t commutators into the argument.We note that including the whole set of commutators is no problem. As in-dicated in Remark 7.10 using the arguments of Remark 6.20 the perturbationTheorem 7.4 also hold using the larger set of commutators if the notion of disper-sive solution in Definition 7.1 is modified appropriately to include the full set ofcommutators from Z .We note that this issue with the ∂ t commutator arising can also be circumventedin a di ff erent manner. Instead of commuting with ∂ t , we can exploit an “ellip-tic” estimate for ∂ t . Noting that Φ m solves a wave equation, one can control ∂ tt S m − Φ m in terms of ∂ t LS m − Φ m , LLS m − Φ m , and the inhomogeneity/nonlinearity (cid:3) S m − Φ m , none of which involves second order t derivatives. In the present situ-ation the first method seems simpler. (cid:4) Proof (of Lemma 8.14) We check here the terms involving the ( K + t ) operator,which is the most complicated in view of (8.9d). The claim follows for the ∂ t and L terms straightforwardly in a similar manner from the computations (8.9b),(8.9c), (8.11), and (8.13). Without loss of generality we consider only the casewhere ℓ = 0 (no commutators). Each commutation incurs some additional termsfor which we have point-wise comparison by virtue of (8.9a), (8.9b), and (8.9c),and which only contribute additional constant factors.Starting from (8.9d) we can write | ( K + t ) S m Φ m | ≤ | S ( K + t ) S m − Φ m | + 2 X(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S x i t L i S m − Φ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S [( τ t + 1) ∂ t S m − Φ m ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + | S S m − Φ m | . Let us first consider the point-wise estimate for ( K + t ) S m Φ m = ( K + t ) S m S m − Φ m .Applying the point-wise estimates for S m − Φ m we get | ( K + t ) S m Φ m | ≤ M + 4 MS ( τ − ) + 4 MS (1 + τ − ) + MS (ln( τ ) τ − ) . On the support of S m − Φ m when t ≥ 2, we have that τ ≥ √ 3, and ln( τ ) /τ ≤ /e . Sowe conclude pointwise | ( K + t ) S m Φ m | ≤ ( 8 √ e + 4 + 12 ) M ≈ . M. S m Φ m has, by construction and finitespeed of propagation, support within the set { S t > max(1+ | S x | , } , we have thaton its support r τ ≤ S τ ≤ τ. Similarly, we also have t ≤ S t ≤ t. This implies that ln( τ ) τt ≤ S (ln τ ) S ( τ ) S ( t ) . So Z ln( τ ) τ t ([ K + t ] S m Φ m ) dvol ≤ Z S ln( τ ) τ t (cid:18) | ( K + t ) S m − Φ m | + 4 | LS m − Φ m | + 4 (cid:16) τ t + 1 (cid:17) | ∂ t S m − Φ m | + | S m − Φ m | (cid:19) dvolUndoing the scaling (which we observe to map the set { t ≥ max(1 + | x | , } intoitself by construction) we get ≤ Z ln( τ ) τ t (cid:18) | ( K + t ) S m − Φ m | + 4 | LS m − Φ m | + 4 (cid:16) τ t + 1 (cid:17) | ∂ t S m − Φ m | + | S m − Φ m | (cid:19) dvol . The integral of the first three terms can be bounded by 512 M using the assumed L bound. (Notice that τ ( τ /t + 1) ≤ S m − Φ m to get Z ln( τ ) τ cosh ρ | S m − Φ m | dvol . Z ln( τ ) τ cosh ρ | LS m − Φ m | dvol . Z ln( τ ) τ cosh ρ | LS m − Φ m | dvol . M as needed. (The implicit constant here given by the Hardy inequality.) (cid:3) 36 simple induction argument using Theorem 7.4 yields then the following su ffi -cient condition for global existence: Let C be the constant of Lemma 8.14 and let λ be the constant from Theorem 7.4.Then if the initial data ( ˚ φ, ˚ ϕ ) is such thatsup m ≥ (cid:16)(cid:13)(cid:13)(cid:13) ˚ S m ˚ φ m (cid:13)(cid:13)(cid:13) H k +1 + 2 m (cid:13)(cid:13)(cid:13) ˚ S m ˚ ϕ m (cid:13)(cid:13)(cid:13) H k (cid:17) exp λ (cid:18) C m − C − (cid:19) ≤ (cid:4) We can rephrase the smallness and localization of the initial data in terms of aweighted Sobolev space that requires super-exponential decay of data. There exists constants κ, ν > φ, ˚ ϕ ) is su ffi -cient small in the weighted Sobolev space H k +1 ( R , exp( ν (1 + | x | ) κ ) d x ) × H k ( R , exp( ν (1 + | x | ) κ ) d x )there exists a global solution to the initial value problem for (1.1). (cid:4) References [1] S. 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