Nonlinear interaction of three impulsive gravitational waves I: main result and the geometric estimates
NNonlinear interaction of three impulsive gravitational waves I:main result and the geometric estimates
Jonathan Luk ∗ Department of Mathematics, Stanford University,450 Serra Mall Building 380, Stanford CA 94305-2125, United States of America
Maxime Van de Moortel † Department of Mathematics, Princeton University,Fine Hall, Washington Road, Princeton NJ 08544, United States of America
January 22, 2021
Abstract
Impulsive gravitational waves are (weak) solutions to the Einstein vacuum equations such that theRiemann curvature tensor admits a delta singularity along a null hypersurface. The interaction ofimpulsive gravitational waves is then represented by the transversal intersection of these singular nullhypersurfaces.This is the first of a series of two papers in which we prove that for all suitable U (1) -symmetric initialdata representing three “small amplitude” impulsive gravitational waves propagating towards each othertransversally, there exists a local solution to the Einstein vacuum equations featuring the interaction ofthese waves. Moreover, we show that the solution remains Lipschitz everywhere and is H loc ∩ C , − loc away from the impulsive gravitational waves. This is the first construction of solutions to the Einsteinvacuum equations featuring the interaction of three impulsive gravitational waves.In this paper, we focus on the geometric estimates, i.e. we control the metric and the null hypersurfacesassuming the wave estimates. The geometric estimates rely crucially on the features of the spacetimewith three interacting impulsive gravitational waves, particularly that each wave is highly localized andthat the waves are transversal to each other. In the second paper of the series, we will prove the waveestimates and complete the proof. Contents Σ
255 Precise statement of the main theorems 286 Approximation argument and the proof of Theorem 5.2 31 ∗ [email protected] † [email protected] a r X i v : . [ g r- q c ] J a n Bootstrap argument and the proof of Theorem 5.6 378 Preliminary estimates resulting from the bootstrap assumptions 419 Estimates for the metric components in elliptic gauge 4410 Estimates for the Ricci coefficients and related geometric quantities 6211 Conclusion of the proof of Theorem 7.1 68A Solving the constraint equations 68
It is well-known that the Einstein vacuum equations
Ric ( g ) = 0 (1.1)admit (weak) solutions ( M , g ) in (3 + 1) dimensions for which the Riemann curvature tensor admits deltasingularities on a null hypersurface; see for instance [60]. These are interpreted as impulsive gravitationalwaves .Remarkably, an explicit solution has been discovered by Khan–Penrose [38] (see also [65]), in whichthere are two transversally intersecting null hypersurfaces on which (different) components of the Riemanncurvature tensor admit delta singularities. This was interpreted as representing the interaction of twoimpulsive gravitational waves. The Khan–Penrose solution also exhibits interesting global properties in thatthe spacetime remains smooth locally beyond the interaction of the two impulsive gravitational waves buteventually a stronger Kasner-like spacelike singularity develops in the future [73]. This is thought of as anidealized situation representing two very strong gravitational waves coming together from (infinitely) far-away strongly gravitating objects such that the interaction of the gravitational waves gives rise to a focusingeffect and ultimately leads to a (more severe) singularity. After Khan–Penrose, there are many other explicitconstructions of solutions featuring the interaction of impulsive gravitational waves, all of which rely onintroducing a high degree of symmetry; see Section 1.2.1.In [50, 51], Luk–Rodnianski initiated the study of the propagation and interaction of impulsive gravita-tional waves without any symmetry assumptions. Even though the impulsive gravitational waves have lowerregularity than that required for general local existence results [39, 42, 64], Luk–Rodnianski developed ageneral local theory for solutions to the Einstein vacuum equations incorporating not only the propagationof one impulsive gravitational wave, but also the interactions of two impulsive gravitational waves. Theirresults can be summarized as follows: Theorem 1.1 (Luk–Rodnianski [50, 51]) . Consider the characteristic initial value problem for the Einsteinvacuum equations with characteristic initial data posed on two null hypersurfaces H and H transversallyintersecting at a spacelike -sphere. Suppose on the initial hypersurface H (respectively H ), the null secondfundamental form has a jump discontinuity across the -sphere S ∗ (respectively S ∗ ) but smooth otherwise.Then, assuming S ∗ and S ∗ are sufficiently close to each other, there exists a unique local solution to theEinstein vacuum equations with two singular hypersurfaces emanating from the initial singularities S ∗ and S ∗ , and intersecting in the future. Moreover, the spacetime metric is everywhere Lipschitz and is smoothaway from the union of null hypersurface emanating from S ∗ and the null hypersurface emanating from S ∗ . Theorem 1.1 shows that at least locally near the interaction, the structure of the spacetime (in terms ofsmoothness) is similar to that of the Khan–Penrose solution. It moreover provides the setup to understandmore generally the global structure of spacetimes.However, all the existing examples and results cover only the interaction of two impulsive gravitationalwaves (despite the fact that the Luk–Rodnianski theory applies without any symmetries and allows for verygeneral wave fronts). The question remains as to what is the structure of the spacetime singularities — evenlocally! — associated with the interaction of three impulsive gravitational waves coming in from differentdirections. In fact, there is not even a single example of a solution to (1.1) featuring the transversal interaction2f three impulsive gravitational waves. In particular, a symmetry assumption such as T -symmetry is toorestrictive to allow for the construction of such examples. The purpose of this work is to go beyond the interaction of two impulsive gravitationalwaves and to consider the interaction of three impulsive gravitational waves. Our main resultis a local theory for vacuum spacetime solutions under polarized U (1) symmetry which featurethe transversal interaction of three small amplitude impulsive gravitational waves. To see the difference between two and three impulsive gravitational waves, first recall that in the proof ofTheorem 1.1, one fundamental insight is that even though the spacetime metric necessarily very singular inthe directions transversal to each of the impulsive gravitational waves, one can find two vector fields whichare linearly independent at every spacetime point , such that the spacetime metric is more regular when (Lie-)differentiated in the direction of these two vector fields. These vector fields are constructed with the use ofa so-called double null foliation. As a result of the strong reliance of the double null foliation, the methodsof Theorem 1.1 cannot be extended to in the case of three impulsive gravitational waves, which necessarilyrequires new techniques. Moreover, known results on much weaker singularities for semi-linear problemssuggest that the local singularity structure after the interaction of three impulsive gravitational waves mayeven be qualitatively different from that for two impulsive gravitational waves [62]; see further discussionsin Remark 1.8 and Section 1.2.4.To make the problem slightly more tractable, we impose the simplifying assumption that the spacetimeis polarized U (1) symmetric, i.e. we consider an ambient manifold ( I × R , (4) g ) , where I ⊆ R is an interval,and stipulate that the metric takes the following ansatz (4) g = e − φ g + e φ ( dx ) , where φ : I × R → R is a scalar function and g is a Lorentzian metric on I × R . The Einstein vacuumequations then reduce to the (2 + 1) -dimensional Einstein–scalar field problem (cid:40) Ric ( g ) = 2d φ ⊗ d φ (cid:3) g φ = 0 , (1.2)which simplifies the analysis. Notice that unlike T -symmetry, in our setting the symmetry group is one-dimensional and therefore the transversal interaction of three impulsive gravitational waves is still allowed.The setup of the problem (see the precise statements in Section 4.2) is the following. Consider initial datawhich are compactly supported such that ∂ i φ and ∂ t φ are small in L ∞ and are smooth except along threelines { (cid:96) k } k =1 where they have a (small) jump discontinuity. Moreover, prescribe the jump discontinuity in amanner such that locally they propagate towards each other, and, assuming that the metric remains C -closeto Minkowski, arrange them to interact before time t = 1 . Prescribe g by solving the constraint equationsand imposing suitable gauge conditions (using modifications of methods in [34]).The following is an informal version of our main theorem (see Section 5 for a more precise statement): Theorem 1.2.
Given a polarized U (1) symmetric initial data set corresponding to three (non-degenerate)small-amplitude impulsive gravitational waves propagating towards each other, there exists a weak solution tothe Einstein vacuum equations corresponding to the given data up to and beyond the transversal interactionof these waves. In particular, in the solution, the metric is everywhere Lipschitz and is H loc ∩ C ,θloc for some θ ∈ (0 , ) away from the three null hypersurfaces corresponding to the impulsive gravitational waves. A few remarks regarding Theorem 1.2 are in order:
Remark 1.3 (More than three waves) . While Theorem 1.2 only explicitly treats the case of three transversallyinteracting impulsive gravitational waves, the techniques that are introduced can handle initial data featuringany finite number of impulsive gravitational waves propagating in different directions.When there are more than three waves in the initial data, generically at each interaction point only threewaves interact. Moreover, even when four waves are arranged to interact at the same point in the reduced -dimensional spacetime, in the original -dimensional spacetime, the waves interact at a one-dimension curve, which should be considered as a non-generic case. Put differently, to understand genuineinteraction of four impulsive gravitational waves, it seems necessary to relax the symmetry assumption.
Remark 1.4 (Anisotropic estimates in L spaces) . Even though Theorem 1.2 is most conveniently statedin terms of isotropic spaces (Lipschitz, C ,α and H ), when we prove the Lipschitz and C ,α estimates, weneed to first obtain higher regularity estimates in L based spaces with respect to some geometrically definedvector fields; see Section 7.2.2. Remark 1.5 ( δ -impulsive gravitational waves) . Impulsive gravitational waves should be viewed as an ideal-ized description of very strong and localized gravitational waves. It may be argued that instead of having asolution whose curvature has a delta singularity, a more physically relevant description would be a smoothsolution whose curvature scales like an approximate delta singularity. To capture this more general class ofsolutions, we introduce the notion of δ -impulsive gravitational waves (for δ > ), which roughly speakingcorresponds to solutions to the Einstein vacuum equations whose Riemann curvature tensor is of amplitude O ( δ − ) in a δ -neighborhood of a null hypersurface (and of O (1) otherwise). The class of δ -impulsive gravi-tational waves also includes the particular cases where the curvature profile has zero average (which is notpossible with a delta function) as discussed [38, 66].In this paper, we will also prove a version of Theorem 1.2 for the nonlinear interaction of three δ -impulsivegravitational waves (for all small δ > ); see Theorem 5.6. In fact, our approach in proving Theorem 1.2for the impulsive gravitational waves proceeds by first approximating the impulsive wave data by those of δ -impulsive waves, and then passing to the δ → limit; see Section 1.1.1.We remark also that the non-degeneracy condition in Theorem 1.2 (see 7 in Definition 4.3) will onlybe used for solving the constraint equations, in order to show that the impulsive wave data can indeed beapproximated by δ -impulsive wave data. Remark 1.6 (Uniqueness) . We note explicitly that our proof does not give a uniqueness statement as werely on a compactness argument.
Remark 1.7 (Relation to low regularity problem) . The main difficulty of Theorem 1.2 (and of understandingimpulsive gravitational waves in general) is the low regularity of the initial data. Without any symmetryassumptions, the best-known general local result is the celebrated bounded L curvature theorem, which requiresthe initial data to have curvature in L [42], while impulsive gravitational waves have much lower regularity.In this paper, we impose polarized U (1) symmetry, and under such a symmetry assumption, local well-posedness holds in a lower regularity than in the general (3+1) -dimensional case without symmetry. For U (1) symmetry (even without polarization), the results of [64] imply that local well-posedness can be obtained forin H + (cid:15) . While the optimal regularity in polarized U (1) symmetry is not explicitly discussed in the literature,some interesting progress has been made on a related quasilinear model problem [5, 33].Note that the initial data that we consider barely fail to be in the H space, and are below the thresholdfor any standard theorem. More importantly, our main focus is not just to obtain a local existence result.Instead, we also obtain control of the Lipschitz norm and a finer description of the singularity structure. Remark 1.8 (Higher regularity) . In the case of the interaction of two impulsive gravitational waves (recallTheorem 1.1), the spacetime metric is smooth away from the impulsive gravitational waves. In our setting, he improved regularity we obtain away from the impulsive gravitational waves is only in class C ,α . However,in view of some examples for even much weaker singularities for some simpler model semilinear problems(see Section 1.2.4), one may conjecture that in our setting the spacetime metric is not smooth away fromthe union of impulsive gravitational waves, and that there is a weaker singularity that emanated from theintersection point of the three impulsive gravitational waves. It would be interesting to understand what isthe optimal regularity that can be obtained. Our proof of Theorem 1.2 has three main components, which are highly coupled to each other.1. Control the geometric quantities, including the metric components, the null hypersurfaces and thecommuting vector fields, assuming suitable bounds on the scalar wave.2. Show that the L -based wave energy estimates for the scalar wave imply via an anisotropic Sobolevembedding theorem that the scalar field is everywhere Lipschitz with improved Hölder regularity awayfrom the singular hypersurfaces.3. Prove L -based wave energy estimates for the scalar wave with appropriately chosen commutators.The three parts are of somewhat different nature, and are further discussed in Section 7.2.1–7.2.3 respectively.In this paper, we will discuss the relation between the three steps, and carry out Step 1. Steps 2 and 3 willbe performed in the companion paper [53].The remainder of the introduction is structured as follows: In Section 1.1 we will give a brief indicationof the ideas used in the proof of Theorem 1.2, emphasizing the ideas for the geometric estimates. In
Section 1.2 , we will discuss some related works. In
Section 1.3 , we then give a list of some relatedproblems. Finally, in
Section 1.4 , we will outline the remainder of the paper. δ -impulsive gravitational waves Rather than directly constructing a solution with the impulsive wave data, our strategy will be to consider a δ -approximate problem and the pass to the δ → limit. Recall that in the reduction (1.2), the original Einsteinvacuum equations reduce to a lower dimensional Einstein–scalar field system. This naturally separates theestimates into the scalar field part and the geometry part. For the impulsive gravitational wave problem, thescalar field φ (in the reduced system) is only Lipschitz, and ∂φ has jump discontinuities along three differenthypersurfaces. We will instead first regularize the initial data, so that the data are smooth, and while ∂φ remains O ( (cid:15) ) , we only have ∂ φ = O ( (cid:15)δ − ) in a δ -neighborhood of three curves (for δ (cid:28) (cid:15) (cid:28) ). We willcall these regularized waves the δ -impulsive gravitational waves; see precise conditions in Section 4.3.The advantage of first considering the δ -impulsive waves before passing to the limit is that we can boundsome norms which blow up in a controlled manner in terms of δ − . This is useful in the analysis becauseafter introducing a suitable decomposition (see (1.5) below), some quantities are small in terms of δ , andcan compensate for the large δ − powers in the estimates. This is reminiscent of Christodoulou’s short pulsemethod; see [24].The challenge will now be to show that for all δ > sufficiently small, (1) there is a uniform time ofexistence of the solutions, that (2) we can prove some estimates that are independent of δ , and that (3) theestimates are sufficiently strong for us to pass to the δ → limit to obtain a solution.We rely on a compactness argument to extract the δ → ; for this reason, we do not prove uniqueness (seeRemark 1.6). Note the importance to have strong convergence of ( φ, g ) in H since general weak H limitsof solutions to the Einstein vacuum equations are not necessarily (weak) vacuum solutions (see [16, 36]). The choice of gauge plays a fundamental role for low-regularity problems in general relativity. In the presentwork, we will in fact need to choose multiple gauges: one global system of coordinates determined by anelliptic gauge and three sets of null coordinates. The global elliptic gauge is chosen to maximize the regularityof the (reduced (2 + 1) -dimensional) metric coefficients given the low regularity setting, and each set of null5oordinates is adapted to each propagating impulsive gravitational wave. Because we use multiple sets ofcoordinates, it is also important to control the transformation between any two sets of coordinates.
Elliptic gauge.
Since the U (1) -reduced problem (1.2) is effectively (2 + 1) -dimensional, the scalar field,which determines the Ricci curvature tensor, completely determines the Riemann curvature tensor. In orderto maximize the gain in regularity when reconstructing the metric from the curvature tensor, we use anelliptic gauge. More precisely, we foliate the spacetime by maximal hypersurfaces { Σ t } t ∈ [0 ,T ] and we choosespatial coordinates such that the induced metric is conformal to the flat metric on each t ∈ [0 , T ] . In doingso, each metric component g obeys a spatial elliptic equation schematically of the form ∆ g = ( ∂φ ) + ( ∂ x g ) , (1.3)where ∆ is the flat Laplacian and we use the convention that ∂ x denotes a spatial derivative, while ∂ denoteseither a spatial or a time derivative. Eikonal functions, null frames, and geometric coordinates.
To understand the propagation of the δ -impulsive waves, a crucial role is played by the eikonal functions u k , for k = 1 , , . Each u k is defined sothat its level sets correspond to the null hypersurfaces along which one of the δ -impulsive wave propagates.Associated with each eikonal function u k , we will introduce• a null frame ( L k , E k , X k ) such that L k and E k are tangential to constant- u k (null) hypersurfaces, and• a system of geometric coordinates ( u k , t k , θ k ) with u k as above, t k = t and θ k transported by L k θ k = 0 .The significance of the eikonal functions and the null frames lie in that• u k captures the location of each δ -impulsive wave. In particular, for the k -th wave , the most singularbehavior is only expected in u k ∈ [ − δ, δ ] .• The L k and E k vector fields corresponds to regular directions for the k -th wave. In other words, the L k and E k derivatives are better behaved than a generic derivative.While the geometric constructions associated with u k are important for capturing the propagation of the δ -impulsive waves, in order for them to be useful, we need to obtain the relevant geometric control, includingestimating the connection coefficients such as ∇ L k E k etc. All the connection coefficients can algebraicallydetermined from χ k := g ( ∇ E k L k , E k ) , η k := g ( ∇ X k L k , E k ) and spatial derivatives of the metric coefficients g in the elliptic gauge coordinates. A bound for χ k is particular means that the constant- u k null hypersurfaceare regular without conjugate points.The Einstein equations imply that χ k and η k satisfy nonlinear transport equations (see (2.93), (2.94)) L k χ k = − L k φ ) + · · · , L k η k = − L k φ )( E k φ ) + · · · , (1.4)where · · · are lower order terms. The reduced equations (1.2) naturally divide the estimates into those for the wave part and for the geometricpart. This paper is focused on the geometric part; we refer the reader to the introduction to [53] for thediscussion of the proof of the wave estimates. Nevertheless, since the geometric estimates are highly coupledwith the wave estimates, we first point out the main wave estimates, before we explain how they dictate thegeometric estimates that we prove.The wave estimates we state here are natural from the point of view of propagation of singularities forlinear wave equations. The much less obvious part, which will be addressed in [53], is that these estimatescontinue to hold in the quasilinear setting, particularly under the low regularity of the metric that weestablish in this paper. To make precise the notion of the three different propagating δ -impulsive waves require a decomposition of φ ; see (1.5)below.
6n order to capture the three propagating singularities, we show that φ admits a decomposition φ = φ reg + (cid:88) k =1 (cid:101) φ k , (1.5)where φ reg is a “regular” part, and (cid:101) φ k are the “singular” parts, each corresponding to one of the impulsivewaves. Each of these parts are defined to satisfy the wave equation, i.e. (cid:3) g φ reg = 0 and (cid:3) g (cid:101) φ k = 0 .The following are the most important features of φ reg and (cid:101) φ k .1. On constant- t hypersurfaces Σ t , φ obeys the following isotropic bounds:(a) For fixed s (cid:48) ∈ (0 , ) , (cid:107) φ (cid:107) H s (cid:48) (Σ t ) + (cid:107) ∂φ (cid:107) H s (cid:48) (Σ t ) (cid:46) (cid:15) .(b) (cid:107) ∂φ (cid:107) L ∞ (Σ t ) (cid:46) (cid:15) . Importantly, this also holds with a Besov improvement .2. The second derivatives of φ is not better than (cid:107) ∂ φ (cid:107) L (Σ t ) (cid:46) (cid:15)δ − , but have the following features:(a) The regular part is better: (cid:107) ∂φ reg (cid:107) H s (cid:48) (Σ t ) (cid:46) (cid:15) .(b) The bad part for ∂ (cid:101) φ k is only localized to S kδ = { u k ∈ [ − δ, δ ] } : in fact (cid:107) ∂ (cid:101) φ k (cid:107) L (Σ t \ S kδ ) (cid:46) (cid:15) .(c) L k and E k are better than general derivatives on (cid:101) φ k : (cid:107) ∂L k (cid:101) φ k (cid:107) L (Σ t ) + (cid:107) ∂E k (cid:101) φ k (cid:107) L (Σ t ) (cid:46) (cid:15) .3. The following flux estimates on constant- u k (cid:48) null hypersurfaces C k (cid:48) u k (cid:48) : (cid:88) Z k (cid:48) ∈{ L k (cid:48) ,E k (cid:48) } (cid:107) Z k (cid:48) ∂ x (cid:101) φ k (cid:107) L ( C k (cid:48) uk (cid:48) ) (cid:46) (cid:15)δ − , (cid:88) Z k (cid:48) ∈{ L k (cid:48) ,E k (cid:48) } (cid:107) Z k (cid:48) ∂ x φ reg (cid:107) L ( C k (cid:48) uk (cid:48) ) (cid:46) (cid:15). (1.6)There are two additional improvements to (1.6):(a) The bounds for (cid:101) φ k improve away from S kδ = { u k ∈ [ − δ, δ ] } : (cid:88) Z k (cid:48) ∈{ L k (cid:48) ,E k (cid:48) } (cid:107) Z k (cid:48) ∂ x (cid:101) φ k (cid:107) L ( C k (cid:48) uk (cid:48) \ S kδ ) (cid:46) (cid:15). (1.7)(b) When k = k (cid:48) , there is a δ -independent bound for all u k for a more restricted choice of derivatives: (cid:107) L k ∂ x (cid:101) φ k (cid:107) L ( C kuk ) + (cid:107) E k E k (cid:101) φ k (cid:107) L ( C kuk ) (cid:46) (cid:15). (1.8)4. Singular parts of different impulsive waves (cid:101) φ k and (cid:101) φ k (cid:48) ( k (cid:54) = k (cid:48) ) are transversal in a quantitative manner.For brevity, we have suppressed some additional wave estimates for φ which are proven and used in our ar-guments. Moreover, in order to obtain the Lipschitz bound for φ , we need further bound (cid:107) ∂E k φ reg (cid:107) H s (cid:48)(cid:48) (Σ t ) (cid:46) (cid:15) (for fixed s (cid:48)(cid:48) ∈ (0 , s (cid:48) ) ). We will defer all these discussions to [53]. Terms related to the elliptic gauge are in principle the most regular due to the ellipticity of the equations(1.3). There are two main technical issues:• In order to close our estimates, we need to bound ∂ ij g in L ∞ . Since ( ∂φ ) ∈ L ∞ , this correspondsexactly to an end-point elliptic estimate that fails.• A priori, ellipticity only gains in spatial, but not temporal regularity. Notice that the Besov estimate implies that ∂φ is continuous, and thus fails for the (rough) impulsive gravitational waves.Importantly for our argument, the Besov estimate nonetheless holds for the δ -impulsive waves, uniformly for all sufficientlysmall δ > . he easy estimates. Since φ ∈ W , ∞ ∩ H s (cid:48) (1 in Section 1.1.3) and is compactly supported, standardelliptic estimates immediately imply that g ∈ H s (cid:48) and g ∈ W ,p (with weights) for any p ∈ [1 , + ∞ ) . Theonly subtlety concerns the weights at infinity: ∂ x g are no better than (cid:104) x (cid:105) − at ∞ , and thus to handle the ( ∂ x g ) term on the RHS of (1.3) requires using the precise structure of the nonlinear terms. These weightissues can be handled in a similar manner as [34, 35]. The endpoint Besov space elliptic estimates.
To close our estimates we need further that ∂ x g ∈ L ∞ .This corresponds to the p = + ∞ case for the L p Calderon–Zygmund elliptic theory, which does not hold.Hence to obtain the W , ∞ estimate we need a slightly stronger Besov type estimate for ∂φ (recall 1(a)in Section 1.1.3). Our anisotropic Sobolev embedding theorem (see Theorem 7.3), which we use to obtainLipschitz bounds for φ , naturally gives a Besov strengthening. However, the Besov estimates we get are withrespect to ( u k , u k (cid:48) ) coordinates ( k (cid:54) = k (cid:48) ), and for this reason we introduce an extra physical space argumentto obtain a good endpoint elliptic estimates for the operator ∆ defined in the elliptic gauge coordinates. Estimates for ∂ t g . It is slightly more delicate to control second derivatives of the metric coefficientswith one spatial and one ∂ t derivative. Differentiating (1.3) by ∂ t , we obtain an equation with the followmain term: ∆ ∂ t g = ( ∂ φ )( ∂φ ) + . . . (1.9)In terms of W s,p spaces, the RHS is no better than being in L . This by itself would only give an estimatefor ∂ t g which is much worse than that for ∂ i g .Here is the key idea: we decompose ∂ t as linear combination of a good null derivative and spatialderivatives. Take for instance a contribution from two parallel waves, i.e. a term ∂ t [( ∂ (cid:101) φ k ) ] on the RHS of(1.9). With some (well-controlled) coefficients α and σ i , we have ∂ t [( ∂ (cid:101) φ k ) ] = αL k [( ∂ (cid:101) φ k ) ] + σ i ∂ i [( ∂ (cid:101) φ k ) ] = αL k [( ∂ (cid:101) φ k ) ] + ∂ i [ σ i ( ∂ (cid:101) φ k ) ] + . . . . The L k derivative is a good derivative for (cid:101) φ k which then gives better estimates. The other term is essentiallya total spatial derivative so that we gain with the ellipticity of the equation (1.9).The above argument becomes more subtle when there is an interaction of two waves propagating indifferent directions. Nevertheless, for a term such as ∂ t [( ∂ (cid:101) φ j )( ∂ (cid:101) φ k )] , (with j (cid:54) = k ) we exploit precisely thatthe waves are transversal (point 4 in Section 1.1.3) and decompose ∂ t is a spatially-dependent manner. Theresulting decomposition of ∂ t is not regular so that we are not able to control ∂ x ∂ t g in L ∞ , yet the errorgenerated is sufficiently lower order that we still bound ∂ x ∂ t g in sufficiently high L p spaces, as well as in H s (cid:48) . We now turn to the bounds for geometric quantities related to the eikonal functions u k , particularly theRicci coefficients χ k and η k , which obey the transport equations (1.4).Using the transport equation (1.4) and the Lipschitz bound for φ , it immediately follows that χ k and η k are bounded in (weighted) L ∞ .The first derivative estimates for χ k and η k are more subtle. Clearly, L k χ k and L k η k are in L ∞ by (1.4)and the above discussions. For the other first derivative, consider first χ k . We differentiate (1.4) by ∂ q toget the following schematic equation: L k ∂ q χ k = − L k ∂ q φ )( L k φ ) + · · · , (1.10)where · · · are lower order terms as before.To control the second derivative term L k ∂ q φ (recall that the first derivative L k φ is bounded), we use theflux estimates (3 in Section 1.1.3). Decompose the top derivative L k ∂ q φ = L k ∂ q φ reg + (cid:80) j =1 , , L k ∂ q (cid:101) φ j . Theregular part is well under control by (1.6), but for the singular parts, (1.6) itself (which has a δ − weight)is too weak, and we need to separately consider the cases k = j or k (cid:54) = j .• First, when k = j , we use the fact L k is a good derivative for φ k and the term L k ∂ q (cid:101) φ k is controlled by(1.8) without δ − weights. 8 Second, when k (cid:54) = j , the energy flux does not give good estimates for L k ∂ q (cid:101) φ j . Here, we use cruciallythe transversality of the δ -impulsive waves (4 in Section 1.1.3). While the L norm of L k ∂ q (cid:101) φ k is largeit is concentrated in S kδ , which is a small region of length scale δ . On the other hand, integrating (1.10)requires an L — instead of L — estimate along the integral curve of L k . We can thus gain a powerof δ (by the Cauchy–Schwarz inequality) using the smallness of the length scale.This gives a good estimate for ∂ q χ k in a mixed L ∞ u k L θ k type space. A similar argument controls E k η k .However, for a general spatial derivative ∂ x η k , when running the above argument for the k = j case, there isa second derivative term which is not controlled by (1.8), which results in the L ∞ u k L θ k norm of ∂ x η k blowingup as δ − . Instead, we can only control ∂ x η k in the L u k L θ k space.Finally, we have some bounds for special combinations of second derivatives such as L k χ k , L k ∂ q χ k , etc.,by virtue of the equations (1.4) and the already established bounds. Ultimately, the geometric estimates are important because they are needed to close the wave estimates. In[53], we will show indeed that the geometric estimates we obtain are sufficient.Since we will need some anisotropic bounds for the wave variables up to s (cid:48)(cid:48) derivatives, as well assome δ − -dependent bounds up to derivatives, all the geometric estimates that we mentioned above (forinstance the L ∞ bound for ∂ x g , the L p ∩ H s (cid:48) ( p ∈ [4 , ∞ ) ) bound for ∂ x ∂ t g and the L ∞ u k L θ k bound for χ k ,etc.) are necessary to carry out the energy estimates for the scalar wave.On the other hand, it is quite remarkable that even though for some geometric quantities we only haveweaker estimates (for instance we do not put ∂ x ∂ t g in L ∞ or obtain a δ − -independent L ∞ u k L θ k bound for ∂ x η k or control general second derivatives of χ k and η k ), theses bounds are sufficient in the commutatorestimates that we need to bound the wave part in [53]. This is for instance because certain potentiallydangerous terms do not appear due to the structure of the commutators.Finally, in order to close the argument, we need to control the change between the elliptic gauge coordi-nates and the geometric coordinates, as well as the commutators for various vector fields. It will turn outthat the control we establish for the geometric quantities will just be sufficient to justify that the eikonalfunction u k is a W , ∞ function in terms of the elliptic gauge coordinates, and that the second derivativesof the commutation vector fields ( L k , E k , X k ) with respect to the elliptic gauge coordinate derivatives are in L . Both of these statements are used in order to close the geometric and wave estimates. Beyond [38, 65], there are further examples of interactions of two impulsive gravitational waves, see forinstance [17, 18, 27, 30, 31, 32, 59]. All these constructions rely heavily on symmetry assumptions. Thesingularity structures in these examples and their stability were further discussed in [68, 72, 73]. We referthe readers to the books [6, 29] for further details and related examples (including those where matter fieldsare present).In terms of mathematical results, priori to the works [50, 51], there were low-regularity existence resultsin T -symmetry [47, 48] which in particular included impulsive gravitational waves and their interactions.Relatedly, Christodoulou [23] constructed solutions in the BV class to the Einstein–scalar field system inspherical symmetry. This can be thought of as including as a particular case a scalar field analogue ofimpulsive gravitational waves. Finally, very recently, a class of spacetimes featuring the interactions of twoimpulsive gravitational waves without any symmetry but still possessing a piece of future null infinity hasbeen constructed in [2]. Our problem can be viewed in the larger context of low-regularity problems in general relativity. In theSobolev H s spaces, this has attracted much interest [3, 4, 39, 40, 41, 64, 67], culminating in the seminalproof of the bounded L curvature theorem [42], which requires the initial data to only be in H .9ow-regularity problems are interesting for quasilinear wave equations beyond the Einstein equations, seefor example [26, 64, 70]. We highlight particularly the work [5] of Bahouri–Chemin on the high-dimensionallow-regularity well-posedness of a coupled wave-elliptic system similar to the structure of the polarized U (1) -reduced Einstein vacuum equations in an elliptic gauge. We compare our result with the work of Huneau–Luk [36] on high-frequency limits in polarized U (1) sym-metry. In both [36] and this paper, a local existence result is proved where φ is Lipschitz but not better.On the one hand, the use of the elliptic gauge and (approximate) eikonal functions plays an important rolein both papers. On the other hand, however, the analysis is quite different as one needs to rely on precisefeatures of the problems (either that the waves are of high frequency in [36] or are highly localized in oursetting). The present work can be viewed in the larger context of interaction of conormal singularities for hyperbolicequations. There is a large literature for weak conormal singularities, beginning with the pioneering worksof Bony [11, 13, 14]. In particular, these singularities are sufficiently weak so that classical well-posednessresults hold . For the interaction of two conormal singularities in the quasilinear case, see [1, 37].The interaction of three conormal singularities — even for very weak singularities — has only been studiedfor semilinear model problems. It has been shown [12, 15, 55, 56] that in this case the only possible newsingularity after the triple interaction must be weaker and lies in the cone emanating from the intersection.Moreover, it has been demonstrated that in general a new singularity could indeed arise in various differentmodels [62, 8, 7].For a sample of further related works, see [9, 10, 19, 28, 46, 57, 61, 63, 74] and the references therein.See also [20, 43, 44, 45, 69, 71] for related more recent works concerning inverse problems.
We discuss some open problems and possible future directions related to our work.1. (
Non-compactly supported initial data ) Our main theorem assumes that φ is initially compactlysupported. It could be expected that the compact support can be replaced by fast decay of theinitial data, but this creates a few technical issues in view of the fact that the metric coefficients growlogarithmically as | x | → ∞ .2. ( Large data ) The theory of [51] allows also for the interaction of impulsive gravitational waves of large amplitude. Among other things, our theory is limited to the small amplitude region due to theglobal elliptic gauge.
3. (
Beyond polarized U (1) symmetry ) The present work restricts to polarized U (1) symmetry. Whilethe completely general case seems out of reach at the moment, the natural next step would be tostudy the interaction still under the U (1) symmetry assumption but without polarization. In this case,the equations reduce to an Einstein–wave map system (as opposed to simply the Einstein–scalar fieldsystem) in (2 + 1) dimensions. It seems plausible that the extra “wave map” part can be controlledafter choosing an elliptic (e.g. Coulomb) gauge, so that one can use some of the techniques in thispaper. We hope to return to this problem in the future.4. ( More singular initial data ) In [51], a more general theorem was proven, which allows the interactionof not only impulsive gravitational waves, but also of more singular data where the worst Christoffelsymbol is only L (instead of being in L ∞ ∩ BV ). This stronger result has various other applications[25, 49, 52], and is in particular related to the interaction of null dust shells [52]. It is therefore naturalto ask whether we can extend our results in the present paper on the interaction of three impulsivegravitational waves to more singular initial data. We remark that a smallness assumption is already needed in the smooth theory in such a gauge [35].
10. (
Uniqueness ) As already mentioned in Remark 1.6, our main theorem does not give uniqueness. Itis of interest to appropriately formulate and prove a uniqueness result for these solutions.
6. (
Higher regularity ) Ideally one would like to prove stronger regularity statements away from theunion of the impulsive waves, or better yet to understand the optimal regularity.7. (
Lower bounds and creation of new singularities ) Related to the last point, it would be interestingto show that the jump in the data persists along null characteristics, or even to derive a transportequation for the jump. More ambitiously, one can study whether new (but weaker) singularities appearin the cone emanating from the intersection point of the three impulsive gravitational waves as in thesemilinear model problems [62].8. (
Interaction of four impulsive gravitational waves ) While our work allows for the transversalinteraction of any number of impulsive gravitational waves under the polarized U (1) symmetry assump-tion , it would be of interest to study the generic transversal interaction of four impulsive gravitationalwaves, where four waves interact at a point in (3 + 1) dimensions. See Remark 1.3. The remainder of the paper is structured as follows.We begin with definitions for the geometric setup and the norms in
Section 2 and 3 respectively.In
Section 4 , we then define the class of data corresponding to both impulsive gravitational wavesand δ -impulsive gravitational waves (recall Remark 1.5). The precise statements of the main theorem forimpulsive gravitational waves (Theorem 5.2) and for δ -impulsive gravitational waves (Theorem 5.6) are thengiven in Section 5 . In
Section 6 , we prove Theorem 5.2 assuming Theorem 5.6. In
Section 7 , we proveTheorem 5.6 by reducing it to three theorems on a priori estimates (Theorems 7.1, 7.3 and 7.4).The remainder of this paper is devoted to the proof of Theorem 7.1 (Theorems 7.3 and 7.4 will be provenin [53]). After proving preliminary estimates in
Section 8 , we obtain geometric estimates associated to theelliptic gauge in
Section 9 and geometric estimates associated to the eikonal functions in
Section 10 . In
Section 11 , we then conclude the proof Theorem 7.1.Finally, in
Appendix A , we handle all issues regarding initial data and constraint equations.
Acknowledgements
We are grateful to Igor Rodnianski for inspiring conversations. We particularly thank him for some ideasthat greatly simplified our original arguments. Those ideas now appear as part of Section 9.4.2. We alsothank Haydée Pacheco for Figure 1.Most of this work was carried out when M. Van de Moortel was a visiting student at Stanford University.During the time that this work was pursued, J. Luk has been supported by a Terman fellowship and theNSF grants DMS-1709458 and DMS-2005435.
In this section, we introduce the basic geometric setup. This plays a fundamental role for the whole seriesof papers.In
Section 2.1 , we introduce the polarized U (1) symmetry and our elliptic gauge condition. In Sec-tion 2.2 , we discuss the Einstein vacuum equations under these symmetry and gauge conditions.In
Section 2.3 , we introduce the eikonal functions and the related null frames ( L k , X k , E k ) , which areimportant to capture the propagating impulsive waves. In Section 2.4 , we introduce a system of geometriccoordinates associated to the eikonal functions. In
Section 2.5 , we derive transformation formulas betweenthe null frames, and the coordinate vector fields in various different coordinate system.In
Section 2.6 , we compute all the connection coefficients with respect to the null frames ( L k , X k , E k ) .In Section 2.7 , we compute the derivatives of the coefficients of ( L k , X k , E k ) in the ( ∂ t , ∂ , ∂ ) basis. Note that on the other hand, uniqueness for the δ -impulsive waves of course follows from standard theory. Section 2.8 and
Section 2.9 , we compute respectively the transport equations for the frame coeffi-cients and the connection coefficients.Finally, in
Section 2.10 , we compute the initial values of all the eikonal quantities.
Definition 2.1 (Polarized U (1) symmetry) . We say that a (3+1) Lorentzian manifold ( M = I × R × S , (4) g ) ,where I ⊆ R is an interval, has polarized U (1) symmetry if the metric (4) g can be expressed as: (4) g = e − φ g + e φ ( dx ) , (2.1) where φ is a scalar function on I × R and g is a (2 + 1) Lorentzian metric on I × R . Definition 2.2 (The foliation Σ t ) . Given a spacetime as in Definition 2.1, we foliate the spacetime ( I × R , g ) with hypersurfaces { Σ t } t ∈ I , where each Σ t is spacelike. We will later make a particular choice of t ; see Definition 2.5.The metric can then be written as g = − N dt + ¯ g ij ( dx i + β i dt )( dx j + β j dt ) , (2.2) for some function N > and Riemannian metric ¯ g ij .Here, and the remainder of the paper, we use the convention the lower case Latin indices refer tothe spatial coordinates ( x , x ) , and repeated indices are summed over. In contrast, we use lower caseGreek indices to refer to spacetime coordinates ( x , x , x ) := ( t, x , x ) . Definition 2.3 (Coordinate derivatives) . From now on, we use ∂ i = ∂ x i ( i = 1 , ) to denote the spatialcoordinate partial derivatives, and ∂ α ( α = 0 , , ) to denote spacetime coordinate partial derivatives withrespect to the ( x , x , x ) := ( t, x , x ) coordinate system in (2.2) . Definition 2.4.
Given ( I × R , g ) and { Σ t } t ∈ I in Definition 2.2.1. (Spacetime connection) Denote by ∇ the Levi–Civita connection of the spacetime metric g .2. (Induced metric) Denote by ¯ g the induced metric on the two-dimensional hypersurface Σ t .3. (Normal to Σ t ) Denote by (cid:126)n the future-directed unit normal to Σ t ; (cid:126)n = ∂ t − β i ∂ i N . (2.3) satisfying g ( (cid:126)n, (cid:126)n ) = − . Define also e to be the vector field e = ∂ t − β i ∂ i = N · (cid:126)n. (2.4)
4. (Second fundamental form) Define K to be the second fundamental form on Σ t : K ( Y, Z ) = g ( ∇ Y (cid:126)n, Z ) , (2.5) for every Y, Z ∈ T Σ t . Definition 2.5 (Gauge conditions) . We define our gauge conditions (assuming already (2.1) ) as follows:1. For every t ∈ I , Σ t is required to be maximal, i.e. (¯ g − ) ij K ij = 0 . (2.6) Note that (2.6) defines the coordinate t . Note that since φ and g are defined on I × R , they do not depend on x , the coordinate on S . We remark that the definition of K differs by a factor of − from that in [35]. . We choose the coordinate system on Σ t so that ¯ g ij is conformally flat, i.e. ¯ g ij = e γ δ ij , (2.7) where, from now on, δ ij (or δ ij ) denotes the Kronecker delta. We collect some simple computations:
Lemma 2.6.
The following holds for g of the form (2.2) satisfying Definition 2.5:1. The inverse metric g − is given by g − = 1 N − β β β N e − γ − β β − β β β − β β N e − γ − β β . (2.8)
2. The following commutation formula holds: [ (cid:126)n, ∂ q ] = ∂ q log( N ) · (cid:126)n + 1 N ( ∂ q β i ) · ∂ i . (2.9)
3. The spacetime volume form associated to g is given by dvol = N e γ dx dx dt. (2.10) The induced volume form on the spacelike hypersurface Σ t associated to g is given by dvol Σ t = e γ dx dx . (2.11)
4. The wave operator (cid:3) g is defined to be the Laplace–Beltrami operator associated to g , which is given by (cid:3) g f = − e fN + e − γ δ ij ∂ ij f + e NN e f + e − γ N δ ij ∂ i N ∂ j f = − (cid:126)n f + e − γ δ ij ∂ ij f + e − γ N δ ij ∂ i N ∂ j f. (2.12)
5. The condition (2.6) can be rephrased as ∂ q β q = 2 e ( γ ) , (2.13)
6. The second fundamental form is given by K ij = e γ N · ( ∂ q β q · δ ij − ∂ i β q · δ qj − ∂ j β q · δ iq ) =: − e γ N ( L β ) ij , (2.14) where L is the conformal Killing operator ( L β ) ij := − ∂ q β q · δ ij + ∂ i β q · δ qj + ∂ j β q · δ iq . Finally, we compute the connection coefficients with respect to { e , ∂ , ∂ } : Lemma 2.7.
Given g of the form (2.2) satisfying Definition 2.5, g ( ∇ e e , e ) = − N · e N, (2.15) g ( ∇ e e , ∂ i ) = − g ( ∇ ∂ i e , e ) = g ( ∇ e ∂ i , e ) = N · ∂ i N, (2.16) g ( ∇ ∂ j e , ∂ i ) = g ( ∇ e ∂ j , ∂ i ) − e γ · ∂ j β l δ il = − g ( ∇ ∂ j ∂ i , e )= e γ · (2 e γ · δ ij − ∂ i β q · δ jq − ∂ j β q · δ iq ) = e γ · ( ∂ q β q · δ ij − ∂ i β q · δ jq − ∂ j β q · δ iq ) . (2.17) Moreover, ∇ ∂ i ∂ j = e γ N · ( ∂ q β q · δ ij − ∂ i β q · δ jq − ∂ j β q · δ iq ) (cid:126)n + (cid:0) δ qi ∂ j γ + δ qj ∂ i γ − δ ij δ ql ∂ l γ (cid:1) ∂ q , (2.18) ∇ e e = e NN · e + e − γ δ ij N ∂ i N ∂ j , (2.19) ∇ e ∂ i = ∇ ∂ i e + ∂ i β j e j = ∂ i NN · e + 12 · (cid:16) ∂ q β q · δ ji + ∂ i β j − δ iq δ jl ∂ l β q (cid:17) ∂ k . (2.20)13 .2 Einstein equations With the polarized U (1) symmetry (2.1), the Einstein equation Ric µν ( (4) g ) = 0 can be re-written in termsof the (2 + 1) -dimensional metric g and the scalar field φ as: Ric αβ ( g ) = 2 ∂ α φ∂ β φ, (2.21) (cid:3) g φ = 0 . (2.22)Additionally, given the form of the metric (2.2) and the gauge conditions in Definition 2.5, (2.21) impliesthe following elliptic equations (see [36, (4.26)–(4.28)], but note the sign difference in definitions of K ): δ ik ∂ k K ij = 2 e γ · (cid:126)nφ · ∂ j φ, (2.23) ∆ N = e γ N | L β | + 2 N e γ · ( (cid:126)nφ ) , (2.24) ∆ γ = − δ il ( ∂ i φ )( ∂ l φ ) − e γ N | L β | − e γ · ( (cid:126)nφ ) , (2.25) ∆ β j = δ ik δ jl ( ∂ k NN − ∂ k γ )( L β ) il − N δ jl · (cid:126)nφ · ∂ l φ, (2.26)where ∆ denotes the Euclidean
Laplacian ∆ = (cid:80) i =1 ∂ ii .We remark that the above equations are not independent, as (2.26) can be derived from taking thedivergence of (2.14) and using (2.23).The equations (2.2), Definition 2.5 and (2.21) also imply (cid:126)n ( K ij ) − N − ∂ i ∂ j N + 12 · δ ij · N − · ∆ N = 2 e − γ K li K jl + N − · ( ∂ j β k K ki + ∂ i β k K kj ) − N − · ( δ ki ∂ j γ + δ kj ∂ i γ − δ ij δ lk ∂ l γ ) · ∂ k N + 2 ∂ i φ · ∂ j φ − δ ij δ kl ∂ k φ · ∂ l φ. (2.27) We will define three eikonal functions together with null hypersurfaces and null frames. Each of these willlater be chosen to adapted to one propagating wave.
Definition 2.8 (Eikonal functions) . Given a spacetime ( I × R , g ) of the form (2.2) satisfying Definition 2.5,define three eikonal functions u k , k = 1 , , , corresponding to the three impulsive waves, as the uniquesolutions to ( g − ) αβ ∂ α u k ∂ β u k = 0 , (2.28) ( u k ) | Σ = a k + c kj x j , (2.29) which satisfies e u k > . Here, a k , c kj ∈ R are constants obeying the following conditions : (cid:113) c k + c k = 1 , (2.30) | − c k · c k (cid:48) + c k · c k (cid:48) | = (cid:112) − | c k · c k (cid:48) + c k · c k (cid:48) | ≥ κ , (2.31) for some fixed constant κ ∈ (0 , π ) , and for every k (cid:54) = k (cid:48) ∈ { , , } . Definition 2.9 (Sets associated with the eikonal functions) . Let u k ( k = 1 , , ) satisfying (2.28) and (2.29) in ( I × R , g ) be given. For simplicity, we stipulate here that the initial wavefront ( u k ) | Σ are exact lines in the ( x , x ) coordinates. This caneasily be relaxed so that they are only approximate lines. The identity | − c k · c k (cid:48) + c k · c k (cid:48) | = 1 − | c k · c k (cid:48) + c k · c k (cid:48) | is an immediate consequence of (2.30). . For all w ∈ R , define C kw := { ( t, x ) : u k ( t, x ) = w } , C k ≤ w := (cid:91) u k ≤ w C ku k , C k ≥ w := (cid:91) u k ≥ w C ku k , (2.32) and, for every T ∈ I , define C kw ([0 , T )) := C kw ∩ ( ∪ t ∈ [0 ,T ) Σ t ) . (2.33)
2. For all w , w ∈ R , define S k ( w , w ) := (cid:91) w ≤ u k ≤ w C ku k . (2.34) For δ > , define also S kδ := S k ( − δ , δ ) . (2.35) We will later understand S kδ as “the singular zone” for (cid:101) φ k . Definition 2.10 (Definition of the null frame) .
1. Define the null vector L geok associated to the eikonalfunction u k by L geok = − ( g − ) αβ ∂ β u k · ∂ α . (2.36)
2. Define L k to be the vector field parallel to L geok which satisfies L k t = N − , i.e. L k = µ k · L geok , µ k = ( N · L geok t ) − . (2.37)
3. Define the vector field X k to be the unique vector field tangential to Σ t which is everywhere orthogonal(with respect to ¯ g ) to C ku k ∩ Σ t and such that g ( X k , L k ) = − .4. Define E k to be the unique vector field which is tangent to C ku k ∩ Σ t , satisfies g ( E k , E k ) = 1 , and suchthat ( X k , E k ) has the same orientation as ( ∂ , ∂ ) . Lemma 2.11. L geok is null and geodesic, i.e. g ( L geok , L geok ) = 0 , ∇ L geok L geok = 0 . (2.38)
2. The following holds: L k u k = E k u k = 0 , E k t = X k t = 0 , L k t = N − , X k u k = µ − k . (2.39)
3. The normal (cid:126)n can be expressed in terms of X k and L k as: (cid:126)n = L k + X k . (2.40)
4. The triplet ( X k , E k , L k ) forms a null frame, i.e. it satisfies g ( L k , X k ) = − , g ( E k , L k ) = g ( E k , X k ) = g ( L k , L k ) = 0 , g ( E k , E k ) = g ( X k , X k ) = 1 . (2.41) g − can be given in terms of the ( X k , E k , L k ) frame by g − = − L k ⊗ L k − L k ⊗ X k − X k ⊗ L k + E k ⊗ E k . (2.42) Proof. (2.38) is an immediate consequence of (2.28).For (2.39), the first two chains of equalities simply follow from tangential properties of the vector fields.That L k t = N − follows from Definition 2.10.1. Finally, using Definition 2.10, X k u k = g σρ ( g − ) αρ ∂ α u k X σk = − g ( L geok , X k ) = µ − k .To establish (2.40), we need to show that − L k + (cid:126)n satisfies all the defining properties of X k in Def-inition 2.10. First, (2.3) and (2.37) imply ( − L k + (cid:126)n ) t = 0 , i.e. L k − (cid:126)n is tangent to Σ t . Moreover,15 ( − L k + (cid:126)n, E k ) = − g ( L k , E k ) + g ( (cid:126)n, E k ) = 0 , and also g ( − L k + (cid:126)n, X k ) = − g ( L k , X k ) = 1 . Hence − L k + (cid:126)n = X k , i.e. (2.40) holds.Turning to (2.41), first note that g ( L k , X k ) = − , g ( E k , E k ) = 1 and g ( E k , X k ) = 0 by Definition 2.10,and g ( L k , L k ) = 0 can be derived using additionally (2.38).Next, g ( E k , L geok ) = 0 (and hence g ( E k , L k ) = 0 ) follows from E k u k = 0 and (2.36). Finally, note thatusing g ( L k , L k ) = 0 , (2.40) and the fact that (cid:126)n is the unit normal to Σ t , we have g ( L k , L k ) = g ( (cid:126)n − X k , (cid:126)n − X k ) = g ( (cid:126)n, (cid:126)n ) − g ( (cid:126)n, X k ) + g ( X k , X k ) = − g ( X k , X k ) , which gives g ( X k , X k ) = 1 .Now that we have established (2.41), the equation (2.42) follows as an immediate consequence. ( u k , θ k , t k ) We now introduce the coordinate θ k such that ( u k , θ k , t k ) is a regular coordinate system on I × R . Definition 2.12.
1. Given u k satisfying (2.28) – (2.29) , and fixing some constants b k , define θ k by L k θ k = 0 , (2.43) ( θ k ) | Σ = b k + c ⊥ kj x j , (2.44) where c ⊥ k = − c k and c ⊥ k = c k , and c ki are the constants in (2.29) .2. Let t k = t .3. Denote by ( ∂ u k , ∂ θ k , ∂ t k ) the coordinate vector fields in the ( u k , θ k , t k ) coordinate system. (Note that wecontinue to use ∂ t to denote the coordinate derivative in the ( x , x , t ) coordinate system of Section 2.1.) Lemma 2.13.
Defining Θ k = ( E k θ k ) − and Ξ k = X k θ k , we have L k = 1 N · ∂ t k , E k = Θ − k · ∂ θ k , X k = µ − k · ∂ u k + Ξ k · ∂ θ k . (2.45) Proof.
This follows from combining (2.39), (2.43) with the definitions of Θ k and Ξ k . Lemma 2.14.
1. The metric g in the ( t k , u k , θ k ) coordinate system is given by g = Θ k dθ k − µ k N dt k du k − µ k Ξ k Θ k du k dθ k + µ k (1 + Ξ k Θ k ) du k . (2.46)
2. The volume form induced by g and ¯ g in the ( t k , u k , θ k ) coordinate system are given by dvol = µ k · N · Θ k dt k du k dθ k , dvol Σ t = µ k Θ k du k dθ k . (2.47)
3. Letting dvol C uk be the volume form on C u k such that du k ∧ dvol C uk = dvol . Then dvol C uk = µ k · N · Θ k dt k dθ k . (2.48) Proof. (2.46) follows from (2.45) and (2.41); (2.47) and (2.48) follow from (2.46) directly.We establish using Proposition 2.13 a first set of relations between the frame coefficients.
Lemma 2.15.
We have the following relations between the commutator and the frame coefficients: [ E k , L k ] = L k log( Θ k ) · E k − E k log( N ) · L k . (2.49) [ L k , X k ] = − L k log( µ k ) · X k + ( L k log( µ k ) · Ξ k + L k Ξ k ) · Θ k · E k + X k log( N ) · L k , (2.50) [ E k , X k ] = − E k log( µ k ) · X k + (cid:0) E k Ξ k − X k Θ − k + Ξ k · E k log( µ k ) (cid:1) · Θ k · E k . (2.51)16 roof. (2.49) follows directly from (2.45).For (2.50), we first compute [ L k , X k ] = (cid:2) N − ∂ t k , µ − k ∂ u k + Ξ k · ∂ θ k (cid:3) = X k log( N ) · L k − L k log( µ k ) µ − k ∂ u k + L k Ξ k · ∂ θ k , and then use L k log( µ k ) · µ − k ∂ u k = L k log( µ k ) · X k − Ξ k · L k log( µ k ) · ∂ θ k , which gives (2.50).Finally (2.51) is an easy consequence of [ E k , X k ] = Θ − k ( ∂ θ k µ − k ) ∂ u k + Θ − k ( ∂ θ k Ξ k ) ∂ θ k − ( X k Θ − k ) ∂ θ k , and E k log µ k · µ − k ∂ u k = E k log µ k · X k − Ξ k · E k log µ k · ∂ θ k . Σ t between ( X k , E k ) and the elliptic coordinate vector fields ( ∂ , ∂ ) Lemma 2.16.
The following identities between E ik and X ik hold: E k = − X k , E k = X k . (2.52) Moreover, the coordinate vector fields ( ∂ , ∂ ) can be expressed in terms of ( E k , X k ) as follows: ∂ = e γ · (cid:0) − X k · E k + E k · X k (cid:1) , ∂ = e γ · (cid:0) X k · E k − E k · X k (cid:1) . (2.53) Proof.
Since E k and X k are orthonormal for ¯ g (see (2.41)), it follows that δ ij X ik X jk = δ ij E ik E jk = e − γ .Now set ˜ E k = e γ · E k and ˜ X k = e γ · X k . Then, using also point 4 of Definition 2.10, ( ˜ X k , ˜ E k ) is anorthonormal basis for the Euclidean metric δ ij on R with the same orientation as ( ∂ , ∂ ) . Hence thereexists ϕ ∈ R such that ˜ E k = cos( ϕ ) · ∂ + sin( ϕ ) · ∂ , ˜ X k = sin( ϕ ) · ∂ − cos( ϕ ) · ∂ , This, in particular, gives (2.52). Inverting the rotation matrix, we obtain ∂ = cos( ϕ ) · ˜ E k − sin( ϕ ) · ˜ X k , ∂ = − sin( ϕ ) · ˜ E k − cos( ϕ ) · ˜ X k , which gives directly (2.53). u k Now we compute the derivatives of u k and θ k with respect to ( ∂ t , ∂ i ) . Lemma 2.17.
The following identities hold: ∂ i u k = e γ · µ − k · δ ij X jk , (2.54) ∂ i θ k = e γ δ ij · (cid:16) Θ − k · E jk + Ξ k · X jk (cid:17) , (2.55) ∂ t u k = β q ∂ q u k + N · µ − k , (2.56) ∂ t θ k = β i ∂ i θ k + N · Ξ k = e γ · β j · (cid:16) Θ − k · E jk + Ξ k · X jk (cid:17) + N · Ξ k . (2.57) Moreover, for all vector field Y in the tangent space of Σ t , we have Y u k = µ − k · g ( Y, X k ) . (2.58) Proof.
We use (2.53) to compute ∂ i u k and ∂ i θ k , and apply also (2.45) and (2.52) to obtain (2.54) and (2.55).Both (2.56) and (2.57) can be derived by L k u k = L k θ k = 0 (by (2.45)), the identity L k = N − · ( ∂ t − β q ∂ q ) − X k (by (2.40) and (2.3)), and (2.45).Finally, the identity (2.58) follows from (2.54) and (2.7).17 .5.3 Spatial coordinate system ( u k , u k (cid:48) ) on Σ t Fix k, k (cid:48) ∈ { , , } with k (cid:54) = k (cid:48) . Introduce the spatial coordinate system ( u k , u k (cid:48) ) . So as to distinguish itfrom other coordinate derivatives, we define the coordinate vector fields on Σ t in the ( u k , u k (cid:48) ) coordinatesystem by ( /∂ u k , /∂ u k (cid:48) ) .We now express ( /∂ u k , /∂ u k (cid:48) ) in terms of ( X k , E k ) in the following lemma: Lemma 2.18.
The vector fields X k and E k can be expressed in the ( u k , u k (cid:48) ) coordinate system as follows: X k = µ − k · /∂ u k + µ − k (cid:48) · g ( X k , X k (cid:48) ) · /∂ u k (cid:48) , (2.59) E k = µ − k (cid:48) · g ( E k , X k (cid:48) ) · /∂ u k (cid:48) . (2.60) The above transformation can be inverted to give /∂ u k (cid:48) = µ k (cid:48) · g ( E k , X k (cid:48) ) − E k , (2.61) /∂ u k = µ k X k − µ k · g ( X k , X k (cid:48) ) g ( E k , X k (cid:48) ) · E k . (2.62) Proof.
We start to define a , b , c , d as X k = a · /∂ u k + b · /∂ u k (cid:48) ,E k = c · /∂ u k + d · /∂ u k (cid:48) . Since E k u k = 0 we know that c = 0 . To determine d we compute d = E k ( u k (cid:48) ) = µ − k (cid:48) · g ( E k , X k (cid:48) ) by (2.58).We also know by (2.45) that a = X k u k = µ − k . To determine b we compute b = X k ( u k (cid:48) ) = µ − k (cid:48) · g ( X k , X k (cid:48) ) by (2.58) again. We now define some Ricci coefficients in terms of the frame ( X k , E k , L k ) : χ k = g ( ∇ E k L k , E k ) = − g ( ∇ E k E k , L k ) , (2.63) η k = g ( ∇ X k L k , E k ) = − g ( ∇ X k E k , L k ) . (2.64)All the other Ricci coefficients can, in fact, be determined from χ k , η k , N , µ k and the contractions of K . Lemma 2.19.
The following identities hold: g ( ∇ E k X k , E k ) = − g ( X k , ∇ E k E k ) = K ( E k , E k ) − χ k , (2.65) g ( ∇ E k L k , X k ) = − g ( ∇ E k X k , L k ) = K ( E k , X k ) , (2.66) g ( ∇ L k E k , X k ) = − g ( ∇ L k X k , E k ) = K ( E k , X k ) − E k log( N ) , (2.67) g ( ∇ X k L k , X k ) = − g ( ∇ X k X k , L k ) = K ( X k , X k ) , (2.68) g ( ∇ X k X k , E k ) = − g ( ∇ X k E k , X k ) = K ( E k , X k ) − η k , (2.69) g ( ∇ L k X k , L k ) = − g ( ∇ L k L k , X k ) = L k log( µ k ) = K ( X k , X k ) − X k log( N ) . (2.70) All the other Ricci coefficients that have not been mentioned in (2.63) – (2.70) are zero.As a consequence, we have the following covariant derivatives and commutators: ∇ E k L k = χ k · E k − K ( E k , X k ) L k , (2.71) ∇ L k E k = ( E k log( N ) − K ( E k , X k )) · L k , (2.72) [ E k , L k ] = χ k · E k − E k log( N ) · L k . (2.73) ∇ E k X k = K ( E k , X k ) X k + ( K ( E k , E k ) − χ k ) · E k + K ( E k , X k ) L k , (2.74)18 X k E k = η k X k + K ( E k , X k ) L k , (2.75) [ E k , X k ] = ( K ( E k , X k ) − η k ) · X k + ( K ( E k , E k ) − χ k ) · E k , (2.76) ∇ L k X k = ( − K ( E k , X k )+ E k log N ) · E k − ( K ( X k , X k ) − X k log( N )) · X k − ( K ( X k , X k ) − X k log( N )) · L k , (2.77) ∇ X k L k = η k · E k − K ( X k , X k ) · L k , (2.78) [ L k , X k ] = − ( K ( E k , X k ) − E k log N + η k ) · E k − ( K ( X k , X k ) − X k log( N )) · X k + X k log( N ) · L k , (2.79) ∇ E k E k = χ k · X k + K ( E k , E k ) · L k , (2.80) ∇ X k X k = K ( X k , X k ) · X k + ( K ( E k , X k ) − η k ) · E k + K ( X k , X k ) · L k , (2.81) ∇ L k L k = ( K ( X k , X k ) − X k log( N )) · L k . (2.82) Proof.
Recall from (2.40) that L k = (cid:126)n − X k , an identity we will use throughout the proof. In particular(2.65) follows immediately from this identity, (2.63) and the definition of K ; so does (2.66), after noticingthat g ( ∇ E k X k , X k ) = · E k ( g ( X k , X k )) = 0 as X k is g -unitary.(2.67) follows from (2.66) and the observation that, by (2.49), g ([ E k , L k ] , X k ) = E k log( N ) . (2.68) follows from L k = (cid:126)n − X k and the fact that X k is g -unit; so does (2.69), using also the definition(2.64).For (2.70), first note using (2.50) and the fact that L k is null, we have g ( ∇ L k X k , L k ) = g ([ L k , X k ] , L k ) = L k log( µ k ) . Then, by (2.78), (2.41) and (2.50), K ( X k , X k ) = g ( ∇ X k L k , X k ) = g ([ X k , L k ] , X k ) = L k log( µ k ) + X k log( N ) , which implies the last equation in (2.70) after rearranging.The fact that all the other Ricci coefficients vanish is mostly trivial, except for g ( ∇ L k E K , L k ) = 0 , whichholds by (2.49).Finally, the covariant derivatives and commutators follow straightforwardly from the Ricci coefficientsand the frame conditions (2.41). Details are left to the reader. XEL vector fields in the elliptic gauge
The goal of this section is to compute ∂ ( Y αk ) for Y k ∈ { X k , E k , L k } and Y k = Y αk ∂ α in the coordinate system ( t, x , x ) of section 2.1. Proposition 2.20.
The derivatives of E ik with respect to ( L k , X k , E k ) can be expressed as follows: L k ( E ik ) = − ( E k log( N ) − K ( E k , X k )) · X ik − N · (cid:16) E ik ∂ q β q + E jk ∂ j β i − δ jq E jk · δ il ∂ l β q (cid:17) − (cid:16) X ik E jk ∂ j γ + E ik X jk ∂ j γ (cid:17) , (2.83a) X k ( E ik ) = − ( X ik · E k γ + E ik · X k γ ) + ( η k − K ( E k , X k )) · X ik , (2.83b) E k ( E ik ) = − E ik · E k γ + e − γ · δ il · ∂ l γ + ( − K ( E k , E k ) + χ k ) · X ik . (2.83c) The derivatives of X ik with respect to Y ∈ { L k , X k , E k } can be expressed as follows: Y ( X k ) = Y ( E k ) , Y ( X k ) = − Y ( E k ) . (2.84) The derivatives of L tk with respect to Y ∈ { L k , X k , E k } can be expressed as follows: Y ( L tk ) = − Y log( N ) N . (2.85)
The derivatives of L ik with respect to Y ∈ { L k , X k , E k } can be expressed as follows: Y ( L ik ) = − Y ( X ik ) − Y ( β i N ) . (2.86)19 roof. Step 1: Proof of (2.83a) . We start with the elementary ∇ L k E k = ∇ L k ( E jk ∂ j ) = L k ( E jk ) ∂ j + E jk ∇ L k ( ∂ j ) . (2.87)Writing L k = N e − X lk ∂ l (by (2.4) and (2.40)). Hence, using (2.17), (2.18), we have g ( ∇ L k ( ∂ j ) , ∂ i ) = e γ · N · ( ∂ q β q · δ ij + ∂ j β q · δ qi − ∂ i β q · δ jq ) − e γ · X lk · ( δ li ∂ j γ − δ lj ∂ i γ + δ ij ∂ l γ ) . (2.88)Hence, combining (2.87) and (2.88), and using (2.7), we obtain g ( ∇ L k E k , ∂ i ) = e γ · δ il · L k ( E l ) + e γ · E jk N · ( ∂ q β q · δ ij + ∂ j β q · δ qi − ∂ i β q · δ jq ) − e γ · X lk E jk · ( δ li ∂ j γ − δ lj ∂ i γ + δ ij ∂ l γ ) . (2.89)Now, by (2.72) (and (2.4) and (2.40)), we also know that g ( ∇ L k E k , ∂ i ) = − e γ · ( E k log( N ) − K ( E k , X k )) · δ il X lk . Hence, we get L k ( E ik ) = − ( E k log( N ) − K ( E k , X k )) · X ik − N · (cid:16) E ik ∂ q β q + E jk ∂ j β i − δ jq E jk · δ il ∂ l β q (cid:17) − (cid:16) X ik E jk ∂ j γ + E ik X jk ∂ j γ (cid:17) , using the fact that δ lj X lk E jk = 0 (by (2.41)). This gives (2.83a). Step 2: Proof of (2.83b) and (2.83c) . We use (2.18) and (2.7) to deduce that for Z k ∈ { X k , E k } : g ( ∇ Z k ( ∂ j ) , ∂ i ) = e γ · Z lk · ( δ li ∂ j γ + δ ij ∂ l γ − δ lj ∂ i γ ) . (2.90)We now combine (2.90) with X k ( E jk ) ∂ j = ∇ X k E k − E jk ∇ X k ∂ j and (2.75), and use additionally (2.7),(2.4) and (2.40), to obtain ( η k − K ( E k , X k )) · X lk · e γ · δ il = g ( ∇ X k E k , ∂ i )= X k ( E jk ) · e γ · δ ij + E jk · X lk · e γ · ( δ li ∂ j γ + δ ij ∂ l γ − δ lj ∂ i γ )= X k ( E jk ) · e γ · δ ij + e γ · (cid:16) X lk · δ li · E k γ + E jk · δ ij · X k γ (cid:17) , where in the last line we used δ lj X lk E jk = 0 (by (2.41)). We obtain (2.83b) after rearranging.To obtain (2.83c), we argue similarly. Combining (2.90) with E k ( E jk ) ∂ j = ∇ E k E k − E jk ∇ E k ∂ j and (2.80),and using also (2.7), (2.4) and (2.40), we obtain ( χ k − K ( E k , E k )) · X lk · e γ · δ il = g ( ∇ E k E k , ∂ i )= E k ( E jk ) · e γ · δ ij + E jk · E lk · e γ · ( δ li ∂ j γ + δ ij ∂ l γ − δ lj ∂ i γ )= E k ( E jk ) · e γ · δ ij + 2 e γ · E lk · δ li · E k γ − ∂ i γ, where in the last line we used δ lj E lk E jk = e − γ (by (2.7) and (2.41)). Step 3: Proof of (2.84) . This is an immediate consequence of (2.52).
Step 4: Proof of (2.85) and (2.86) . Finally, we get (2.85) and (2.86) from the formulas L tk = N − , L ik = − X ik − N − · β i , which in turn follow from (2.40) and (2.3). 20 .8 Transport equations for the frame coefficients We now derive transport equations for µ k and Θ k . Lemma 2.21.
The frame coefficients µ k and Θ k satisfy the following transport equations: L k log( µ k ) = K ( X k , X k ) − X k log( N ) , (2.91) L k (log( Θ k )) = χ k . (2.92) Proof. (2.91) has already been proven in (2.70). To obtain (2.92), it suffices to compare the expressions in(2.73) with (2.49).
We now derive transport equations for χ k and η k . These equations will involve the Ricci curvature, whichcan then be expressed in terms of derivatives of the scalar field using the Einstein equations (2.21). Lemma 2.22.
Given ( I × R , g ) in the gauge of Definition 2.5 and solving the Einstein equations (2.21) , itholds that for k = 1 , , , L k η k = − L k φ · E k φ − χ k · ( K ( E k , X k ) − E k log N + η k ) , (2.93) L k χ k = − L k φ ) − χ k + ( K ( X k , X k ) − X k log( N )) · χ k . (2.94) Proof. Step 1: Proof of (2.93) . By (2.64), L k η k = g ( ∇ X k L k , ∇ L k E k ) + g ( ∇ L k ( ∇ X k L k ) , E k ) = g ( ∇ L k ( ∇ X k L k ) , E k ) , where for the second equality we used g ( ∇ X k L k , ∇ L k E k ) = 0 coming directly from (2.78) and (2.72).Then, by the definition of the Riemann curvature tensor, we obtain g ( ∇ L k ( ∇ X k L k ) , E k ) = g ( ∇ X k ( ∇ L k L k ) , E k ) + R ( L k , X k , L k , E k ) + g ( ∇ [ L k ,X k ] L k , E k ) . (2.95)Using (2.82) and g ( E k , L k ) = 0 , we rewrite the first term in (2.95) as g ( ∇ X k ( ∇ L k L k ) , E k ) = ( K ( X k , X k ) − X k log( N )) · g ( ∇ X k L k , E k ) = ( K ( X k , X k ) − X k log( N )) · η k . For the second term in (2.95), we use (2.42) to deduce that − R ( L k , X k , L k , E k ) = − =0 (cid:122) (cid:125)(cid:124) (cid:123) R ( L k , L k , L k , E k ) − R ( L k , X k , L k , E k ) − =0 (cid:122) (cid:125)(cid:124) (cid:123) R ( L k , L k , X k , E k ) + =0 (cid:122) (cid:125)(cid:124) (cid:123) R ( L k , E k , E k , E k )= Ric ( L k , E k ) . The third term in (2.95) can be computed using (2.79) and also the definition (2.63) and (2.64) as g ( ∇ [ L k ,X k ] L k , E k ) = − χ k · ( K ( E k , X k ) − E k log N + η k ) + η k · ( X k log( N ) − K ( X k , X k )) . Combining the three terms and using (2.21) give (2.93).
Step 2: Proof of (2.94) . A similar computation as in Step 1, but using (2.63) instead, gives L k χ k = g ( ∇ E k ( ∇ L k L k ) , E k ) + R ( L k , E k , L k , E k ) + g ( ∇ [ L k ,E k ] L k , E k ) . (2.96)The first term in (2.96) can be written using (2.82) and (2.72) as g ( ∇ E k ( ∇ L k L k ) , E k ) = ( K ( X k , X k ) − X k log( N )) · χ k , and for the second term in (2.96), we get from (2.42) that R ( L k , E k , L k , E k ) = − R ( L k , E k , E k , L k ) − =0 (cid:122) (cid:125)(cid:124) (cid:123) R ( L k , X k , L k , L k ) − =0 (cid:122) (cid:125)(cid:124) (cid:123) R ( L k , L k , X k , L k ) − =0 (cid:122) (cid:125)(cid:124) (cid:123) R ( L k , L k , L k , L k )= − Ric ( L k , L k ) . Since g ( ∇ [ L k ,E k ] L k , E k ) = − χ k by (2.73), we get (2.94) using (2.21).21 .10 Initial values of eikonal quantities on Σ Lemma 2.23.
Let ( u k ) | Σ = a k + c kj x j and ( θ k ) | Σ = b k + c ⊥ kj x j , where a k , b k and c ki are as in (2.29) and (2.44) , c ki satisfies (2.30) – (2.31) , and c ⊥ k = − c k and c ⊥ k = c k .Then the following identities hold on Σ : (Ξ k ) | Σ = 0 , (2.97) ( µ k ) | Σ = e γ , (2.98) ( Θ k ) | Σ = e γ , (2.99) ( X ik ) | Σ = e − γ · δ iq · c kq , (2.100) ( E ik ) | Σ = e − γ · δ iq · c ⊥ kq . (2.101) The two Ricci coefficients χ k and η k are given initially by: ( χ k ) | Σ = e − γ · δ ii (cid:48) δ jj (cid:48) K i (cid:48) j (cid:48) c ⊥ ki c ⊥ kj − X k γ = e − γ · δ ii (cid:48) δ jj (cid:48) K i (cid:48) j (cid:48) c ⊥ ki c ⊥ kj − e − γ c kq δ iq ∂ i γ, (2.102) ( η k ) | Σ = e − γ · δ ii (cid:48) δ jj (cid:48) K i (cid:48) j (cid:48) c ki c ⊥ kj + E k γ = e − γ · δ ii (cid:48) δ jj (cid:48) K i (cid:48) j (cid:48) c ki c ⊥ kj + e − γ · c ⊥ kq · δ jq ∂ j γ. (2.103) Proof. Step 1: Proof of (2.97) – (2.101) . First, we notice that, for our choice of ( u k ) | Σ and ( θ k ) | Σ , ∂ u k and ∂ θ k are g -orthogonal. Since E k is proportional to ∂ θ k and g -orthogonal to X k , it means that X k isproportional to ∂ u k , hence (2.97).We expand (2.28) to find the following equation: ( g − ) (( ∂ t u k ) | Σ ) + 2( g − ) i ( ∂ i u k ) | Σ · ( ∂ t u k ) | Σ + ( g − ) ij ( ∂ i u k ) | Σ ( ∂ j u k ) | Σ = 0 . Solving the quadratic equation in ∂ t u k , and using also (2.8), we obtain that on Σ , ∂ t u k = β i ∂ i u k ± N · e − γ . Since e u k = ∂ t u k − β i ∂ i u k > and N > (by Definitions 2.8 and 2.2), we have ∂ t u k = β i ∂ i u k + N e − γ .Comparing this with (2.56), we obtain ∂ t u k = β i c ki + N e − γ = β i c ki + N µ − k , which gives (2.98).To prove (2.100), we combine ( ∂ i u k ) | Σ = c ki with (2.54) and (2.98). By (2.52), this also gives (2.101).Then we use (2.45) and (2.101), (2.55) to get E k θ k = Θ − k = e − γ · δ ij · c ⊥ ki · c ⊥ kj = e − γ , which is (2.99). Step 2: Proof of (2.102) – (2.103) . By (2.65), (2.69), (2.100) and (2.101), we find ( χ k ) | Σ = e − γ · δ ii (cid:48) δ jj (cid:48) K i (cid:48) j (cid:48) c ⊥ ki c ⊥ kj − g ( ∇ E k X k , E k ) , η k = e − γ · δ ii (cid:48) δ jj (cid:48) K i (cid:48) j (cid:48) c ki c ⊥ kj − g ( ∇ X k X k , E k ) . (2.104)Since g ( E k , X k ) = 0 , we use (2.100)–(2.101) to obtain that for any vector field Y , g ( ∇ Y X k , E k ) | Σ = e − γ δ iq c kq Y j E lk · g ( ∇ ∂ j ∂ i , ∂ l ) . By (2.18), we find the following formula: Y j E lk · g ( ∇ ∂ j ∂ i , ∂ l ) = e γ · Y j E lk · ( δ il ∂ j γ + δ jl ∂ i γ − δ ij ∂ l γ ) . (2.105)Using (2.105) for Y = E k and Y = X k , and applying (2.100) and (2.101), we then find that E jk E lk · g ( ∇ ∂ j ∂ i , ∂ l ) = ∂ i γ, X jk E lk · g ( ∇ ∂ j ∂ i , ∂ l ) = ( c ⊥ ki c kq − c ki c ⊥ kq ) · δ jq ∂ j γ, which ultimately implies, using the orthonormality of c : e − γ · δ iq c kq · E jk E lk · g ( ∇ ∂ j ∂ i , ∂ l ) = e − γ · c kq · δ jq ∂ j γ, e − γ δ iq c kq · X jk E lk · g ( ∇ ∂ j ∂ i , ∂ l ) = − e − γ · c ⊥ kq · δ jq ∂ j γ. (2.106)Combining (2.104) with (2.106) gives (2.102) and (2.103).22 Function spaces and norms
This section is devoted to the definition of all the function spaces and norms that are used throughout theremainder of the paper.
Definition 3.1.
Define the following pointwise norms in the coordinate system ( t, x , x ) associated to theelliptic gauge (see 2.1):1. Given a scalar function f , define | ∂ x f | := (cid:88) i =1 ( ∂ i f ) , | ∂f | := (cid:88) α =0 ( ∂ α f ) .
2. Given a higher order tensor field, define its norm and the norms of its derivatives componentwise, e.g. | β | := (cid:88) i =1 | β i | , | ∂ x β | := (cid:88) i,j =1 | ∂ i β j | , | K | := (cid:88) i,j =1 | K ij | , | ∂K | := (cid:88) α =0 2 (cid:88) i,j =1 | ∂ α K ij | etc.
3. Higher derivatives are defined analogously, e.g. | ∂ f | := (cid:88) α,σ =0 ( ∂ ασ f ) , | ∂∂ x K | := (cid:88) α =0 , , i,j,l =1 , | ∂ α ∂ i K jl | , etc. Σ t Unless otherwise stated, all Lebesgue spaces are defined with respect to the measure dx dx (which is ingeneral different from the volume form induced by ¯ g ).Before we define the norms, we define the following weight function. Definition 3.2 (Japanese brackets) . Define (cid:104) x (cid:105) := (cid:112) | x | for x ∈ R and (cid:104) s (cid:105) := √ s for s ∈ R . Definition 3.3 ( C k and Hölder norms) . For k ∈ N ∪ { } and s ∈ (0 , , define C k (Σ t ) to be the spaceof continuously k -differentiable functions with respect to elliptic gauge coordinate vector fields ∂ x with norm (cid:107) f (cid:107) C k (Σ t ) := (cid:80) | α |≤ k sup Σ t | ∂ αx f | , and define C k,s (Σ t ) ⊆ C k (Σ t ) with Hölder norm defined with respect tothe elliptic gauge coordinates as (cid:107) f (cid:107) C k,s (Σ t ) := (cid:107) f (cid:107) C k (Σ t ) + sup x,y ∈ Σ x (cid:54) = y (cid:80) | α | = k | ∂ αx f ( x ) − ∂ αx f ( y ) || x − y | s . Definition 3.4 (Standard Lebesgue and Sobolev norms) .
1. For k ∈ N ∪ { } and p ∈ [1 , + ∞ ) , definethe (unweighted) Sobolev norms (cid:107) f (cid:107) W k,p (Σ t ) := (cid:88) | α |≤ k (cid:18)(cid:90) Σ t | ∂ αx f | p ( t, x , x ) dx dx (cid:19) p . For k ∈ N ∪ { } , define (cid:107) f (cid:107) W k, ∞ (Σ t ) := (cid:88) | α |≤ k ess sup ( x ,x ) ∈ R | ∂ αx f | p ( t, x , x ) .
2. Define L p (Σ t ) := W ,p (Σ t ) and H k (Σ t ) := W k, (Σ t ) . Definition 3.5 (Fractional Sobolev norms) . For s ∈ R \ ( N ∪ { } ) , define H s (Σ t ) by (cid:107) f (cid:107) H s (Σ t ) := (cid:107)(cid:104) D x (cid:105) s f (cid:107) L (Σ t ) . where (cid:104) D x (cid:105) s is defined via the (spatial) Fourier transform F (in the x coordinates) by F ( (cid:104) D x (cid:105) s f ) := (cid:104) ξ (cid:105) s F . efinition 3.6 (Weighted norms) .
1. For k ∈ N ∪ { } , p ∈ [1 , + ∞ ) and r ∈ R , define the weightedSobolev norms by (cid:107) f (cid:107) W k,pr (Σ t ) = (cid:88) | α |≤ k (cid:18)(cid:90) Σ t (cid:104) x (cid:105) p · ( r + | α | ) | ∂ αx f | p ( t, x , x ) dx dx (cid:19) p , with obvious modifications for p = ∞ .2. Define also L pr (Σ t ) := W ,pr (Σ t ) and H kr (Σ t ) := W k, r (Σ t ) . Moreover, define C kr (Σ t ) as the closure ofSchwartz functions under the L ∞ r (Σ t ) norm. Definition 3.7 (Mixed norms) . We will use mixed Sobolev norms, mostly in the ( u k , θ k , t k ) coordinates inspacetime or the ( u k , u k (cid:48) ) coordinates on Σ t . Our convention is that the norm on the right is taken first.For instance, (cid:107) f (cid:107) L uk (cid:48) L ∞ uk (Σ t ) = ( (cid:90) u k (cid:48) ∈ R ( sup u k ∈ R f ( t, u k , u k (cid:48) )) du k (cid:48) ) , and analogously for other combinations. Definition 3.8 (Norms for derivatives) . We combine the notations in Definition 3.1 with those in Defini-tions 3.4–3.7. For instance, given a scalar function f , (cid:107) ∂f (cid:107) L (Σ t ) := ( (cid:90) R (cid:88) α =0 | ∂ α f | dx dx ) , and similarly for (cid:107) ∂ x f (cid:107) L (Σ t ) , (cid:107) ∂∂ x f (cid:107) L (Σ t ) , etc. ( u k , u k (cid:48) ) coordinates Assume for this subsection that k (cid:54) = k (cid:48) , so that ( u k , u k (cid:48) ) forms a coordinate system on Σ t . Definition 3.9 (Littlewood–Paley projection) . Define the Fourier transform in the ( u k , u k (cid:48) ) coordinates by ( F u k ,u k (cid:48) f )( ξ k , ξ k (cid:48) ) = (cid:90) (cid:90) R f ( u k , u k (cid:48) ) e − πi ( u k ξ k + u k (cid:48) ξ k (cid:48) ) du k du k (cid:48) . Let ϕ : R → [0 , be radial, smooth such that ϕ ( ξ ) = (cid:40) for | ξ | ≤ for | ξ | ≥ , where | ξ | = (cid:112) | ξ k | + | ξ k (cid:48) | .Define P u k ,u k (cid:48) by P u k ,u k (cid:48) f := ( F u k ,u k (cid:48) ) − ( ϕ ( ξ ) F u k ,u k (cid:48) f ) , and for q ≥ , define P u k ,u k (cid:48) q f by P u k ,u k (cid:48) q f := ( F u k ,u k (cid:48) ) − (( ϕ (2 − q ξ ) − ( ϕ (2 − q +1 ξ )) F u k ,u k (cid:48) f ( ξ )) . Definition 3.10 (The Besov space B u k ,u k (cid:48) ∞ , ) . Define the Besov norm B u k ,u k (cid:48) ∞ , (Σ t ) by (cid:107) f (cid:107) B uk,uk (cid:48)∞ , (Σ t ) := (cid:88) q ≥ (cid:107) P u k ,u k (cid:48) q f (cid:107) L ∞ (Σ t ) . C ku k and Σ t ∩ C ku k Recall the definition of C ku k from Definition 2.9. The L norm on C ku k is defined with respect to the measure dθ k dt k . Definition 3.11 ( L norm on C ku k ) . For every fixed u k , define the L ( C ku k ([0 , T ))) norm by (cid:107) f (cid:107) L ( C kuk ([0 ,T ))) := ( (cid:90) T (cid:90) R | f | ( u k , θ k , t k ) dθ k dt k ) . L norm Σ t ∩ C ku k is defined with respect to the measure dθ k . Definition 3.12 ( L norm on Σ t ∩ C ku k ) . For every fixed t and u k (and recall t = t k ), define the L θ k (Σ t ∩ C ku k ) norm by (cid:107) f (cid:107) L θk (Σ t ∩ C kuk ) := ( (cid:90) R | f | ( u k , θ k , t k ) dθ k ) . Σ In this section, we give the precise assumptions on the initial data for our theorem. Recall from Section 1.1.1that we will consider both impulsive wave data and δ -impulsive wave data, which are approximation ofimpulsive wave data on a length scale δ > .In Section 4.1 , we first recall the notion of initial data in [35], which in particular involves the constraintequations. The precise assumptions on the impulsive wave data and the δ -impulsive data will be stated in Section 4.2 and
Section 4.3 respectively.
Before we proceed, we need to fix a cutoff function for the rest of the paper:
Definition 4.1 (Cutoff function ω ) . From now on, fix a smooth cutoff function ω : R → [0 , such that ω ( τ ) ≡ for τ ≤ and ω ( τ ) ≡ for τ ≥ We are now ready to define the notion of an admissible initial data set (c.f. [35]).
Definition 4.2 (Admissible initial data) . An admissible initial data set with respect to the elliptic gaugefor the system (2.21) , (2.22) is a quadruple ( φ, φ (cid:48) , γ, K ) , where1. ( φ, φ (cid:48) ) ∈ W , ( R ) × L ( R ) (where φ (cid:48) is the prescribed initial value for (cid:126)nφ ) are a pair of real-valuedcompactly supported functions,2. γ is a real-valued function with the decomposition γ = − γ asymp · ω ( | x | ) · log( | x | ) + (cid:101) γ, where γ asymp ≥ is a constant, ω ( | x | ) is as in Definition 4.1, and (cid:101) γ ∈ H − ( R ) , and3. K ij ∈ H ( R ) is a symmetric traceless (with respect to δ ij ) -tensor,which satisfy1. the constraint equations (2.23) and ∆ γ = − δ il ( ∂ i φ )( ∂ l φ ) − e − γ | K | − e γ · ( (cid:126)nφ ) , (4.1) and2. the integral compatibility condition (cid:90) Σ e γ φ (cid:48) · ∂ j φ = 0 . (4.2) In this subsection, we define the precise notion of impulsive wave data. That such initial data sets existrequire solving the constraint equations; this will be carried out in Appendix A.3; see Lemma A.4.In the statement of the following theorem, u k , X k and E k are to be understood as their values on Σ ,according to (2.29), (2.100), and (2.101). Note that (4.1) is simply a restatement of (2.25), but in terms of K (instead of β ). efinition 4.3. Let (cid:15) > , < s (cid:48)(cid:48) < s (cid:48) < with s (cid:48) − s (cid:48)(cid:48) < , R ≥ and κ > . We say that ( φ, φ (cid:48) , γ, K ) is an admissible initial data set featuring three impulsive waves with parameters ( (cid:15), s (cid:48) , s (cid:48)(cid:48) , R, κ ) if the following conditions are satisfied:1. ( φ, φ (cid:48) , γ, K ) is an admissible initial data set according to Definition 4.2.2. The transversality condition (2.31) hold with the parameter κ .3. We have the decomposition φ = φ reg + (cid:80) k =1 (cid:101) φ k and φ (cid:48) = φ (cid:48) reg + (cid:80) k =1 (cid:101) φ (cid:48) k on Σ , where for every k =1 , , , supp( φ reg ) , supp( φ (cid:48) reg ) , supp( φ k ) , supp( φ (cid:48) k ) ⊆ B (0 , R ) := { ( x , x ) ∈ Σ , (cid:112) ( x ) + ( x ) < R } . Moreover, for every k = 1 , , , supp( (cid:101) φ k ) ∪ supp( (cid:101) φ (cid:48) k ) ⊆ { u k ≥ } .4. φ reg and φ (cid:48) reg satisfy the following estimates: (cid:107) φ reg (cid:107) H s (cid:48) (Σ ) + (cid:107) φ (cid:48) reg (cid:107) H s (cid:48) (Σ ) ≤ (cid:15). (4.3)
5. For k = 1 , , , (cid:101) φ k and (cid:101) φ (cid:48) k satisfy the following estimates: (cid:107) (cid:101) φ k (cid:107) W , ∞ (Σ ) + (cid:107) (cid:101) φ k (cid:107) H s (cid:48) (Σ ) + (cid:107) (cid:101) φ (cid:48) k (cid:107) L ∞ (Σ ) + (cid:107) (cid:101) φ (cid:48) k (cid:107) H s (cid:48) (Σ ) ≤ (cid:15), (4.4a) (cid:107) E k (cid:101) φ k (cid:107) H s (cid:48)(cid:48) (Σ ) + (cid:107) E k (cid:101) φ (cid:48) k (cid:107) H s (cid:48)(cid:48) (Σ ) + (cid:107) (cid:101) φ (cid:48) k − X k (cid:101) φ k (cid:107) H s (cid:48)(cid:48) (Σ ) ≤ (cid:15). (4.4b)
6. For k = 1 , , , there exist signed Radon measures T ij,k , T (cid:48) i,k , T ijE,k , T (cid:48) iE,k and T ijL,k such that (cid:107) ∂ ij (cid:101) φ k − T ij,k (cid:107) L (Σ ) + (cid:107) ∂ i (cid:101) φ (cid:48) k − T (cid:48) i,k (cid:107) L (Σ ) + (cid:107) ∂ ij E k (cid:101) φ k − T ijE,k (cid:107) L (Σ ) + (cid:107) ∂ i E k (cid:101) φ (cid:48) k − T (cid:48) iE,k (cid:107) L (Σ ) + (cid:107) ∂ ij ( (cid:101) φ (cid:48) k − X k (cid:101) φ k ) − T ijL,k (cid:107) L (Σ ) ≤ (cid:15), (4.5) supp( T ij,k ) ∪ supp( T (cid:48) i,k ) ∪ supp( T ijE,k ) ∪ supp( T (cid:48) iE,k ) ∪ supp( T ijL,k ) ⊆ { u k = 0 } , (4.6) and T.V. ( T ij,k ) + T.V. ( T (cid:48) i,k ) + T.V. ( T ijE,k ) + T.V. ( T (cid:48) iE,k ) + T.V. ( T ijL,k ) ≤ (cid:15), (4.7) where T.V. is the total variation norm of Radon measures.7. The following lower bound holds: (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ φ − (cid:104) ∂ φ, ∂ φ (cid:105) L (Σ ,dx ) (cid:107) ∂ φ (cid:107) L (Σ ) ∂ φ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H − (Σ ) × (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ φ − (cid:104) ∂ φ, ∂ φ (cid:105) L (Σ ,dx ) (cid:107) ∂ φ (cid:107) L (Σ ) ∂ φ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H − (Σ ) ≥ (cid:15) . (4.8)The following remarks clarify Definition 4.3. Remark 4.4.
It may be helpful to rephrase the main points of our assumptions in words:1. φ reg and φ (cid:48) reg are the regular parts of φ and φ (cid:48) .2. (cid:101) φ k and (cid:101) φ (cid:48) k are singular. In particular, the second derivatives of (cid:101) φ k and first derivatives of (cid:101) φ (cid:48) k are Radonmeasures with singular parts supported on { u k = 0 } .3. However, E k (cid:101) φ k , E k (cid:101) φ (cid:48) k and L k (cid:101) φ k are better behaved. (Note that (cid:101) φ (cid:48) k − X k (cid:101) φ k corresponds to L k (cid:101) φ k .)Notice in particular that all these bounds are consistent with X k (cid:101) φ k having a jump discontinuity of ampli-tude O ( (cid:15) ) across { u k = 0 } . Remark 4.5.
In Definition 4.3, we assumed that supp( (cid:101) φ k ) ∪ supp( (cid:101) φ (cid:48) k ) ⊆ { u k ≥ } . This is not a severerestriction: we can remove this condition as long as we assume instead that ( (cid:101) φ k , (cid:101) φ (cid:48) k ) |{ u k < } can be extendedto a pair of functions with H s (cid:48) (Σ ) × H s (cid:48) (Σ ) norms of O ( (cid:15) ) . In this case, it is easy to redefine thedecomposition so that the new decomposition obeys the assumptions in Definition 4.3 (including those for thesupport properties), after allowing (cid:15) and R to increase by a constant multiplicative factor. emark 4.6. As part of the proof, we will show that the three singularities propagate along u k = 0 . As aresult, the three impulsive waves interact at the spacetime point characterized by u = u = u = 0 . As wewill see, defining u k = t + a k + c kj x j , we have u k = u k + O ( (cid:15) ) on the support of φ . Therefore, while ourtheorem applies to any choice of a k ’s and c kj ’s, in the particular case where u = u = u = 0 corresponds toa spacetime point in [0 , × R within ∪ k =1 supp( (cid:101) φ k ) , the theorem indeed features the interaction of threeimpulsive waves within the time interval t ∈ [0 , . Remark 4.7.
The condition (4.8) can be thought of as a non-degeneracy assumption, which is used only tosolve the constraints to obtain data for δ -impulsive waves; see the proof of Lemma 6.1. Notice that while ∂ φ and ∂ φ are O ( (cid:15) ) in L (Σ ) , we only need a weaker lower bound of order (cid:15) (as opposed to (cid:15) ). In particular,it can be checked that (4.8) can always be guaranteed after adding, say, an O ( (cid:15) ) smooth perturbation.We remark also that for any non-zero compactly supported H (Σ ) function φ , LHS of (4.8) is non-zero;see Lemma A.2. δ -impulsive waves data In this section we present a choice of smooth data, which are not strictly speaking impulsive, but whichobey scaled estimates consistent with the data being a smooth approximation of the data of Definition 4.3.Note (c.f. introduction) that such data are less idealized, perhaps more realistic representations of impulsivegravitational waves. Most of the paper concerns the propagation of low regularity norms for such smoothdata, a result from which we ultimately obtain local existence for the rough data of section 4.2.
Definition 4.8.
Let (cid:15) > , < s (cid:48)(cid:48) < s (cid:48) < with s (cid:48) − s (cid:48)(cid:48) < , R ≥ , κ > and δ > . We say that ( φ, φ (cid:48) ) ∈ C ∞ (Σ ) × C ∞ (Σ ) is admissible initial data set featuring three δ -impulsive waves withparameters ( (cid:15), s (cid:48) , s (cid:48)(cid:48) , R, κ ) if the following holds: • The conditions 1 and 2 of Definition 4.3 are satisfied. • φ and φ (cid:48) admit decompositions as in 3 of Definition 4.3, and φ reg , φ (cid:48) reg , φ k , φ (cid:48) k are supported in B (0 , R ) for k = 1 , , . Unlike in Definition 4.3, however, for each k = 1 , , , supp( (cid:101) φ k ) ∪ supp( (cid:101) φ (cid:48) k ) ⊆{ u k ≥ − δ } . • The estimates in conditions 4 and 5 of Definition 4.3 are satisfied. • For k = 1 , , , ( (cid:101) φ k , (cid:101) φ (cid:48) k ) satisfy the following additional bounds (recall definitions from (2.34) , (2.35) ): (cid:107) (cid:101) φ k (cid:107) H (Σ ) + (cid:107) (cid:101) φ (cid:48) k (cid:107) H (Σ ) + (cid:107) E k (cid:101) φ k (cid:107) H (Σ ) + (cid:107) E k (cid:101) φ (cid:48) k (cid:107) H (Σ ) + (cid:107) (cid:101) φ (cid:48) k − X k (cid:101) φ k (cid:107) H (Σ t ) ≤ (cid:15) · δ − , (4.9) (cid:107) (cid:101) φ k (cid:107) H (Σ \ S k ( − δ, + (cid:107) (cid:101) φ (cid:48) k (cid:107) H (Σ \ S k ( − δ, ≤ (cid:15). (4.10) Remark 4.9.
The conditions (4.9) and (4.10) can be viewed as a smoothed-out version of (4.5) – (4.7) . Here,the second derivatives of (cid:101) φ k (and first derivatives of (cid:101) φ (cid:48) k , etc.) can be thought of as a Radon measure smoothedout at a length scale δ .The δ -impulsive waves of Definition 4.8 can indeed by constructed by smoothing out the impulsive wavesof Definition 4.3; see Lemma 6.1. Remark 4.10.
Notice that the condition 7 in Definition 4.3 is only needed for solving the constraint equationsin the approximation argument. In particular, the class of data we can handle for interaction of δ -impulsivewaves (as defined in Definition 4.8) do not require such a condition. Notice that the corresponding spacetime point ( t, x , x ) can be given explicitly by tx x = − c c c c c c − a a a whenever inverse matrix above is well-defined. Precise statement of the main theorems
In this section, we present the precise version of the main results.In parallel with the definitions in Section 4, we give two versions of the main theorem. The first ver-sion (
Theorem 5.2 ) concerns existence of impulsive waves (see Definition 4.3), while the second version(
Theorem 5.6 ) concerns existence and uniqueness of δ -impulsive waves (see Definition 4.8). We recallagain that (see Section Section 1.1.1) our proof of existence for impulsive waves relies on first understanding δ -impulsive waves and taking limits. We first begin with a notion of weak solutions. Definition 5.1.
Let γ, β , β , N, φ be functions on [0 , T ) × R , where N is everywhere non-vanishing.1. We say that ( γ, β j , N, φ ) is a weak solution to the Einstein vacuum equations in polarized U (1) symmetry under elliptic gauge if(a) The following regularity conditions hold: γ, β, N ∈ ( C loc ) t,x , ∂ i γ, ∂ i β j , ∂ i N ∈ ( C loc ) t,x , ∂ t γ ∈ ( C loc ) t,x , ∂ t β j , ∂ t N ∈ ( L ∞ loc ) t,x ,φ ∈ ( C loc ) t ( H loc ) x , ∂ t φ ∈ ( C loc ) t ( L loc ) x . (b) The following maximality condition holds pointwise: − e γ + ∂ i β i = 0 . (5.1) (c) The elliptic equations (2.23) – (2.25) hold weakly in the sense that for every t ∈ [0 , T ) , and every ς ∈ C ∞ c ( R ) , we have : − (cid:90) { t }× R δ ik K ij ∂ k ς dx = (cid:90) { t }× R ( RHS of (2.23) ) × ς dx. (5.2) − (cid:90) { t }× R δ ik ∂ i N ∂ k ς dx = (cid:90) { t }× R ( RHS of (2.24) ) × ς dx, (5.3) − (cid:90) { t }× R δ ik ∂ i γ∂ k ς dx = (cid:90) { t }× R ( RHS of (2.25) ) × ς dx, (5.4) (d) The evolution equation (2.27) for K ij (where K ij is defined in terms of γ , β j , N by (2.14) )holds weakly in the sense that for every ς ∈ C ∞ c ((0 , T ) × R ) : (cid:90) (0 ,T ) × R ( − K ij ( e ς ) + K ij ( ∂ l β l ) ς + 12 ∂ i N ∂ j ς + 12 ∂ j N ∂ i ς − δ ij δ lk ∂ l N ∂ k ς ) dx = (cid:90) (0 ,T ) × R N · ( RHS of (2.27) ) × ς dx. (5.5) (e) The wave equation (2.22) holds weakly in the sense that for every ς ∈ C ∞ c ((0 , T ) × R ) , (cid:90) (0 ,T ) × R ( g − ) ασ ∂ α ς∂ σ φ N e γ dx dx dt = 0 . (5.6) It should be remarked that there is a geometric notion of weak solutions which requires only the metric to be continuousand the Christoffel symbols to be L loc ; see for instance [52, Definition 2.1]. In particular, this notion does not require anysymmetry and gauge assumptions. We will however take advantage of the symmetry and gauge conditions in our definition.Note that weak solutions in the sense of Definition 5.1 are automatically weak solutions in the sense of [52, Definition 2.1]. We remark that these conditions correspond respectively to
Ric ( (cid:126)n, ∂ i ) = 2( (cid:126)nφ )( ∂ i φ ) , Ric ( (cid:126)n, (cid:126)n ) = 2( (cid:126)nφ ) and δ ij Ric ( ∂ i , ∂ j )+ e γ Ric ( (cid:126)n, (cid:126)n ) = 2 δ ij ( ∂ i φ )( ∂ j φ ) + 2 e γ ( (cid:126)nφ ) . We remark that this corresponds to
Ric ( ∂ i , ∂ j ) − δ ij δ kl Ric ( ∂ k , ∂ l ) = 2( ∂ i φ )( ∂ j φ ) − δ ij δ kl ( ∂ k φ )( ∂ l φ ) . . Given a weak solution ( γ, β j , N, φ ) to the Einstein vacuum equations in polarized U (1) symmetry un-der elliptic gauge (as defined in part 1), we moreover say that the solution achieves initial data ( γ , K , φ , φ (cid:48) ) if(a) γ , K and φ converges to the prescribed initial value pointwise, i.e. lim t → ( γ, K, φ )( t, x ) = ( γ, K, φ )(0 , x ) . (b) The initial data for (cid:126)nφ is achieved in an L sense, i.e. lim t → (cid:107) (cid:126)nφ ( t, · ) − φ (cid:48) ( · ) (cid:107) L ( R ) = 0 . (5.7)We now state our main result for the interaction of three impulsive waves (recall the definition for thedata in Definition 4.3). Theorem 5.2.
For every < s (cid:48)(cid:48) < s (cid:48) < with s (cid:48) − s (cid:48)(cid:48) < , R ≥ and κ > , there exists (cid:15) = (cid:15) ( s (cid:48) , s (cid:48)(cid:48) , R, κ ) > such that the following holds.Let ( φ , φ (cid:48) , γ , K ) be an admissible initial data set featuring three impulsive waves with parameters ( (cid:15), s (cid:48) , s (cid:48)(cid:48) , R, κ ) as in Definition 4.3.Then, whenever (cid:15) ∈ (0 , (cid:15) ] , there exists a Lorentzian metric gg = − N dt + e γ δ ij ( dx i + β i dt )( dx j + β j dt ) on the manifold M := [0 , × R and a scalar function φ : M → R such that ( γ, β , β , N, φ ) is a weaksolution to the Einstein vacuum equation in polarized U (1) symmetry under elliptic gauge, with the initialdata ( φ , φ (cid:48) , γ , K ) (see Definition 5.1).Moreover, φ = φ reg + (cid:80) k =1 (cid:101) φ k , where each of φ reg and (cid:101) φ k is defined to satisfy the wave equation weaklyin the sense of (5.6) , with initial data as given in Definition 4.3 (understood as in part 2 of Definition 5.1).Furthermore, each of φ reg and (cid:101) φ k is supported in B (0 , R ) for every t ∈ [0 , .Additionally, the following estimates are satisfied for all k = 1 , , and all t ∈ [0 , , for some implicitconstants depending only on s (cid:48) , s (cid:48)(cid:48) , R and κ , and for α = 10 − :1. The following L estimates for φ hold: (cid:107) φ reg (cid:107) H s (cid:48) (Σ t ) + (cid:107) ∂ t φ reg (cid:107) H s (cid:48) (Σ t ) (cid:46) (cid:15), (5.8a) (cid:107) (cid:101) φ k (cid:107) H s (cid:48) (Σ t ) + (cid:107) ∂ t (cid:101) φ k (cid:107) H s (cid:48) (Σ t ) (cid:46) (cid:15), (5.8b) (cid:107) L k (cid:101) φ k (cid:107) H s (cid:48) (Σ t ) + (cid:107) E k (cid:101) φ k (cid:107) H s (cid:48) (Σ t ) + (cid:107) ∂ t L k (cid:101) φ k (cid:107) H s (cid:48) (Σ t ) + (cid:107) ∂ t E k (cid:101) φ k (cid:107) H s (cid:48) (Σ t ) (cid:46) (cid:15). (5.8c)
2. There exist signed Radon measures T µν,k ∈ M (Σ t ) for all t ∈ [0 , such that (cid:107) ∂ µν (cid:101) φ k − T µν,k (cid:107) L (Σ t ) (cid:46) (cid:15), (5.9) Moreover, the following holds for T µν,k : supp( T µν,k ) ⊆ { u k = 0 } , T.V. | Σ t ( T µν,k ) (cid:46) (cid:15). (5.10)
3. The following Lipschitz and improved Hölder estimates hold for φ : (cid:107) ∂φ (cid:107) L ∞ (Σ t ) (cid:46) (cid:15), (cid:107) ∂φ reg (cid:107) C , s (cid:48)(cid:48) (Σ t ) (cid:46) (cid:15). (5.11) Moreover, the following improved Hölder estimates in each half space away from { ( t, x ) : u k = 0 } : (cid:107) ∂ (cid:101) φ k (cid:107) C , s (cid:48)(cid:48) (Σ t ∩{ u k > } ) + (cid:107) ∂ (cid:101) φ k (cid:107) C , s (cid:48)(cid:48) (Σ t ∩{ u k < } ) (cid:46) (cid:15). (5.12) Note that it then follows easily that ∂ (cid:101) φ k admits an extension from { u k > } to { u k ≥ } , which is moreover Höldercontinuous on the closed subset { u k ≥ } . Similarly, there is a Hölder continuous extension from { u k < } to { u k ≤ } . Thesetwo extensions are in general different on { u k = 0 } . . The wavefronts u k of the waves (cid:101) φ k are ( C , loc ) t,x with the following estimates: (cid:107) ∂ i u k (cid:107) W , ∞ (Σ t ) (cid:46) , (5.13) and the components of the vector fields ( L k , E k , X k ) adapted to the wavefronts are ( C , loc ) t,x and satisfy | L βk | + | X ik | + | E ik | (cid:46) (cid:104) x (cid:105) (cid:15) , | ∂ t L tk | + (cid:104) x (cid:105) ( | ∂ t L ik | + | ∂ i L βk | + | ∂X ik | + | ∂E ik | ) (cid:46) (cid:15) · (cid:104) x (cid:105) α . (5.14)
5. Finally, the metric components γ and N admit a decomposition for every ( t, x ) ∈ [0 , × R γ ( t, x ) = − γ asymp ω ( | x | ) log | x | + (cid:101) γ ( t, x ) ,N ( t, x ) = 1 + N asymp ( t ) ω ( | x | ) log | x | + (cid:101) N ( t, x ) , (5.15) where γ asymp ≥ is a constant, N asymp ( t ) ≥ is a Lipschitz function of t , and ω is the cutoff functionin Definition 4.1. Moreover, γ , β i and N satisfy the following estimates for all t ∈ [0 , : | γ asymp | + | N asymp | ( t ) (cid:46) (cid:15) , (cid:88) (cid:101) g ∈{ (cid:101) γ,β i , (cid:101) N } ( (cid:107) (cid:101) g (cid:107) W , ∞ − α (Σ t ) + (cid:107) ∂ x (cid:101) g (cid:107) L ∞ − α (Σ t ) ) (cid:46) (cid:15) , (5.16) and the following estimates hold for a.e. t ∈ [0 , : | ∂ t N asymp | ( t ) + (cid:88) (cid:101) g ∈{ (cid:101) γ,β i , (cid:101) N } (cid:107) ∂ t (cid:101) g (cid:107) W , s (cid:48)− s (cid:48)(cid:48) − s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) (cid:46) (cid:15) . (5.17)The proof of Theorem 5.2 can be found in Section 6. We will show there that Theorem 5.2 follows fromthe theorem on δ -impulsive gravitational waves (Theorem 5.6 below), after a suitable approximation andlimiting argument.We give a few technical remarks regarding the estimates. Remark 5.3.
The proof of the theorem gives a few other estimates, which are not stated explicitly inTheorem 5.2. For instance, we have additional bounds for the commuting vector fields L k , E k , as well as forthe metric components. Remark 5.4.
Notice that while we assume ∂ ij E k (cid:101) φ k , ∂ i (cid:126)nE k (cid:101) φ k and ∂ ij L k (cid:101) φ k to be Radon measures initially,the theorem does not guarantee that this is propagated. We only propagate (see point 2 in Theorem 5.2) that ∂ ij (cid:101) φ k is a Radon measure. Remark 5.5.
We can impose, in addition to (4.4a) – (4.4b) , the stronger assumption that for all s ∈ (0 , ) , (cid:107) (cid:101) φ k (cid:107) H s (Σ ) + (cid:107) (cid:101) φ (cid:48) k (cid:107) H s (Σ ) + (cid:107) E k (cid:101) φ k (cid:107) H s (Σ ) + (cid:107) E k (cid:101) φ (cid:48) k (cid:107) H s (Σ ) + (cid:107) (cid:101) φ (cid:48) k − X k (cid:101) φ k (cid:107) H s (Σ ) ≤ (cid:15) √ − s . (Note that this is still consistent with X k (cid:101) φ k having a jump discontinuity.) In this case, one can in principlealso show a posteriori that the stronger estimate is propagated. Moreover, using Theorem 7.3, one also havethat ∂ (cid:101) φ k ∈ ∩ θ ∈ [0 , ) C ,θ in the sets { u k > } and { u k < } (as opposed to only being in C , s (cid:48)(cid:48) ). δ -impulsive impulsive waves Now we present our result for smooth , quantitatively impulsive data as in Definition 4.8. It is, in fact, thefollowing theorem that we prove in most of the paper, and we use this theorem to obtain Theorem 5.2eventually. 30 heorem 5.6.
For every < s (cid:48)(cid:48) < s (cid:48) < with s (cid:48) − s (cid:48)(cid:48) < , R ≥ and κ > , there exists (cid:15) = (cid:15) ( s (cid:48) , s (cid:48)(cid:48) , R, κ ) > such that the following holds.Let ( φ , φ (cid:48) , γ , K ) be an admissible initial data set featuring three δ -impulsive waves with parameters ( (cid:15), s (cid:48) , s (cid:48)(cid:48) , R, κ ) as in Definition 4.8 for some (cid:15) > and δ > .Then, whenever (cid:15) ∈ (0 , (cid:15) ] , there exists δ = δ ( (cid:15), s (cid:48) , s (cid:48)(cid:48) , R, κ ) > such that for all < δ < δ , thereexists a unique smooth Lorentzian metric gg = − N dt + e γ δ ij ( dx i + β i dt )( dx j + β j dt ) on the manifold M := [0 , × R and a unique smooth scalar function φ : M → R such that ( g, φ ) satisfy theEinstein vacuum equations in polarized U (1) symmetry under elliptic gauge (2.22) , (2.23) – (2.25) and (2.27) all hold, with initial data ( φ , φ (cid:48) , γ , K ) , in the classical sense.Moreover, φ = φ reg + (cid:80) k =1 (cid:101) φ k , where φ reg and (cid:101) φ k are defined to satisfy (cid:3) g φ reg = 0 , (cid:3) g (cid:101) φ k = 0 , (5.18) with initial data as prescribed by the corresponding decomposition in Definition 4.8. Furthermore, each of φ reg and (cid:101) φ k is supported in B (0 , R ) for every t ∈ [0 , .Additionally, the following estimates hold for all k = 1 , , and all t ∈ [0 , , for some implicit constantsdepending only on s (cid:48) , s (cid:48)(cid:48) , R and κ :1. The wave estimates in (5.8a) – (5.8c) , the wavefront estimate (5.13) , the vector fields estimates (5.14) ,and the metric estimates (5.15) – (5.17) all hold.2. The following higher-order estimates hold: (cid:107) ∂ (cid:101) φ k (cid:107) L (Σ t ) + (cid:88) Y (1) k ,Y (2) k ,Y (3) k ∈{ X k ,E k ,L k }∃ i,Y ( i ) k (cid:54) = X k (cid:107) Y (1) k Y (2) k Y (3) k (cid:101) φ k (cid:107) L (Σ t ) (cid:46) (cid:15) · δ − , (5.19) (cid:107) φ (cid:107) H (Σ t ) + (cid:107) (cid:126)nφ (cid:107) H (Σ t ) (cid:46) (cid:15) · δ − + (cid:107) φ (cid:107) H (Σ ) + (cid:107) (cid:126)nφ (cid:107) H (Σ ) , (5.20) together with the following improvement away from S kδ (recall S kδ = S k ( − δ, δ ) in Definition 2.9): (cid:107) ∂ (cid:101) φ k (cid:107) L (Σ t \ S kδ ) (cid:46) (cid:15). (5.21)
3. The Lipschitz and improved Hölder estimates (5.11) hold. Moreover, (cid:107) ∂ (cid:101) φ k (cid:107) C s (cid:48)(cid:48) (Σ t \ S kδ ) (cid:46) (cid:15). (5.22)The high-level proof of Theorem 5.6 can be found in Section 7.3. (It relies in particular on the a prioriestimates stated in Section 7.2, whose proof will occupy most of the remainder of the series.) In this section, we assume the validity of Theorem 5.6 and prove Theorem 5.2. We first approximate theimpulsive wave data in Theorem 5.2 by δ -impulsive waves data, then use Theorem 5.6 to obtain solutionsfor δ -impulsive waves, and finally pass to the δ → limit. For the remainder of this section, we assume the validity of Theorem 5.6. .1 Approximating the initial data Our first step is to show that data in Definition 4.3 can be approximated by data in Definition 4.8. This isgiven by the following lemma, whose proof will be postponed to Section A.4. Notice that both the size andthe support are allowed to be slightly larger for the approximate data.
Lemma 6.1.
For every < s (cid:48)(cid:48) < s (cid:48) < , R ≥ and κ > , there exists (cid:15) = (cid:15) ( s (cid:48) , s (cid:48)(cid:48) , R, κ ) ∈ (0 , suchthat the following holds for all (cid:15) ∈ (0 , (cid:15) ] .Let ( φ, φ (cid:48) , γ, K ) be an initial data set featuring three impulsive waves with parameters ( (cid:15), s (cid:48) , s (cid:48)(cid:48) , R, κ ) asin Definition 4.3 . Then, for every δ ∈ (0 , (cid:15) s (cid:48) ] , there exists an initial data set ( φ ( δ ) , ( φ (cid:48) ) ( δ ) , γ ( δ ) , K ( δ ) ) suchthat1. ( φ ( δ ) , ( φ (cid:48) ) ( δ ) , γ ( δ ) , K ( δ ) ) correspond to data for three δ -impulsive waves with parameters (3 (cid:15), s (cid:48) , s (cid:48)(cid:48) , R, κ ) in Definition 4.8, and2. ( φ ( δ ) , ( φ (cid:48) ) ( δ ) , γ ( δ ) , K ( δ ) ) is an approximation of ( φ, φ (cid:48) , γ, K ) in the sense that as δ → , φ ( δ ) reg → φ reg , (cid:101) φ ( δ ) k → (cid:101) φ k in H s (cid:48) (Σ ) , ( φ (cid:48) reg ) ( δ ) → φ (cid:48) reg , ( (cid:101) φ (cid:48) k ) ( δ ) → (cid:101) φ (cid:48) k in H s (cid:48) (Σ ) , and, after writing γ ( δ ) = − γ ( δ ) asymp ω ( | x | ) log | x | + (cid:101) γ ( δ ) and γ = − γ asymp ω ( | x | ) log | x | + (cid:101) γ , it holds that ( γ ( δ ) asymp , (cid:101) γ ( δ ) , K ( δ ) ) → ( γ asymp , (cid:101) γ, K ) in R × H − (Σ ) × H (Σ ) . Since in what follows we will need to consider the data and the solution for both the limit and theapproximations, let us introduce the following conventions for the remainder of the section: (1) we willuse subscripts to denote data quantities, and quantities without the subscripts correspondsto those in the solution , and (2) a superscript ( δ ) denotes quantities from the approximating δ -impulsive waves, while a superscript (0) denotes quantities from the limit impulsive waves(for both the data and the solution). Suppose now we are given data ( φ , φ (cid:48) , γ , K ) as in Definition 4.3 with parameters (cid:15) , s (cid:48) , s (cid:48)(cid:48) , R , κ . Takealso α = 10 − . Assuming (cid:15) ∈ (0 , (cid:15) ] for a sufficiently small (cid:15) , we apply Lemma 6.1 to obtain a -parameterfamily of δ -impulsive wave data ( φ ( δ )0 , ( φ (cid:48) ) ( δ )0 , γ ( δ )0 , K ( δ )0 ) .By Theorem 5.6 (local existence for δ -impulsive waves), there exists δ > such that for all δ ∈ (0 , δ ] ,the initial data set ( φ ( δ )0 , ( φ (cid:48) ) ( δ )0 , γ ( δ )0 , K ( δ )0 ) gives rise to a unique solution in [0 , × R , which we denote as ( γ ( δ ) , ( β i ) ( δ ) , N ( δ ) , φ ( δ ) ) . For the remainder of the section, consider such one-parameter family of δ -impulsive wave solutions ( γ ( δ ) , ( β i ) ( δ ) , N ( δ ) , φ ( δ ) ) . In order to prove Theorem 5.2, our goal will be to show that there exists a sequence δ j → such that1. one can take appropriate limits of ( γ ( δ j ) , ( β i ) ( δ j ) , N ( δ j ) , φ ( δ j ) ) as j → ∞ ( Section 6.2 ),2. and the limit is a weak solution which satisfies all the bounds stated in Theorem 5.2 (
Section 6.3 ).See
Section 6.4 for the conclusion of the proof of Theorem 5.2.
Let ( γ ( δ ) , ( β i ) ( δ ) , N ( δ ) , φ ( δ ) ) to be as in the end of Section 6.1.The goal of this subsection is to extract a suitable limit. We will combine the bounds for the δ -impulsivewaves with various compactness results and the following standard Aubin–Lions lemma: Lemma 6.2 (Aubin–Lions lemma) . Let X ⊆ X ⊆ X be three Banach spaces such that the embedding X ⊆ X is compact and the embedding X ⊆ X is continuous. For T > and q > , let W := { v ∈ L ∞ ([0 , T ]; X ) : ˙ v ∈ L q ([0 , T ]; X ) } , where ˙ denotes the (weak) derivative in the variable on [0 , T ] .Then W embeds compactly into C ([0 , T ]; X ) . We now begin extracting a limit of ( γ ( δ ) , ( β i ) ( δ ) , N ( δ ) , φ ( δ ) ) . For the remainder of the subsection, we willrepeatedly extract subsequences of δ , which will always be denoted by δ j without relabelling.32 roposition 6.3 (Limiting metric) . There exists a sequence δ j → and γ (0) = − γ (0) asymp ω ( | x | ) log | x | + (cid:101) γ (0) , ( β i ) (0) , and N (0) = 1+ N (0) asymp ( t ) ω ( | x | ) log | x | + (cid:101) N (0) such that after writing γ ( δ ) = − γ ( δ ) asymp ω ( | x | ) log | x | + (cid:101) γ ( δ ) and N ( δ ) = 1 + N ( δ ) asymp ( t ) ω ( | x | ) log | x | + (cid:101) N ( δ ) , it holds that ( γ ( δ j ) asymp , (cid:101) γ ( δ j ) , ( β i ) ( δ j ) , N ( δ j ) asymp , (cid:101) N ( δ j ) ) → ( γ (0) asymp , (cid:101) γ (0) , ( β i ) (0) , N (0) asymp , (cid:101) N (0) ) in R × C ([0 , C ( R )) × C ([0 , C ( R )) × C ([0 , R ) × C ([0 , C ( R )) . Moreover, the following addi-tional estimates hold for all t ∈ [0 , : ≤ γ (0) asymp (cid:46) (cid:15) , | N (0) asymp | ( t ) (cid:46) (cid:15) , (6.1) (cid:88) (cid:101) g (0) ∈{ (cid:101) γ (0) , ( β i ) (0) , (cid:101) N (0) } ( (cid:107) (cid:101) g (0) (cid:107) W , ∞ − α (Σ t ) + (cid:107) ∂ x (cid:101) g (0) (cid:107) L ∞ − α (Σ t ) ) (cid:46) (cid:15) , (6.2) and the following holds for a.e. t ∈ [0 , : | ∂ t N (0) asymp | ( t ) + (cid:107) ∂ t (cid:101) g (0) (cid:107) W , s (cid:48)− s (cid:48)(cid:48) − s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) (cid:46) (cid:15) . (6.3) Proof.
By Theorem 5.6, γ ( δ ) asymp is a set of bounded numbers obeying (6.1), which by the Bolzano–Weierstrasstheorem has a limit for some δ j . In particular, the bound for γ (0) asymp in (6.1) holds.By Theorem 5.6, N ( δ j ) asymp are bounded C functions of t , and thus by the Arzelà–Ascoli theorem on C ([0 , , has a subsequential strong limit in C ([0 , . Hence, the bound for N (0) asymp in (6.1) holds. Finally,the C bound and the uniform convergence imply the L ∞ bound for | ∂ t N (0) asymp | ( t ) in (6.3).For the convergences of (cid:101) γ , β i and (cid:101) N , we apply Lemma 6.2 with T = 1 , X = C ( R ) , X = C ( R ) , X = W , − ( R ) , and W as in Lemma 6.2. (The compactness of X ⊆ X follows from the Arzelà–Ascolitheorem.) The uniform boundedness of (cid:101) γ ( δ j ) , ( β i ) ( δ j ) and (cid:101) N ( δ j ) in W is an immediate consequence of theestimates in Theorem 5.6. Finally, note that by Theorem 5.6, for all sufficiently small δ , γ ( δ ) , β ( δ ) and N ( δ ) satisfy analogous bounds as (6.1)–(6.3), which imply the estimates in (6.1)–(6.3). Proposition 6.4 (Limiting wavefront) . There exists a subsequence δ j (of that in Proposition 6.3, but notrelabelled) such that for k = 1 , , , the following holds:1. The eikonal functions u ( δ j ) k converges in ( C loc ) t,x ([0 , × R ) to a limit u (0) k . Moreover, u (0) k is a ( C , loc ) t,x function satisfying the following estimate: (cid:107) ∂ i u (0) k (cid:107) W , ∞ (Σ t ) (cid:46) . (6.4)
2. The vector fields L ( δ j ) k , E ( δ j ) k and X ( δ j ) k converge uniformly on compact sets to limiting vector fields L (0) k , E (0) k and X (0) k . Moreover, on any compact set, L (0) k , E (0) k and X (0) k are ( C , loc ) t,x vector fields with the following estimates (recall that α = 10 − ): | L βk | + | X ik | + | E ik | (cid:46) (cid:104) x (cid:105) (cid:15) , | ∂ t L tk | + (cid:104) x (cid:105) ( | ∂ t L ik | + | ∂ i L βk | + | ∂X ik | + | ∂E ik | ) (cid:46) (cid:15) · (cid:104) x (cid:105) α . (6.5) Proof. Step 1: Limit for u k . By the Arzelà–Ascoli Theorem, in order to prove ( C loc ) t,x convergence, it sufficesto prove a C t,x bound for u ( δ ) j on any compact set, uniformly in δ . The bounds for the spatial derivativesfollow from the fact that (5.13) holds for all small enough δ by Theorem 5.6. To obtain the bounds for ∂ t u ( δ ) k Here, we use the obvious notation that for every δ , L ( δ ) k , E ( δ ) k and X ( δ ) k denote respectively the L k , E k and X k vector fieldsarising from the initial data set ( φ ( δ ) , ( φ (cid:48) ) ( δ ) , γ ( δ ) , K ( δ ) ) . Notice that higher regularity holds if we separately consider different components. For instance, E (0) k and X (0) k are in W , ∞ loc in spacetime; and while our estimates do not necessarily show that L (0) k is W , ∞ loc in spacetime, it does show that ∂ i ( L jk ) (0) , ∂ t ( L jk ) (0) and ∂ i ( L tk ) (0) are spacetime-Lipschitz, and that ∂ t ( L tk ) (0) is Lipschitz in space. ∂ x ∂ t u ( δ ) k , we combine L ( δ ) k u ( δ ) k = 0 and ( L ( δ ) k ) t = N ( δ ) to write ∂ t u ( δ ) k = − N ( δ ) · ( L ( δ ) k ) i · ∂ i u ( δ ) k , and usethe bounds in (5.16), (5.17) and (5.14) (which hold for small δ by Theorem 5.6) with the estimates on ∂ i u ( δ ) k .The estimate (6.4) then follows from the ( C loc ) t,x convergence and the estimate (5.13) (for δ > ). Step 2: Limit for L k , E k and X k . Using the Arzelà–Ascoli Theorem, in order to prove the convergencestatement, we need uniform C t,x bounds for L ( δ ) k , E ( δ ) k and X ( δ ) k on compact sets. These bounds are conse-quences of Theorem 5.6, which states that (5.14) holds for small δ > . Using the estimate (5.14) again thenimplies (6.5). Proposition 6.5 (Limiting φ ) . There exists a subsequence δ j (of that in Proposition 6.4, but not relabelled)such that the following holds:1. There exists φ (0) reg such that ( φ ( δ j ) reg , ∂ t φ ( δ j ) reg ) → ( φ (0) reg , ∂ t φ (0) reg ) in the C ([0 , , H s ( R )) × C ([0 , , H s ( R )) norm for all s < s (cid:48) . Moreover, for every t ∈ [0 , , (cid:107) φ (0) reg (cid:107) H s (cid:48) (Σ t ) + (cid:107) ∂ t φ (0) reg (cid:107) H s (cid:48) (Σ t ) (cid:46) (cid:15). (6.6)
2. For each k = 1 , , , there exists (cid:101) φ (0) k such that ( (cid:101) φ ( δ j ) k , ∂ t (cid:101) φ ( δ j ) k ) → ( (cid:101) φ (0) k , ∂ t (cid:101) φ (0) k ) in the C ([0 , , H s ( R )) × C ([0 , , H s ( R )) norm for all s < s (cid:48) . Moreover, for every t ∈ [0 , , (cid:107) (cid:101) φ (0) k (cid:107) H s (cid:48) (Σ t ) + (cid:107) ∂ t (cid:101) φ (0) k (cid:107) H s (cid:48) (Σ t ) (cid:46) (cid:15). (6.7)
3. For each k = 1 , , , the second distributional derivatives of (cid:101) φ (0) k satisfies the following properties:(a) For every t ∈ [0 , , ∂ µν (cid:101) φ (0) k = T µν,k + f µν,k , where T µν,k is a signed Radon measure with supp( T µν,k ) ⊆ { u (0) k = 0 } , and T.V. | Σ t ( T µν,k ) + (cid:107) f µν,k (cid:107) L (Σ t ) (cid:46) (cid:15). (b) For the vector fields L (0) k and E (0) k as in part 2 of Proposition 6.4, L (0) k (cid:101) φ k and E (0) k (cid:101) φ k are moreregular and satisfy (cid:107) L (0) k (cid:101) φ (0) k (cid:107) H s (cid:48)(cid:48) (Σ t ) + (cid:107) E (0) k (cid:101) φ (0) k (cid:107) H s (cid:48)(cid:48) (Σ t ) + (cid:107) ∂ t L (0) k (cid:101) φ (0) k (cid:107) H s (cid:48)(cid:48) (Σ t ) + (cid:107) ∂ t E (0) k (cid:101) φ (0) k (cid:107) H s (cid:48)(cid:48) (Σ t ) (cid:46) (cid:15).
4. For each k = 1 , , , φ (0) reg and (cid:101) φ (0) k satisfy the following Lipschitz estimates for every t ∈ [0 , : (cid:107) ∂ (cid:101) φ (0) k (cid:107) L ∞ (Σ t ) (cid:46) (cid:15), (cid:107) ∂φ (0) reg (cid:107) C , s (cid:48)(cid:48) (Σ t ) (cid:46) (cid:15). (6.8) Moreover, for each k = 1 , , and for every t ∈ [0 , , (cid:107) ∂ (cid:101) φ (0) k (cid:107) C , s (cid:48)(cid:48) (Σ t ∩{ u k > } ) + (cid:107) ∂ (cid:101) φ (0) k (cid:107) C , s (cid:48)(cid:48) (Σ t ∩{ u k < } ) (cid:46) (cid:15). (6.9)
5. For each k = 1 , , , φ (0) reg and (cid:101) φ (0) k are supported in B (0 , R ) for every t ∈ [0 , .Proof. Step 1: Limits for φ reg , ∂ t φ reg , (cid:101) φ k and ∂ t (cid:101) φ k and support properties (Statements 1, 2, 5). All of φ reg , ∂ t φ reg , (cid:101) φ k and ∂ t (cid:101) φ k can be treated in a similar manner, except for the different regularity (i.e. H s (cid:48) for φ reg , H s (cid:48) for ∂ t φ reg and (cid:101) φ k , and H s (cid:48) for ∂ t (cid:101) φ k ). We will thus only discuss φ reg in detail.Denote, for the purpose only on this proof, by H s c ( R ) the closed subspace of H s ( R ) where thefunction is compactly supported in B (0 , R ) ; similarly for H s (cid:48) c ( R ) . With this definition, for s ∈ (0 , s (cid:48) ) , H s (cid:48) c ( R ) ⊆ H s c ( R ) is compact (for instance using [21, Lemma 2.1] and Plancherel’s theorem).For s ∈ (0 , s (cid:48) ) , we thus apply Lemma 6.2 with T = 1 , X = H s (cid:48) c ( R ) , X = H s c ( R ) , X = H s (cid:48) ( R ) ,together with the fact that Theorem 5.6 guarantees the bound (5.8a) holds for all small δ , to obtain thedesired convergence. 34o show that the first estimate in (6.6) for φ (0) reg holds. Note that since (5.8a) holds uniformly for allsmall δ , it admits a weak limit which also satisfies (5.8a). The weak limit necessarily coincides with φ (0) reg ,implying the desired bound.Finally, it is clear from the definition that the limits are indeed supported in B (0 , R ) as stated. Step 2: Regularity for ∂ µν (cid:101) φ (0) k (Statement 3(a)). Fix k , µ and ν . Let (cid:101) ρ : R → [0 , be a smooth functionwith (cid:101) ρ ≡ on R \ [ − , , and (cid:101) ρ ≡ on [ − , .For each δ > sufficiently small, define ρ ( δ ) k ( u k , θ k , t ) = (cid:101) ρ ( u k δ ) . Introduce the decomposition ∂ µν (cid:101) φ ( δ ) k = ρ ( δ ) k · ∂ µν (cid:101) φ ( δ ) k + (1 − ρ ( δ ) k ) · ∂ µν (cid:101) φ ( δ ) k . (6.10)Using the Cauchy–Schwarz inequality, (5.19), the support properties of ρ ( δ ) k , and Corollary 8.6, we obtain (cid:107) ρ ( δ ) k · ∂ µν (cid:101) φ ( δ ) k (cid:107) L (Σ t ) (cid:46) (cid:107) ρ ( δ ) k (cid:107) L (Σ t ) (cid:107) ∂ µν (cid:101) φ ( δ ) k (cid:107) L (Σ t ) (cid:46) δ · ( (cid:15)δ − ) = (cid:15). Combining this uniform L boundwith the Banach–Alaoglu theorem, it follows that there exists a subsequence of ρ ( δ ) k · ∂ µν (cid:101) φ ( δ j ) k (not relabelled)which converges in weak-* to a signed Radon measure T µν,k with T.V. | Σ t ( T µν,k ) (cid:46) (cid:15) .Note that indeed supp( T µν,k ) ⊆ { ( t, x ) : u (0) k = 0 } . This is because of δ j → , Proposition 6.4, and supp( ρ ( δ j ) k · ∂ µν (cid:101) φ ( δ j ) k ) ⊆ { ( t, x ) : u ( δ j ) k ∈ [ − δ j , δ j ] } .On the other hand, note that (cid:107) (1 − ρ ( δ ) k ) · ∂ µν (cid:101) φ ( δ ) k (cid:107) L (Σ t ) (cid:46) (cid:15) using (5.21) and supp(1 − ρ ( δ ) k ) ∩ S kδ = ∅ .Therefore, by the Banach–Alaoglu theorem, there is a further subsequence δ j such that (1 − ρ ( δ j ) k ) · ∂ µν (cid:101) φ ( δ j ) k converges weakly in L (Σ t ) to some f µν,k satisfying (cid:107) f µν,k (cid:107) L (Σ t ) (cid:46) (cid:15) .Finally, since by definition, T µν,k + f µν,k is the distributional limit of ∂ µν (cid:101) φ ( δ j ) k , it must hold that ∂ µν (cid:101) φ (0) k = T µν,k + f µν,k . Step 3: Regularity for second derivatives of (cid:101) φ k with one good derivative (Statement 3(b)). Arguing as inStep 1, but using that E ( δ j ) k (cid:101) φ ( δ j ) k , L ( δ j ) k (cid:101) φ ( δ j ) k satisfy uniformly the estimates (5.8c), it follows that (for afurther subsequence) E ( δ j ) k (cid:101) φ ( δ j ) k , L ( δ j ) k (cid:101) φ ( δ j ) k , ∂ t E ( δ j ) k (cid:101) φ ( δ j ) k , ∂ t L ( δ j ) k (cid:101) φ ( δ j ) k have (say, distributional) limits whichagain obey again the estimates (5.8c).It thus remains to show that the distributional limit of E ( δ j ) k (cid:101) φ ( δ j ) k (respectively L ( δ j ) k (cid:101) φ ( δ j ) k ) is indeed E (0) k (cid:101) φ (0) k (respectively L (0) k (cid:101) φ (0) k ). To see this, it suffices to note that ( E ik ) ( δ j ) → ( E ik ) (0) uniformly (by part 2of Proposition 6.4) and that ∂ (cid:101) φ ( δ j ) k → ∂ (cid:101) φ (0) k in L ∞ ([0 , L ( R )) (by part 2 of this proposition). Step 4: Lipschitz and improved Hölder bounds (Statement 4).
To show the first estimate in (6.8), notethat Step 1 in particular implies ∂ (cid:101) φ ( δ j ) k → ∂ (cid:101) φ (0) k in L ∞ ([0 , L ( R )) . Thus, after passing to a furthersubsequence, ∂ (cid:101) φ ( δ j ) k → ∂ (cid:101) φ (0) k almost everywhere. In particular, the desired L ∞ bound follows from that for ∂ (cid:101) φ ( δ j ) k .For the second estimates in (6.8) (for the C , s (cid:48)(cid:48) bound for φ reg ), we first note that the uniform C , s (cid:48)(cid:48) estimates, together with Lemma 6.2, imply that (for a further subsequence) ∂φ ( δ j ) reg → ∂φ (0) reg uniformly oncompact sets. As a result, the desired C , s (cid:48)(cid:48) bound follows from that for ∂φ ( δ j ) reg .Finally, we prove the Hölder estimates in (6.8). It suffices to consider u k > since (cid:101) φ k ≡ when u k < by the finite speed of propagation. For every δ > , define R k δ := { ( t, x ) ∈ [0 , × R : u (0) k ( t, x ) ≥ δ } .Now, by part 1 of Proposition 6.4, there exists J ∈ N such that R k δ ⊆ { ( t, x ) ∈ [0 , × R : u ( δ j ) k ( t, x ) ≥ δ j } for all j ≥ J . Thus, using (5.22), we argue as for the improved Hölder estimate in (6.8) to obtain (cid:107) ∂ (cid:101) φ (0) k (cid:107) C , s (cid:48)(cid:48) (Σ t ∩ R k δ ) (cid:46) (cid:15). (6.11)Importantly, since (6.11) is independent of δ , we deduce the estimate in (6.8) for u k > . We continue to take ( φ , φ (cid:48) , γ , K ) , ( φ ( δ )0 , ( φ (cid:48) ) ( δ )0 , γ ( δ )0 , K ( δ )0 ) and ( γ ( δ ) , ( β i ) ( δ ) , N ( δ ) , φ ( δ ) ) to be as in the endof Section 6.1, and let ( γ (0) , ( β i ) (0) , N (0) , φ (0) ) be as given by Propositions 6.3 and 6.5.35 roposition 6.6.
1. The limit ( γ (0) , ( β i ) (0) , N (0) , φ (0) ) given by Propositions 6.3 and 6.5 is a weak so-lution to the weak solution to the Einstein vacuum equations in polarized U (1) symmetry under ellipticgauge (in the sense of part 1 Definition 5.1). Moreover, each of φ (0) reg and (cid:101) φ (0) k satisfies the wave equationweakly in the sense of (5.6) .2. The solution in part 1 achieves the given data ( φ , φ (cid:48) , γ , K ) in Theorem 5.2, and moreover, each of φ (0) reg and (cid:101) φ (0) k achieve the initial data as given in Definition 4.3 (in the sense of part 2 of Definition 5.1).Proof. Step 1: The limit is a weak solution. For every δ > sufficiently small, we have a smooth solution andthus the equations (5.1)–(5.6) all hold (either directly or after integrating by parts). Moreover, Theorem 5.6stated that each of φ (0) reg and (cid:101) φ (0) k satisfies the wave equation. In order to pass to the limit, it suffices to have,for g ∈ { γ, β i , N } , g , ∂ i g and e γ converge uniformly on compact sets, and ∂ t g and ∂φ both converge in L ([0 , L loc ( R )) . These convergences (in fact much stronger ones) follow from Propositions 6.3 and 6.5. Step 2: The limit achieves the given initial data.
In view of part 2 of Lemma 6.1, it suffices to provequantitative convergence for δ > .Note that ∂ t γ ( δ ) , ∂ t K ( δ ) , ∂ t φ ( δ ) reg and ∂ t (cid:101) φ ( δ ) k are all uniformly bounded on B (0 , R ) for any R > , whichimplies that (cid:107) γ ( δ ) | Σ t − γ ( δ )0 (cid:107) L ∞ ( R ∩ B (0 ,R )) + (cid:107) K ( δ ) | Σ t − K ( δ )0 (cid:107) L ∞ ( R ∩ B (0 ,R )) + (cid:107) ( φ ( δ ) reg ) | Σ t − ( φ ( δ ) reg ) (cid:107) L ∞ ( R ∩ B (0 ,R )) + (cid:107) ( (cid:101) φ ( δ ) k ) | Σ t − ( (cid:101) φ ( δ ) k ) (cid:107) L ∞ ( R ∩ B (0 ,R )) (cid:46) (cid:15)t. (6.12)On the other hand, by Lemma 6.1 and Sobolev embedding ( H s (cid:48) loc ( R ) (cid:44) → L ∞ loc ( R ) ), we know that for any R > , lim δ → ( (cid:107) γ ( δ )0 − γ (cid:107) L ∞ (Σ ∩ B (0 ,R )) + (cid:107) K ( δ )0 − K (cid:107) L ∞ (Σ ∩ B (0 ,R )) + (cid:107) φ ( δ )0 − φ (cid:107) L ∞ (Σ ∩ B (0 ,R )) = 0 . (6.13)Combining (6.12) and (6.13), and using the triangle inequality give part 2(a) of Definition 5.1.It remains to check (5.7) for each of φ (cid:48) reg and (cid:101) φ (cid:48) k . First, by Proposition 6.5, lim δ → sup t ∈ [0 , ( (cid:107) ( (cid:126)nφ reg ) (0) − ( (cid:126)nφ reg ) ( δ ) (cid:107) L (Σ t ) + (cid:88) k =1 (cid:107) ( (cid:126)n (cid:101) φ k ) (0) − ( (cid:126)n (cid:101) φ k ) ( δ ) (cid:107) L (Σ t ) ) = 0 , (6.14)where by Lemma 6.1, we also have ( (cid:126)nφ reg ) (0) | Σ = ( φ (cid:48) reg ) , ( (cid:126)n (cid:101) φ k ) (0) | Σ = ( (cid:101) φ (cid:48) k ) . (6.15)Note now that for all δ sufficiently small, we have the uniform bound (cid:107) ( L k (cid:126)n (cid:101) φ k ) ( δ ) (cid:107) L (Σ t ) (cid:46) (cid:15) by Theo-rem 5.6 (which implies (5.8b), (5.8c), (5.16) and (5.17) hold for small δ > ). Working in the ( u ( δ ) k , θ ( δ ) k , t ( δ ) k ) coordinates, and recalling ∂ t ( δ ) k = N ( δ ) L ( δ ) k , the fundamental theorem of calculus gives (cid:126)n (cid:101) φ ( δ ) k ( u k , θ k , t k ) − ( (cid:101) φ (cid:48) k ) ( δ )0 ( u k , θ k ) = (cid:90) t k N ( δ ) · ( L k (cid:126)n (cid:101) φ k ) ( δ ) ( u k , θ k , s ) ds. Thus the bound (cid:107) L k (cid:126)n (cid:101) φ ( δ ) k (cid:107) L (Σ t ) (cid:46) (cid:15) , together with the compact support of (cid:101) φ ( δ ) k and Minkowski’s inequality,imply that (cid:88) k =1 (cid:107) ( (cid:126)n (cid:101) φ k ) ( δ ) | Σ t − ( (cid:126)n (cid:101) φ k ) ( δ ) | Σ (cid:107) L ( R ) (cid:46) (cid:15) t. (6.16)Also, an analogous (and indeed easier) statement as (6.16) holds for φ reg instead of (cid:101) φ k (which can be provenby replacing the use of (5.8b), (5.8c) by that of (5.8a)): (cid:107) ( (cid:126)nφ reg ) ( δ ) | Σ t − ( (cid:126)nφ reg ) ( δ ) | Σ (cid:107) L ( R ) (cid:46) (cid:15) t. (6.17)Combining (6.14), (6.15), (6.16) and (6.17) yields the desired statement lim t → ( (cid:107) ( (cid:126)nφ reg ) (0) | Σ t − ( φ (cid:48) reg ) (cid:107) L (Σ ) + (cid:88) k =1 (cid:107) ( (cid:126)n (cid:101) φ k ) (0) | Σ t − ( (cid:101) φ (cid:48) k ) (cid:107) L ( R ) ) = 0 . .4 Proof of Theorem 5.2 We now conclude the proof of Theorem 5.2 (under the assumption of the validity of Theorem 5.6).
Proof of Theorem 5.2.
We start with an initial data set ( φ , φ (cid:48) , γ , K ) as in Definition 4.3 with param-eters (cid:15) , s (cid:48) , s (cid:48)(cid:48) , R , κ . For (cid:15) sufficiently small, we use the approximation procedure described in Sec-tion 6.1 and the limiting procedure in Proposition 6.3–6.5 to obtain a limiting quadruple ( γ, β i , N, φ ) (called ( γ (0) , ( β i ) (0) , N (0) , φ (0) ) above). By Proposition 6.6, ( γ, β i , N, φ ) is a solution arising from the given initialdata set ( φ , φ (cid:48) , γ , K ) (in the sense of Definition 5.1).We claim that ( γ, β i , N, φ ) is the desired solution as asserted in Theorem 5.2. To see this, it remains tocheck all the estimates.• Support properties of the scalar wave and the wave estimates (5.8a)–(5.12) follow from Proposition 6.5.• The regularity properties for u k , L k , E k , X k and the estimates (5.13)–(5.14) follow from Proposition 6.4.• The estimates in (5.15)–(5.17) follow from Proposition 6.3.This concludes the proof. In this section, we outline the structure for the proof of the main theorem on δ -impulsive gravitational waves,i.e. Theorem 5.6.The proof of Theorem 5.6 is based on a priori estimates proven in a bootstrap argument:• The bootstrap assumptions will be set up in Section 7.1 .• Under the bootstrap assumptions of Section 7.1, we prove a priori estimates in
Section 7.2 . – The a priori estimates are split into three steps: the metric estimates will be stated in
Sec-tion 7.2.1 , the Lipschitz and improved Hölder estimates for the wave variables will be stated in
Section 7.2.2 , and finally, the L -based energy estimates can be found in Section 7.2.3 .The proof of these estimates will occupy most of the remainder of this paper and [53].• In
Section 7.3 , we conclude the proof of Theorem 5.6 assuming the a priori estimates of Section 7.2.
Let (cid:15) , δ , s (cid:48) , s (cid:48)(cid:48) , R and κ be as in Theorem 5.6. We continue to take α = 10 − as in Theorem 5.2.We now introduce the main bootstrap assumptions. In the setting of a bootstrap argument (see Sec-tion 7.2), we will assume that there is a T B ∈ (0 , such that all the estimates below hold on [0 , T B ) × R .The definitions of all the norms below can be found in Section 3. Estimates for the metric components in the elliptic gauge.
Let ω be a cutoff function as in Defini-tion 4.1 . Assume that the metric components γ and N admit the following decomposition : γ ( t, x ) = − γ asymp ω ( | x | ) log | x | + (cid:101) γ ( t, x ) ,N ( t, x ) = 1 + N asymp ( t ) ω ( | x | ) log | x | + (cid:101) N ( t, x ) , (7.1) That such a decomposition exists is a consequence of the local theory; see [35, Theorem 5.4]. γ asymp ≥ is a constant, and N asymp ( t ) ≥ is a function of t alone. Moreover, γ , β i and N satisfy | γ asymp | + sup ≤ t
Theorem 7.3.
Under the assumptions of Theorem 7.1, and after choosing (cid:15) and δ smaller, the followingholds for some C = C ( s (cid:48) , s (cid:48)(cid:48) , R, κ ) > and all t ∈ [0 , T B ) :LHSs of (7.8a) – (7.8c) + (cid:107) ∂φ reg (cid:107) C , s (cid:48)(cid:48) (Σ t ) + (cid:107) ∂ (cid:101) φ k (cid:107) C , s (cid:48)(cid:48) (Σ t ∩ C k ≥ δ ) (cid:46) E . The proof of Theorem 7.3 will be carried out in [53].
Under the assumptions of Theorem 7.1, and after choosing (cid:15) and δ smaller, the following holds for some C = C ( s (cid:48) , s (cid:48)(cid:48) , R, κ ) > and all t ∈ [0 , T B ) :1. The wave energy estimates (7.4) – (7.7d) all hold with (cid:15) replaced by C(cid:15) .2. The wave energy estimates stated in (5.8a) – (5.8c) , (5.19) – (5.21) hold.3. In addition, the norm E satisfies the estimate E (cid:46) (cid:15) . The proof of Theorem 7.4 will be carried out in [53]. Note that some bounds in 1 and 2 are repeated. .3 Local existence and the proof of Theorem 5.6 Theorem 7.5 (Local existence) .
1. Given an initial data set in Theorem 5.6, there exists a time T local ∈ (0 , (potentially depending on the initial data profile and not only the parameters involved) and asmooth solution ( γ, β i , N, φ ) to (2.22) , (2.23) – (2.25) and (2.27) on [0 , T local ) × R
2. Moreover, there exists a universal (cid:15) local > such that if [0 , T ∗ ) is the maximal time interval for whichthe solution exists for some T ∗ ∈ (0 , , then at least one of the following holds:(a) lim inf t → T ∗ ( (cid:107) φ (cid:107) H (Σ t ) + (cid:107) (cid:126)nφ (cid:107) H (Σ t ) ) = ∞ ,(b) lim inf t → T ∗ ( (cid:107) (cid:126)nφ (cid:107) L ∞ (Σ t ) + (cid:107) ∂ i φ (cid:107) L ∞ (Σ t ) ) ≥ (cid:15) local .Proof. This is an immediate consequence of the local existence result in [35, Theorem 5.4].
Lemma 7.6.
Taking T local ∈ (0 , smaller if necessary, the solution on (0 , T local ) × R given by Theorem 7.5obeys all the estimates in (7.3a) – (7.8c) with T B replaced by T local .Proof. By continuity, it suffices to check that the estimates (7.3a)–(7.8c) are satisfied initially for t = 0 . Itis easy to verify that the following stronger estimates all hold when t = 0 :1. The wave energy estimates (7.4)–(7.6b) hold at t = 0 with (cid:15) replaced by C(cid:15) . For (7.4)–(7.5d) and(7.6b), this is an immediate consequence of the assumptions in Definition 4.8. The initial estimate for(7.6a) follows from the other estimates; see [53, Proposition 12.1] for the proof of this fact.2. The wave flux estimates (7.7a)–(7.7d) trivially hold when (a) T B is replaced by and (b) (cid:15) is replacedby . That this is the case is because C ku k ( { } ) has measure zero (with respect to du k dθ k ).3. The pointwise wave estimates (7.8a)–(7.8c) hold when t = 0 with (cid:15) replaced by C(cid:15) . For (7.8c), thisis directly assumed in Definition 4.8. For (7.8a) and (7.8b), this is a consequence of the embedding inTheorem 7.3 together with the L -based assumptions in Definition 4.8.4. The estimates (7.2a)–(7.2c) all hold at t = 0 with (cid:15) replaced by C(cid:15) . That this is the case can beproven using the elliptic equations for the metric components, in the same manner as Propositions 9.7and 9.19. Moreover, since the Besov estimates (7.8a)–(7.8b) hold for φ initially (see point 3. above),we have the estimates (cid:80) g ∈{ γ,β l ,N } (cid:107) ∂ ij g (cid:107) L ∞ − α (Σ ) (cid:46) (cid:15) (proved in the same way as Proposition 9.11).(Note that the estimates for C(cid:15) instead of C(cid:15) as in Propositions 9.7, 9.11 and 9.19 because we havethe better initial wave estimates from points 1 and 3 above.)5. Finally, the bounds (7.3a)–(7.3e) for K , χ k , η k , µ k and Θ k hold at t = 0 with (cid:15) replaced by C(cid:15) . Thatthis is the case is due to (a) the formulas (2.14) and Lemma 2.23 for their initial values at t = 0 , and(b) the metric estimates (7.2a)–(7.2c) and (cid:80) g ∈{ γ,β l ,N } (cid:107) ∂ ij g (cid:107) L ∞ − α (Σ ) (cid:46) (cid:15) discussed in point 4 above.We are now ready to combine the local existence results (Theorem 7.5 and Lemma 7.6) with the bootstrapresults (Theorems 7.1, 7.3 and 7.4) to conclude the proof of Theorem 5.6. Proof of Theorem 5.6. Step 1: Solution exists in the time interval [0 , . Suppose for the sake of contradictionthat there exists T ∗ ∈ (0 , such that [0 , T ∗ ) is the maximal time interval for which the solution exists.We claim that the bootstrap assumptions (7.3a)–(7.8c) hold for all t ∈ [0 , T ∗ ) . Suppose not, then bycontinuity and Lemma 7.6, there exists T ∗∗ ∈ (0 , T ∗ ) such that (7.3a)–(7.8c) all hold for t ∈ [0 , T ∗∗ ] , and thatwhen t = T ∗∗ , in at least one of the estimates (7.3a)–(7.8c), “ ≤ ” can be replaced by “ = ”.We now apply Theorems 7.1, 7.3 and 7.4 with T B = T ∗∗ . In particular, (7.3a)–(7.8c) all hold with (cid:15) replaced by C(cid:15) and (cid:15) replaced by C(cid:15) for all t ∈ [0 , T ∗∗ ) . Choosing (cid:15) smaller, we have C(cid:15) ≤ (cid:15) and C(cid:15) ≤ (cid:15) . However, this contradicts that fact that at least one of the estimates (7.3a)–(7.8c) is an equalityat t = T ∗∗ .We have thus established that the bootstrap assumptions (7.3a)–(7.8c) hold for all t ∈ [0 , T ∗ ) . Applynow again Theorems 7.1, 7.3 and 7.4, but with T B = T ∗ , we obtain the following:40a) Since the initial data are smooth, (5.20) implies lim inf t → T ∗ ( (cid:107) φ (cid:107) H (Σ t ) + (cid:107) (cid:126)nφ (cid:107) H (Σ t ) ) < ∞ .(b) By (5.11), (cid:107) (cid:126)nφ (cid:107) L ∞ (Σ t ) + (cid:107) ∂ i φ (cid:107) L ∞ (Σ t ) ≤ C(cid:15) .Therefore, after taking (cid:15) smaller if necessary (so that C(cid:15) ≤ (cid:15) local ), part 2 of Theorem 7.5 implies that [0 , T ∗ ) is not a maximal time interval, which leads to a contradiction. Step 2: Estimates in Theorem 5.6.
Repeating the argument in Step 1, we also show that the bootstrapassumptions (7.3a)–(7.8c) hold for all t ∈ [0 , . As a result, all the estimates stated in the conclusions ofTheorems 7.1, 7.3 and 7.4 hold. This thus implies all the estimates stated in Theorem 5.6. From this section onwards until Section 10, we will prove Theorem 7.1 (see Section 11 for the conclusion ofthe proof). In particular, we work under the assumptions of Theorem 7.1.Moreover, we will allow all constants C or implicit constants in (cid:46) to depend on s (cid:48) , s (cid:48)(cid:48) , R and κ (as in the statement of Theorem 7.1). Whenever necessary, we will also take (cid:15) smaller (dependingon s (cid:48) , s (cid:48)(cid:48) , R and κ ) without further comments. In this section, we prove preliminary estimates which follow directly from the bootstrap assumptions.Before we begin, we prove an easy finite speed of propagation lemma in
Section 8.1 that will be useful forthe remainder of the paper. Turning to the preliminary estimates, the first group of estimates involve thecoefficients of ( X k , E k , L k ) in the ( ∂ t , ∂ , ∂ ) basis; see Section 8.2 . The second group of estimates can beviewed as quantitative bounds on the transversality between the three waves; see
Section 8.3 . Lemma 8.1.
The following holds on Σ t for all t ∈ [0 , T B ) and for k = 1 , , :1. supp( φ reg ) , supp( (cid:101) φ k ) ⊆ B (0 , R ) ,2. supp( (cid:101) φ k ) ⊆ { ( t, x ) : u k ( t, x ) ≥ − δ } .Proof. Both assertions are standard finite speed of propagation statements for the wave equation. The firststatement follows from the facts that the initial supports are in B (0 , R ) and that for (cid:15) sufficiently small,the metric is O ( (cid:15) ) -close to the Minkowski metric in the C t,x norm on compact sets (by (7.2a) and (7.2b)).The second statement follows from the facts that u k ≥ − δ on supp( (cid:101) φ k ) initially, and that { ( t, x ) : u k = − δ } is a null hypersurface with e u k > (by Definition 2.8). ( X k , E k , L k ) in the ( ∂ t , ∂ , ∂ ) basis Lemma 8.2.
The following estimates hold for the coefficients of the vector fields L k , E k , X k : | L βk | + | X ik | + | E ik | (cid:46) (cid:104) x (cid:105) (cid:15) . (8.1) Proof.
Using g ( X k , X k ) = g ( E k , E k ) = 1 (by (2.41)), g ij = e γ δ ij (by (2.7)), (7.2a) and (7.2b), we know that | X ik | + | E ik | (cid:46) e − γ (cid:46) e C · (cid:15) · log(1+ (cid:104) x (cid:105) ) (cid:46) (cid:104) x (cid:105) C(cid:15) . (8.2)Using (2.40) and (2.4), and then (7.2a), (7.2b), (8.2) and the inequality log y ≤ ρ y ρ for all ρ > and y ≥ , (8.3)we obtain | L αk | (cid:46) | N − | + max i =1 , ( | X ik | + | β i N | ) (cid:46) (cid:15) log(1 + (cid:104) x (cid:105) ) + (cid:104) x (cid:105) C(cid:15) (cid:46) (cid:104) x (cid:105) C(cid:15) . (8.4)After choosing (cid:15) to be sufficiently small, (8.2) and (8.4) obviously imply (8.1).41 emma 8.3. For any sufficiently regular function f , | ∂ i f | (cid:46) (cid:104) x (cid:105) (cid:15) ( | E k f | + | X k f | ) , (8.5) | ∂ t f | (cid:46) (cid:104) x (cid:105) (cid:15) ( | L k f | + | ∂ x f | ) , (8.6) | ∂ t f | (cid:46) (cid:104) x (cid:105) (cid:15) ( | L k f | + | X k f | ) + (cid:104) x (cid:105) − (cid:15) | E k f | . (8.7) Proof.
Starting with the formula (2.53), estimating e γ with (7.2a), (7.2b), and bounded X ik , E ik by (8.2),(8.4), we immediately obtain (8.5).By (2.40) and (2.3), ∂ t = N L k + N X k + β i ∂ i . (8.8)Using (8.8) and applying (7.2a), (7.2b), (8.2), (8.4) and (8.3), we obtain | ∂ t f | (cid:46) N · ( | L k f | + | X k f | ) + | β | · | ∂ x f | (cid:46) (cid:104) x (cid:105) (cid:15) ( | L k f | + | ∂ x f | ) , (8.9)which implies (8.6).Finally, arguing as in (8.9) but controlling | β | · | ∂ x f | by (7.2a), (7.2b) and (8.5), we obtain (8.7). Lemma 8.4.
The following estimates hold for the derivatives of the coefficients of the vector fields L k , E k , X k : | L k L tk | + (cid:104) x (cid:105) ( | L k L ik | + (cid:88) Y k , Z k ∈{ L k ,E k ,X k } ( Y k ,Z k ) (cid:54) =( L k ,L k ) | Y k Z βk | ) (cid:46) (cid:15) · (cid:104) x (cid:105) α (8.10) | ∂ t L tk | + (cid:104) x (cid:105) ( | ∂ t L ik | + | ∂ i L βk | + | ∂X ik | + | ∂E ik | ) (cid:46) (cid:15) · (cid:104) x (cid:105) α . (8.11) Proof. Step 1: Proof of (8.10) . The derivatives on the LHS of (8.10) were computed in (2.83a)–(2.86) inLemma 2.20. Using those formulas, and plugging in the bootstrap assumptions (7.3a), (7.3b), (7.2a), (7.2b),as well as the estimates (8.2) and (8.4), we obtain the desired result.
Step 2: Proof of (8.11) . Finally, (8.11) follows from (8.10) and Lemma 8.3.
Proposition 8.5.
The following estimate holds: sup ≤ t To prove (8.12), we will compare the value of ∂ i u k with its initial value alongan integral curve of L k .First, we need an easy bound that (cid:104) x (cid:105) is comparable at any two points along the integral curve of L k .For this, we simply use (8.1) to obtain | L k (cid:104) x (cid:105) | = | δ ij L ik x j | (cid:46) (cid:104) x (cid:105) , and apply Grönwall’s inequality. In what follows, we will silently assume the comparability of (cid:104) x (cid:105) . Step 1: Estimates for ∂ i u k . Recalling the formula for ∂ i u k in (2.54), we compare each of the factors e γ , X ik and µ − k at ( u k , θ k , t k ) (recall the coordinates in Section 2.4) with their (initial) values at ( u k , θ k , . For thispurpose, we will consider the L k derivative of each of these quantities. Recall that by (2.45), ∂ t k = N · L k ,where ∂ t k is the coordinate derivative in the ( u k , θ k , t k ) coordinate system.42or e γ , we use (7.2a), (7.2b) and (8.1) to obtain | e γ ( u k ,θ k ,t k ) − e γ ( u k ,θ k , | = 2 | (cid:90) t k ( e γ ∂ t k γ )( u k , θ k , s ) ds | = 2 | (cid:90) t k ( e γ N L k γ )( u k , θ k , s ) ds | (cid:46) (cid:15) (cid:104) x (cid:105) − (cid:15) . (8.15)For X ik , we use Lemmas 8.2 and 8.4 with (7.2a) and (7.2b) to get that | X ik ( u k , θ k , t k ) − X ik ( u k , θ k , | = | (cid:90) t k ( N L k X ik )( u k , θ k , s ) ds | (cid:46) (cid:15) (cid:104) x (cid:105) − α . (8.16)Finally, we control the difference of µ − k . By the equation (2.91) and the estimates in (7.3a), (7.3d),(7.2a), (7.2b) and Lemma 8.2, we obtain | µ − k ( u k , θ k , t k ) − µ − k ( u k , θ k , | = | (cid:90) t k ( µ − k N L k log µ k )( u k , θ k , s ) ds | (cid:46) (cid:104) x (cid:105) − α . (8.17)By (2.29), ( ∂ i u k ) | Σ = c ik . Thus, we write ∂ i u k ( u k , θ k , t k ) − c ik = δ ij · [ e γ ( u k ,θ k ,t k ) − e γ ( u k ,θ k , ] · µ − k ( u k , θ k , t k ) · X jk ( u k , θ k , t k )+ δ ij · e γ ( u k ,θ k , · [ µ − k ( u k , θ k , t k ) − µ − k ( u k , θ k , · X jk ( u k , θ k , t k )+ δ ij · e γ ( u k ,θ k , · µ − k ( u k , θ k , · [ X jk ( u k , θ k , t k ) − X jk ( u k , θ k , , and combining (8.15)–(8.17) with bootstrap assumptions (7.3d), (7.2a), (7.2b) and Lemma 8.2 yields (8.12). Step 2: Proof of (8.13) . In view of (2.7) and (2.52), we have g ( E k , X k (cid:48) ) = e γ · ( E k · X k (cid:48) + E k · X k (cid:48) ) = e γ · ( − X k · X k (cid:48) + X k · X k (cid:48) ) . Hence, by (2.7), the initial condition (2.100), and the estimates (8.15) and (8.16), we obtain | g ( E k , X k (cid:48) )( u k , θ k , t k ) − g ( E k , X k (cid:48) )( u k , θ k , | = | g ( E k , X k (cid:48) )( u k , θ k , t k ) + c k · c k (cid:48) − c k · c k (cid:48) | (cid:46) (cid:104) x (cid:105) − (cid:15) · (cid:15) . The lower bounds in (8.13) then follow immediately from (2.31), after taking < (cid:15) (cid:28) κ , while the upperbounds follow from (2.30). Step 3: Proof of (8.14) . To fix the notation we take k (cid:48) = 2 and k (cid:54) = 2 . By (2.3), (2.40) and (2.45), we have ∂ t k = N − · ( L + X − X k ) . Hence, since L u = 0 by definition, we have ∂ t k u = N − · ( X i − X ik ) ∂ i u . (8.18)Using (8.16) and (2.100), we have | X ik ( u k , θ k , t k ) − e − γ ( u k ,θ k , δ iq · c kq | | B (0 ,R ) (cid:46) (cid:15) , | X i ( u , θ , t ) − e − γ ( u ,θ , δ iq · c q | | B (0 ,R ) (cid:46) (cid:15) . (8.19)Therefore, combining (8.12), (8.19) with (7.2a), (7.2b) to estimate the RHS of (8.18), we obtain | ( X i − X ik ) ∂ i u − δ iq ( c q − c kq ) · c i | | B (0 , R ) (cid:46) (cid:15) . (8.20)In view of (2.30), we have δ iq · c q · c i = 1 . By (2.30) and the Cauchy–Schwarz inequality, we also have | δ iq · c kq · c i | ≤ . Hence, by the triangle inequality and (2.31), we have | − δ iq · c kq · c i | ≥ − | δ iq · c kq · c i | ≥ (1 + | δ iq · c kq · c i | )(1 − | δ iq · c kq · c i | ) ≥ κ . Hence, after choosing (cid:15) smaller, we obtain (8.14) by using(8.18) and (8.20). 43he following is an immediate consequence of Proposition 8.5. Corollary 8.6. For any k (cid:54) = k (cid:48) , the map ( x , x ) (cid:55)→ ( u k , u k (cid:48) ) is a C -diffeomorphism with entry-wisepointwise estimates independent of δ : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:34) ∂u k ∂x ∂u j ∂x ∂u k ∂x ∂u j ∂x (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:34) ∂u k ∂x ∂u j ∂x ∂u k ∂x ∂u j ∂x (cid:35) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) . Proposition 8.7. For any k = 1 , , , | ∂ ij u k | (cid:46) (cid:15) . Proof. We start with (2.54) and differentiate by ∂ j . Then estimate the resulting terms by (7.3d), (7.2a),(7.2b) and Lemmas 8.2 and 8.4. (Notice that the ∂ j derivative of e γ , µ − k or X k would give sufficient (cid:104) x (cid:105) decay to compensate the growth in weights in the other factors.) We continue to work under the assumptions of Theorem 7.1.Our goal in this section is to prove estimates for the metric components ( γ, β , β , N ) in elliptic gauge.In particular, we improve the bootstrap assumptions (7.2a)–(7.2c).We begin with some preliminaries in Sections 9.1 and Section 9.2 . The main estimates are given inthe following sections:• In Section 9.3 , we prove elliptic estimates for purely spatial derivatives of g .• In Section 9.4 , we prove the elliptic estimates for ∂ t g .• In Section 9.5 , we prove the top (fractional) order elliptic estimates for ∂ t g .• In Section 9.6 , we carry out the elliptic estimates for third derivatives of the metric.• In Section 9.7 , we deduce some estimates for K which follow directly from earlier subsections. Most of the following results can be found in [22]; see also [35, Lemmas A2–A3]. The only part not in [22]is the compactness statement in 1(b), which follows readily from the first part of 1(b) together with theKondrachov compactness theorem (for compact domains). Proposition 9.1. 1. Let m ∈ N ∪ { } , p ∈ (1 , + ∞ ) .(a) If m > p and σ ≤ σ (cid:48) + p , then (cid:107) f (cid:107) C σ ( R ) (cid:46) m,p (cid:107) f (cid:107) W m,pσ (cid:48) ( R ) .(b) If m < p , then for any σ , (cid:107) f (cid:107) L p − mpσ + m ( R ) (cid:46) m,p,σ (cid:107) f (cid:107) W m,pσ ( R ) . For m < p and σ < σ (cid:48) , theembedding W m,pσ (cid:48) ( R ) (cid:44) → L p − mp σ + m ( R ) is moreover compact.2. Let ≤ p ≤ p ≤ ∞ and σ − σ > p − p ) . Then (cid:107) f (cid:107) L p σ ( R ) (cid:46) p ,p ,σ ,σ (cid:107) f (cid:107) L p σ ( R ) . We cite a standard lemma regarding fractional derivatives. Lemma 9.2 ((2.1) in [58]) . For any θ > and ≤ p , p (cid:48) , p , p (cid:48) ≤ + ∞ such that p + p = = p (cid:48) + p (cid:48) , (cid:107)(cid:104) D x (cid:105) θ ( f h ) (cid:107) L ( R ) (cid:46) p ,p ,p (cid:48) ,p (cid:48) ,θ (cid:107)(cid:104) D x (cid:105) θ f (cid:107) L p (Σ t ) (cid:107) h (cid:107) L p ( R ) + (cid:107) f (cid:107) L p (cid:48) ( R ) (cid:107)(cid:104) D x (cid:105) θ h (cid:107) L p (cid:48) ( R ) . .1.3 Standard facts about elliptic estimatesDefinition 9.3. Let p ∈ (1 , + ∞ ) , − p < σ < − p . Define ∆ − : L pσ +2 ( R ) → S (cid:48) ( R ) by ∆ − f ( x ) = 12 π (cid:90) R f ( y ) log | x − y | dy. We need a result regarding mapping properties of ∆ − in weighted Sobolev space, which essentiallyfollows from [34, 54]. Since we cannot find the exact statement we need, we include a reduction to [34, 54]for completeness. Proposition 9.4. Let p ∈ (1 , + ∞ ) and − p < σ < − p . Then for every f ∈ L pσ +2 ( R ) , (∆ − f )( x ) = 12 π ( (cid:90) R f ( y ) d y ) ω ( | x | ) log | x | + (cid:101) v ( x ) , (cid:101) v ∈ W ,pσ ( R ) , where ω : R → [0 , is a cutoff function such that ω ( s ) ≡ for s ≤ and ω ( s ) ≡ for s ≥ .Moreover, there exists C = C ( p, σ ) > such that for every f ∈ L pσ +2 ( R ) , (cid:107) ∆ − f − π ( (cid:90) R f ( y ) d y ) ω ( | x | ) log | x |(cid:107) W ,pσ ( R ) ≤ C (cid:107) f (cid:107) L pσ +2 ( R ) . Proof. By [34, Corollary 2.7] , given f ∈ L pσ +2 ( R ) , there exists a function v such that (∆ v )( x ) = f ( x ) , v ( x ) = 12 π ( (cid:90) R f ( y ) d y ) ω ( | x | ) log | x | + (cid:101) v ( x ) , (cid:101) v ∈ W ,pσ , (9.1)and a constant C = C ( p, σ ) > such that for every f ∈ L pσ +2 ( R ) , (cid:107) (cid:101) v (cid:107) W ,pσ ( R ) ≤ C (cid:107) f (cid:107) L pσ +2 ( R ) . Let (recall Definition 9.3) v ( x ) := ∆ − f = 12 π (cid:90) R f ( y ) log | x − y | d y (9.2)so that ∆ v = f in the sense of distribution. In order to prove the proposition, it suffices to show that v = v .To achieve this, first note that h = v − v is a harmonic function. In particular, it is a bounded functionon | x | ≤ . Moreover, using (9.1) and (9.2), we obtain that for | x | ≥ , h ( x ) = v ( x ) − v ( x ) = − (cid:101) v ( x ) + 12 π (cid:90) R f ( y )(log | x − y | − log | x | ) d y. (9.3)Notice that by [54, Corollary 2], log | x − y | − log | x | = (cid:90) dd t log | x − ty | d t = − (cid:90) y · ( x − ty ) | x − ty | d t =: (cid:101) R ( x, y ) satisfies, for some C > , sup (cid:107) w (cid:107) Lpσ +2( R =1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) R w ( y ) (cid:101) R ( x, y ) d y (cid:13)(cid:13)(cid:13)(cid:13) L pσ ( R ) ≤ C. (9.4)By (9.3), (9.4) and the fact that f ∈ L pσ +2 ( R ) , (cid:101) v ∈ L pσ ( R ) , it follows that h ∈ L pσ ( R ) . Writing in polarcoordinates, this means (cid:90) + ∞ (cid:90) π (cid:104) r (cid:105) σp | h | p ( r, ϑ ) d ϑ r d r < + ∞ . Notice that technically, [34, Corollary 2.7] only gives the result when p = 2 . However, the general case for p ∈ (1 , + ∞ ) follows with an identical proof. 45n particular, by the mean value theorem, there exists a sequence { r i } + ∞ i =1 , r i → + ∞ such that as i → + ∞ , (cid:90) π | h | p ( r i , ϑ ) d ϑ (cid:46) r − σp − i → Let i be sufficiently large so that r i ≥ . Recall that h is harmonic. Then, by Poisson’s integral formula, sup y ∈ B (0 , | h | ( y ) (cid:46) (cid:90) π | h | ( r i , ϑ ) d ϑ (cid:46) ( (cid:90) π | h | p ( r i , ϑ ) d ϑ ) p → . Hence, h is a harmonic function which is identically on B (0 , . Unique continuation implies that h ≡ ,which is what we wanted to show. Proposition 9.5. Let p ∈ (1 , + ∞ ) , σ ∈ ( − p , − p ) and σ (cid:48) > . Then ∂ i ∆ − : L pσ +2 ( R ) → W ,p − p − σ (cid:48) ( R ) is a bounded map.Proof. By Proposition 9.4, (cid:107) ∆ − f − π ( (cid:90) R f ( y ) d y ) ω ( | x | ) log | x |(cid:107) W ,pσ ( R ) (cid:46) (cid:107) f (cid:107) L pσ +2 ( R ) . It follows that (cid:107) ∂ i ∆ − f − π ( (cid:90) R f ( y ) d y ) ∂ i ( ω ( | x | ) log | x | ) (cid:107) W ,pσ +1 ( R ) (cid:46) (cid:107) f (cid:107) L pσ +2 ( R ) . (9.5)Notice now that for p , σ , σ (cid:48) as given, we have (cid:107) ∂ i ( ω ( | x | ) log | x | ) (cid:107) W ,p − p − σ (cid:48) ( R ) (cid:46) , | (cid:90) R f ( y ) d y | (cid:46) (cid:107) f (cid:107) L pσ +2 ( R ) . (9.6)Note that − p − σ (cid:48) < σ + 1 since − p < σ . Therefore, using the triangle inequality, (9.5) and (9.6), weobtain (cid:107) ∂ i ∆ − f (cid:107) W ,p − p − σ (cid:48) ( R ) ≤ (cid:107) ∂ i ∆ − f − π ( (cid:90) R f ( y ) d y ) ∂ i ( ω ( | x | ) log | x | ) (cid:107) W ,pσ +1 ( R ) + | (cid:90) R f ( y ) d y |(cid:107) ∂ i ( ω ( | x | ) log | x | ) (cid:107) W ,p − p − σ (cid:48) ( R ) (cid:46) (cid:107) f (cid:107) L pσ +2 ( R ) , as desired. In the remainder of this section, we obtain the desired bounds using the Poisson equations that the metriccomponents N , β i and γ satisfy. Recall that the metric components admit decompositions as in (7.1) andso they are given by γ = ∆ − ∆ γ, β i = ∆ − ∆ β i , N = 1 + ∆ − ∆ N, (9.7)where ∆ − is as in Definition 9.3. Note particularly the constant term built into the definition of N .Moreover, it is useful to note that in the decomposition in (7.1), β i does not have a logarithmic contributionand γ asymp is independent of t .We introduce some schematic notations to be used in this section.• We will use g to denote a metric component, i.e. g ∈ { γ, β i , N } .46 Denote by Ω( g ) a smooth function of g such that | Ω( g )( x ) | (cid:46) (cid:104) x (cid:105) α , | ∂ t (Ω( g ))( x ) | (cid:46) (cid:15) (cid:104) x (cid:105) α , | ∂ i (Ω( g ))( x ) | (cid:46) (cid:15) (cid:104) x (cid:105) − α . (9.8)This holds for instance for e γ N , e γ N , e − γ N and e − γ log N , etc.• When considering terms on the RHSs of (2.24)–(2.26), if the precise structure of the term is unimpor-tant, we will denote the terms schematically by Ω( g ) ∂ α φ∂ β φ and Ω( g ) ∂ i g ∂ j g , where it is understoodthat Ω( g ) satisfies (9.8).It will also useful introduce the following cutoff function. Definition 9.6. Fix a cutoff function (cid:36) ∈ C ∞ c ( R ) such that (cid:36) ≡ on B (0 , R ) and supp( (cid:36) ) ⊆ B (0 , R ) . g In this subsection, we obtain various estimates for the purely spatial derivatives of g : the simple up to W , estimates (with weights) in Section 9.3.1 , the higher order W , s (cid:48) estimates in Section 9.3.2 , and finally,the most difficult W , ∞ estimate (which is a Besov space end-point elliptic estimate; recall Section 1.1.4) in Section 9.3.3 . W , estimates We begin with the simplest elliptic estimates, which is a direct application of Proposition 9.4. Proposition 9.7. γ , β i and N admit a decomposition as in (7.1) . Moreover, for all t ∈ [0 , T B ) : | γ asymp | + | N asymp | (cid:46) (cid:15) , (cid:107) (cid:101) γ (cid:107) W , − α (Σ t ) + (cid:107) (cid:101) N (cid:107) W , − α (Σ t ) + (cid:107) β (cid:107) W , − α (Σ t ) (cid:46) (cid:15) . (9.9) Furthermore, it holds that (cid:107) (cid:101) γ (cid:107) W , ∞ − α (Σ t ) + (cid:107) (cid:101) N (cid:107) W , ∞ − α (Σ t ) + (cid:107) β (cid:107) W , ∞ − α (Σ t ) (cid:46) (cid:15) . (9.10) Proof. Step 0: The logarithmic term as | x | → ∞ . To show that the decomposition (7.1) holds, we need that γ asymp is a constant, γ asymp , N asymp ≥ , and β i has no logarithmic terms. All these follow from the localexistence result in [35, Theorem 5.4]. From now on, we thus focus on the estimates. Step 1: Estimates in (9.9) for γ and N . Recall that γ and N are given by (9.7). Using Proposition 9.4,it thus suffices show that the RHSs of (2.24) and (2.25) can be bounded in L − α +2 by (cid:46) (cid:15) . We will nowprove such a bound.We start with the scalar field terms. The precise structure of scalar field terms in the RHSs of (2.24) and(2.25) is unimportant, and we control general terms of the schematic form Ω( g ) ∂ α φ∂ β φ (recall Section 9.2).Since supp( φ ) ⊆ B (0 , R ) , we can ignore all the weights and use (9.8) and (7.8c) to obtain (cid:107) Ω( g ) ∂ α φ∂ β φ (cid:107) L − α +2 (Σ t ) (cid:46) (cid:107) ∂φ (cid:107) L ∞ (Σ t ) (cid:46) (cid:15) . (9.11)The remaining terms in (2.24) and (2.25) take the form e γ N | L β | and e γ N | L β | . Note that the signproperties of γ asymp and N asymp means that e γ N and e γ N are favorable in terms of (cid:104) x (cid:105) weights. Hence, byHölder’s inequality and the bootstrap assumptions (7.2b) and (7.2c), we have (cid:107) ( L β )( L β ) (cid:107) L − α +2 (Σ t ) (cid:46) (cid:107) L β (cid:107) L − α +1 (Σ t ) (cid:107) L β (cid:107) L ∞ (Σ t ) (cid:46) (cid:107) β (cid:107) W , − α (Σ t ) (cid:107) β (cid:107) W , ∞ − α (Σ t ) (cid:46) (cid:15) . (9.12)Combining (9.11) and (9.12), we obtain the desired bound (cid:107) ∆ − ( RHS of (2.24) ) (cid:107) L − α +2 (Σ t ) + (cid:107) ∆ − ( RHS of (2.25) ) (cid:107) L − α +2 (Σ t ) (cid:46) (cid:15) . tep 2: Estimates in (9.9) for β . To obtain the bound in (9.9) for β , we argue as in Step 1 to bound the RHSof (2.26) in L − α +2 . Clearly, the scalar field term can be bounded exactly as in (9.11). For the remainingterms, we use Hölder’s inequality and the bootstrap assumptions (7.2b) and (7.2c) to obtain (cid:107) ∂ i β∂ j γ (cid:107) L − α +2 (Σ t ) (cid:46) (cid:107) ∂ i β (cid:107) L − α +1 (Σ t ) (cid:107) ∂ j γ (cid:107) L ∞ (Σ t ) (cid:46) (cid:107) β (cid:107) W , − α (Σ t ) (cid:107) ∂ j γ (cid:107) L ∞ (Σ t ) (cid:46) (cid:15) . (9.13)Note that the estimate (9.13) saturates the (cid:104) x (cid:105) weights. For the ∂ i β∂ j NN , we note that N is favorable in termsof weight, so it suffices to show the following bound (which can be proven as in (9.13)): (cid:107) ∂ i β∂ j N (cid:107) L − α +2 (Σ t ) (cid:46) (cid:15) . (9.14)Combining (9.11), (9.13), (9.14) gives (cid:107) ∆ − ( RHS of (2.26) ) (cid:107) L − α +2 (Σ t ) (cid:46) (cid:15) , which gives the desiredbound for β in (9.9). Step 3: Proof of (9.10) . Finally, (9.10) follows from Sobolev embedding (1(a) of Proposition 9.1) and theestimate (9.9) that we have already obtained. s (cid:48) derivatives of metric coefficients in L Proposition 9.8. The following estimate holds for all t ∈ [0 , T B ) : (cid:88) g ∈{ γ,β i ,N } (cid:107) ∂ x (cid:104) D x (cid:105) s (cid:48) g (cid:107) L (Σ t ) (cid:46) (cid:15) . Proof. Using (9.7) and the L -boundedness of the operator ∂ ij ∆ − , it suffices to prove that (cid:88) g ∈{ γ,β i ,N } (cid:107)(cid:104) D x (cid:105) s (cid:48) (∆ g ) (cid:107) L (Σ t ) (cid:46) (cid:15) . (9.15)(Note in particular that we do not demand weights in this bound.)Let Ω( g ) be a smooth function of g as in (9.8). Schematically , ∆ g = Ω( g ) ∂ α φ∂ σ φ + Ω( g ) ∂ i g ∂ j g . (9.16)Since supp( φ ) ⊆ B (0 , R ) , Ω( g ) ∂ α φ∂ σ φ = (cid:36) Ω( g ) ∂ α φ∂ σ φ (for (cid:36) in Definition 9.6). Thus, by Lemma 9.2,(9.8) and the bootstrap assumptions (7.4), (7.5c) and (7.8c), we have (cid:107)(cid:104) D x (cid:105) s (cid:48) (Ω( g ) ∂ α φ∂ σ φ ) (cid:107) L (Σ t ) (cid:46) (cid:107)(cid:104) D x (cid:105) s (cid:48) ( (cid:36) Ω( g )) (cid:107) L ∞ (Σ t ) (cid:107) ∂φ (cid:107) L (Σ t ) + (cid:107) (cid:36) Ω( g ) (cid:107) L ∞ (Σ t ) (cid:107) ∂ (cid:104) D x (cid:105) s (cid:48) φ (cid:107) L (Σ t ) (cid:107) ∂φ (cid:107) L ∞ (Σ t ) (cid:46) (cid:107) (cid:36) Ω( g ) (cid:107) C (Σ t ) ( (cid:107) ∂φ (cid:107) L ∞ (Σ t ) + (cid:107) ∂ (cid:104) D x (cid:105) s (cid:48) φ (cid:107) L (Σ t ) (cid:107) ∂φ (cid:107) L ∞ (Σ t ) ) (cid:46) (cid:15) . (9.17)Using Lemma 9.2 after distributing the weights, and applying (9.8) and the bootstrap assumptions(7.2a)–(7.2c) (and recalling α = 10 − ), we obtain (cid:107)(cid:104) D x (cid:105) s (cid:48) (Ω( g ) ∂ i g ∂ i g ) (cid:107) L (Σ t ) (cid:46) (cid:107)(cid:104) D x (cid:105) s (cid:48) ( (cid:104) x (cid:105) − α Ω( g )) (cid:107) L ∞ (Σ t ) (cid:107)(cid:104) x (cid:105) α ∂ i g (cid:107) L (Σ t ) + (cid:107)(cid:104) x (cid:105) − α Ω( g ) (cid:107) L ∞ (Σ t ) (cid:107)(cid:104) D x (cid:105) s (cid:48) ( (cid:104) x (cid:105) α ∂ i g ) (cid:107) L (Σ t ) (cid:107)(cid:104) x (cid:105) α ∂ i g (cid:107) L (Σ t ) (cid:46) (cid:107) Ω( g ) (cid:107) W , ∞− α (Σ t ) (cid:107) ∂ i g (cid:107) W , − α (Σ t ) (cid:46) (cid:15) . (9.18)Combining (9.16), (9.17) and (9.18), we obtain (9.15), as desired. We emphasize that this equation is schematic so that the components on the RHS may be different from the componenton the LHS. .3.3 The estimates for second spatial derivatives of g Finally, we control the second spatial derivatives of g . This could be thought of as an L ∞ -endpoint ellipticestimates in Besov space. Notice that the scalar field obeys Besov space estimates in the ( u a , u b ) coordinatesystem, while the elliptic operator that we need to invert is a constant coefficient operator only in the ( x , x ) coordinate system. We will treat this by using the physical space representation of the kernel. Lemma 9.9. For g ∈ { N, β, γ } , ∆ g (c.f. (2.24) – (2.26) ) admits the decomposition ∆ g = (cid:88) ≤ a
The desired estimate relies only on the schematic form of the equation (9.16). After defining thedecomposition in Step 1, we first prove the Besov space estimates in Step 2. The L estimates are simpler,and the decomposition plays no role. This will be carried out in Step 3. Step 1: The decomposition. We now define the decomposition. For definiteness, we put all the metric termsand the quadratic φ reg terms in F (12) g . The other terms require a more precise decomposition:• Ω( g ) ∂ α (cid:101) φ a ∂ β φ reg will be put in F (1 a ) g if a (cid:54) = 1 , and in F ( a g if a = 1 ;• Ω( g ) ∂ α (cid:101) φ a ∂ β (cid:101) φ b will be put in F ( ab ) g if a < b , in F ( ba ) g if b < a , in F (1 a ) g if < a = b , and in F (12) g if a = b = 1 .For concreteness, we explicitly give the decomposition when g = N . In this case, we decompose RHS of(2.24) as F (12) N := e γ N | L β | + 2 N e γ · [( (cid:126)nφ reg ) + ( (cid:126)n (cid:101) φ ) + ( (cid:126)n (cid:101) φ ) ]+ 4 N e γ · { [( (cid:126)n (cid:101) φ ) + ( (cid:126)n (cid:101) φ )] · ( (cid:126)nφ reg ) + ( (cid:126)n (cid:101) φ ) · ( (cid:126)n (cid:101) φ ) } , (9.20) F (13) N := 2 N e γ · ( (cid:126)n (cid:101) φ ) + 4 N e γ · ( (cid:126)n (cid:101) φ ) · [( (cid:126)nφ reg ) + ( (cid:126)n (cid:101) φ )] , (9.21) F (23) N := 4 N e γ · ( (cid:126)n (cid:101) φ ) · ( (cid:126)n (cid:101) φ ) . (9.22) Step 2: The Besov estimates. An important ingredient for the estimate is that the Besov space B u a ,u b ∞ , is analgebra and obeys the estimate (cid:107) f · h (cid:107) B ua,ub ∞ , (cid:46) (cid:107) f (cid:107) B ua,ub ∞ , (cid:107) h (cid:107) B ua,ub ∞ , . (9.23)This is obvious using the definition of B u a ,u b ∞ , and Young’s convolution inequality. Step 2(a): The metric terms. We first bound terms schematically of the form Ω( g ) ∂ i g ∂ j g (which are in F (12) g of the decomposition). Note the standard Sobolev embedding W , ( R ) (cid:44) → B ∞ , ( R ) . Hence, using alsoCorollary 8.6, we have (cid:107) f (cid:107) B ua,ub ∞ , (Σ t ) (cid:46) (cid:107) f (cid:107) W , (Σ t ) . In particular, using (9.8), (7.2a) and (7.2c), it followsthat (for any a (cid:54) = b ) (cid:107)(cid:104) x (cid:105) − α Ω( g ) (cid:107) B ua,ub ∞ , (Σ t ) (cid:46) and (cid:107)(cid:104) x (cid:105) α ∂ x g (cid:107) B ua,ub ∞ , (Σ t ) (cid:46) (cid:15) .Hence, by (9.23), we have, for any a (cid:54) = b (and in particular ( a, b ) = (1 , ), (cid:107) Ω( g ) ∂ i g ∂ j g (cid:107) B ua,ub ∞ , (Σ t ) (cid:46) (cid:107)(cid:104) x (cid:105) − α Ω( g ) (cid:107) B ua,ub ∞ , (Σ t ) (cid:107)(cid:104) x (cid:105) α ∂ x g (cid:107) B ua,ub ∞ , (Σ t ) (cid:46) (cid:15) . (9.24) Step 2(b): The scalar field terms. Since supp( φ ) ⊆ B (0 , R ) , we have Ω( g ) ∂ α φ∂ β φ = (cid:36) Ω( g ) ∂ α φ∂ β φ . Arguingas in Step 2(a), we have (for any a (cid:54) = b ) (cid:107) (cid:36) Ω( g ) (cid:107) B ua,ub ∞ , (Σ t ) (cid:46) . (9.25) Recall the definition of the Besov space in Definition 3.10. 49o proceed, we need to control φ , for which we use the decomposition φ = φ reg + (cid:80) k =1 (cid:101) φ k . The quadraticterm would give the following three types of contributions: ∂φ reg · ∂φ reg (cid:124) (cid:123)(cid:122) (cid:125) =: I , ∂φ reg · ∂ (cid:101) φ a (cid:124) (cid:123)(cid:122) (cid:125) =: II , ∂ (cid:101) φ a · ∂ (cid:101) φ b (cid:124) (cid:123)(cid:122) (cid:125) =: III . Each of these terms has B u a ,u b ∞ , norm (cid:46) (cid:15) after choosing suitable a and b . More precisely, (cid:107) I (cid:107) B ua,ub ∞ , (cid:46) (cid:15) for any ( a, b ) such that a (cid:54) = b (by (9.23) and (7.8a)); (cid:107) II (cid:107) B ua,ub ∞ , (cid:46) (cid:15) for a as in the term and any b (cid:54) = a (by(9.23), (7.8a) and (7.8b)); (cid:107) III (cid:107) B ua,ub ∞ , (cid:46) (cid:15) for a , b as in the term (by (9.23), (7.8a) and (7.8b)).Combining this with (9.25), and using (9.23), it follows that that all the scalar field contributions for F g obey the desired Besov bound. Finally, combining Steps 2(a) and 2(b), we conclude the proof of the Besovbounds in (9.19). Step 3: L estimates. We begin with the estimates for the metric terms. We have more than enoughregularity; the key issue is thus the decay at infinity. Noting that (cid:104) x (cid:105) − − α ∈ L (Σ t ) , we have, by Hölder’sinequality, (7.2a), (7.2b) and (9.8), that (cid:107) Ω( g ) ∂ i g ∂ j g (cid:107) L (Σ t ) (cid:46) (cid:107)(cid:104) x (cid:105) − − α (cid:107) L (Σ t ) (cid:107)(cid:104) x (cid:105) − α Ω( g ) (cid:107) L ∞ (Σ) (cid:107)(cid:104) x (cid:105) + α ∂ x g (cid:107) L ∞ (Σ t ) (cid:46) (cid:15) . Turning to the scalar field terms, we use (9.8), (7.8c), and that supp( φ ) ⊆ B (0 , R ) to obtain (cid:107) Ω( g ) ∂ α φ reg ∂ β φ reg (cid:107) L (Σ t ) , (cid:107) Ω( g ) ∂ α φ reg ∂ β (cid:101) φ a (cid:107) L (Σ t ) , (cid:107) Ω( g ) ∂ α (cid:101) φ a ∂ β (cid:101) φ b (cid:107) L (Σ t ) (cid:46) (cid:107) Ω( g ) (cid:107) L ∞ (Σ t ∩ B (0 ,R )) ( (cid:107) ∂φ reg (cid:107) L ∞ (Σ t ) + (cid:107) ∂ (cid:101) φ k (cid:107) L ∞ (Σ t ) ) (cid:46) (cid:15) . Recalling the decomposition in Step 1, and combining the above estimates, we obtain the desired L bound in (9.19).Using the decomposition in Lemma 9.9, we prove our main elliptic estimate for ∂ ij g : Proposition 9.10. The following estimates hold for all t ∈ [0 , T B ) : (cid:88) g ∈{ N, β l , γ } max i,j (cid:107) ∂ ij g (cid:107) L ∞ (Σ t ) (cid:46) (cid:15) . (9.26) Proof. By Lemma 9.9, each g satisfies a Poisson equation ∆ g = (cid:88) ≤ a cases separately. (See Steps 1 and 2 below.)Before we proceed, note that by (the second term in) (9.19), we also have sup k ≥ (cid:107) P u a ,u b k F ( ab ) g (cid:107) L (Σ t ) (cid:46) (cid:15) . (9.29)50 tep 1: The case k = 0 . This is the easy case. Clearly, (cid:107) P F ( ab ) g (cid:107) L (Σ t ) + (cid:107) ∂ (cid:96) ( P F ( ab ) g ) (cid:107) L (Σ t ) (cid:46) (cid:15) . (9.30)(By definition of P , P F ( ab ) g ∈ W , in the ( u a , u b ) coordinates. (9.30) then follows from Corollary 8.6.)Using the definition (9.28), the bound (9.30), standard L elliptic estimates and then Sobolev embedding,we obtain immediately that max i,j (cid:107) ∂ ij g ( ab )0 (cid:107) L ∞ (Σ t ) (cid:46) (cid:15) . (9.31) Step 2: The case k > .Step 2(a): Extracting information from (9.19) and (9.29) . By (9.19) and the frequency support informationof P u a ,u b k F ( ab ) g , we know that (cid:107) /∂ u a ( P u a ,u b k F ( ab ) g ) (cid:107) L ∞ (Σ t ) + (cid:107) /∂ u b ( P u a ,u b k F ( ab ) g ) (cid:107) L ∞ (Σ t ) (cid:46) k (cid:107) P u a ,u b k F ( ab ) g (cid:107) L ∞ (Σ t ) . (9.32)Hence, by Corollary 8.6, (cid:107) ∂ i ( P u a ,u b k F ( ab ) g ) (cid:107) L ∞ (Σ t ) (cid:46) k (cid:107) P u a ,u b k F ( ab ) g (cid:107) L ∞ (Σ t ) . (9.33)On the other hand, since the Fourier transform of P u a ,u b k F ( ab ) g is by definition supported away from ,we can introduce a partition of unity in the angular Fourier directions to deduce that there exist (cid:101) F ( ab ) g and (cid:101)(cid:101) F ( ab ) g such that P u a ,u b k F ( ab ) g = /∂ u a (cid:101) F ( ab ) g + /∂ u b (cid:101)(cid:101) F ( ab ) g . (9.34)Moreover, the frequency support of P u a ,u b k F ( ab ) g implies that (cid:101) F ( ab ) g and (cid:101)(cid:101) F ( ab ) g can be chosen so that (cid:107) (cid:101) F ( ab ) g (cid:107) L ∞ (Σ t ) + (cid:107) (cid:101)(cid:101) F ( ab ) g (cid:107) L ∞ (Σ t ) (cid:46) − k (cid:107) P u a ,u b k F ( ab ) g (cid:107) L ∞ (Σ t ) (cid:107) (cid:101) F ( ab ) g (cid:107) L (Σ t ) + (cid:107) (cid:101)(cid:101) F ( ab ) g (cid:107) L (Σ t ) (cid:46) − k (cid:15) , (9.35)where we have used Corollary 8.6 (to compare volume forms) and (9.29).We now rewrite /∂ u a (cid:101) F ( ab ) g = (cid:80) (cid:96) =1 { ∂ (cid:96) [( /∂ u a x (cid:96) ) (cid:101) F ( ab ) g ] − [ ∂ (cid:96) ( /∂ u a x (cid:96) )] (cid:101) F ( ab ) g } and similarly for /∂ u b (cid:101)(cid:101) F ( ab ) g . There-fore, using Corollary 8.6 and Proposition 8.7, we deduce from (9.35) that P u a ,u b k F ( ab ) g = (cid:88) (cid:96) =1 , ∂ (cid:96) H ( ab ) g ,k,(cid:96) + (cid:101) H ( ab ) g ,k , (cid:107) H ( ab ) g ,k,(cid:96) (cid:107) L ∞ (Σ t ) (cid:46) − k (cid:107) P u a ,u b k F ( ab ) g (cid:107) L ∞ (Σ t ) , (cid:107) (cid:101) H ( ab ) g ,k (cid:107) L (Σ t ) (cid:46) − k (cid:15) . (9.36) Step 2(b): Estimating the kernel. We now bound (9.28). Differentiating the kernel in Definition 9.3, we have ∂ ij g ( ab ) k = 12 π (cid:90) R δ ij − x − y ) i ( x − y ) j | x − y | | x − y | ( P u a ,u b k F ( ab ) g )( y ) d y = 12 π (cid:90) R ∂ i ( ( x − y ) j | x − y | )( P u a ,u b k F ( ab ) g )( y ) d y. (9.37)We estimate separately the contributions from | x − y | ≤ − k and | x − y | ≥ − k . For | x − y | ≤ − k , weuse the second representation in (9.37), integrate by parts and use (9.33), (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) { y ∈ R : | x − y |≤ − k } ∂ i ( ( x − y ) j | x − y | )( P u a ,u b k F ( ab ) g )( y ) d y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:90) { y ∈ R : | x − y |≤ − k } | x − y | | ∂ i ( P u a ,u b k F ( ab ) g ) | ( y ) d y + (cid:107) P u a ,u b k F ( ab ) g (cid:107) L ∞ (Σ t ) (cid:46) − k (cid:107) ∂ i ( P u a ,u b k F ( ab ) g ) (cid:107) L ∞ (Σ t ) + (cid:107) P u a ,u b k F ( ab ) g (cid:107) L ∞ (Σ t ) (cid:46) (cid:107) P u a ,u b k F ( ab ) g (cid:107) L ∞ (Σ t ) . (9.38)51or | x − y | ≥ − k , we use the first representation in (9.37) together with (9.36). More precisely, we integrateby parts (for the ∂ (cid:96) H ( ab ) g ,k,(cid:96) terms), apply Young’s inequality and use the bounds in (9.36) to obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) { y ∈ R : | x − y |≥ − k } δ ij − x − y ) i ( x − y ) j | x − y | | x − y | ( P u a ,u b k F ( ab ) g )( y ) d y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) { y ∈ R : | x − y |≥ − k } δ ij − x − y ) i ( x − y ) j | x − y | | x − y | ( (cid:88) (cid:96) =1 , ∂ (cid:96) H ( ab ) g ,k,(cid:96) + (cid:101) H ( ab ) g ,k )( y ) d y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:88) (cid:96) =1 , ( (cid:90) { y ∈ R : | x − y |≥ − k } | H ( ab ) g ,k,(cid:96) | ( y ) | x − y | d y + 2 k (cid:107) H ( ab ) g ,k,(cid:96) (cid:107) L ∞ (Σ t ) )+ (cid:90) { y ∈ R : | x − y |≥ − k } | (cid:101) H ( ab ) g ,k | ( y ) | x − y | d y (cid:46) ( (cid:90) { z ∈ R : | z |≥ − k } | z | d z + 2 k )( (cid:88) (cid:96) =1 , (cid:107) H ( ab ) g ,k,(cid:96) (cid:107) L ∞ (Σ t ) ) + ( (cid:90) { z ∈ R : | z |≥ − k } | z | d z ) (cid:107) (cid:101) H ( ab ) g ,k (cid:107) L (Σ t ) (cid:46) k (cid:88) (cid:96) =1 , (cid:107) H ( ab ) g ,k,(cid:96) (cid:107) L ∞ (Σ t ) + 2 k (cid:107) (cid:101) H ( ab ) g ,k (cid:107) L (Σ t ) (cid:46) (cid:107) P u a ,u b k F ( ab ) g (cid:107) L ∞ (Σ t ) + 2 − k (cid:15) . (9.39)Combining (9.37), (9.38) and (9.39), and then using (9.19), we obtain max i,j (cid:88) k ≥ (cid:107) ∂ ij g ( ab ) k (cid:107) L ∞ (Σ) (cid:46) (cid:88) k ≥ ( (cid:107) P u a ,u b k F ( ab ) g (cid:107) L ∞ (Σ t ) + 2 − k (cid:15) ) (cid:46) (cid:15) . (9.40)Finally, combining (9.40) with (9.31), we obtain (cid:107) ∂ ij g (cid:107) L ∞ (Σ) (cid:46) max i,j,a,b (cid:88) k ≥ (cid:107) ∂ ij g ( ab ) k (cid:107) L ∞ (Σ) (cid:46) (cid:15) . (9.41)We now improve Proposition 9.10 to obtain some decaying weights at infinity. This is much easier givenProposition 9.10 since we only need to improve the weights in regions away from the support of φ . Proposition 9.11. The following estimates hold for all t ∈ [0 , T B ) : (cid:88) g ∈{ N, β l , γ } max i,j (cid:107) ∂ ij g (cid:107) L ∞ − α (Σ t ) (cid:46) (cid:15) . (9.42) Proof. Using Proposition 9.10, we only need a bound when | x | ≥ R . By Definition 9.6, it thus suffices tobound | ∂ ij ((1 − (cid:36) ) g ) | . In fact, using Sobolev embedding (1(a) of Proposition 9.1), it in turn suffices to show max i,j (cid:107) ∂ ij (1 − (cid:36) ) g (cid:107) L − α (Σ t ) + max i,j,l (cid:107) ∂ ijl ((1 − (cid:36) ) g ) (cid:107) L − α (Σ t ) (cid:46) (cid:15) . (9.43)The key is now to derive an equation ∆ ∂ l ((1 − (cid:36) ) g ) , and use the fact supp( φ ) ⊆ B (0 , R ) , which guaranteesthat there is no scalar field contribution after multiplying by the cutoff − (cid:36) .We now commute the derivatives with the cutoff. Notice that when at least one derivative falls on (cid:36) , wecan put as much (cid:104) x (cid:105) − weights as we need. Thus, we obtain the pointwise bound | ∆ ∂ l ((1 − (cid:36) ) g ) | (cid:46) (cid:104) x (cid:105) − ( | g | + | ∂ x g | + | ∂ x ∂ x g | ) + (1 − (cid:36) ) | ∆ ∂ l g | . (9.44)Using the estimates in (9.9), it follows that (cid:88) g ∈{ γ,β i ,N } (cid:107)(cid:104) x (cid:105) − ( | g | + | ∂ x g | + | ∂ x ∂ x g | ) (cid:107) L − α (Σ t ) (cid:46) (cid:15) . (9.45)52e now consider (1 − (cid:36) ) | ∆ ∂ l g | . For this, we recall (2.24)–(2.26). Notice that the scalar field terms drop outsince supp( φ ) ∩ supp(1 − (cid:36) ) = ∅ . Hence, we only need to control the derivatives of e γ N | L β | , ∂ x γ∂ x β and ∂ x N∂ x βN . These terms can be controlled in a similar manner as (9.12), (9.13) and (9.14), except that since wehave an additional ∂ l derivative, we control these terms also using (9.9), and get an additional (cid:104) x (cid:105) − weight.In other words, (cid:88) g ∈{ γ,β i ,N } (cid:107) (1 − (cid:36) )∆ ∂ l g (cid:107) L − α (Σ t ) (cid:46) (cid:107) ∂ l ( e γ N | L β | ) (cid:107) L − α (Σ t ) + (cid:107) ∂ l ( ∂ x γ∂ x β ) (cid:107) L − α (Σ t ) + (cid:107) ∂ l ( ∂ x N ∂ x βN ) (cid:107) L − α (Σ t ) (cid:46) (cid:15) . (9.46)Notice that we also have (cid:82) Σ t ∆ ∂ i ((1 − (cid:36) ) g ) dx dx = 0 . (This can be proven by noting that ∆ ∂ i ((1 − (cid:36) ) g ) is an exact divergence, and then using the compact support of φ together with the x -decay given by (9.10).)Hence, by Proposition 9.4 (with p = 4 , σ = − α ) and the estimates in (9.44), (9.45) and (9.46), we obtain (cid:107) ∂ x ((1 − (cid:36) ) g ) (cid:107) W , − α (Σ t ) (cid:46) (cid:15) . (9.47)In particular, (9.47) implies (9.43). ∂ t g We now turn to the elliptic estimates for ∂ t g and its spatial derivatives. The main estimate will be provenin Proposition 9.19. These estimates should be compared to Proposition 9.7 and 9.11. Notice however, thatthe estimate is weak: while we control ∂ ij g in a (weighted) L ∞ (Σ t ) space, we only prove that ∂ it g belongsto a (weighted) L s (cid:48)− s (cid:48)(cid:48) space .As we explained in Section 1.1.4 in the introduction, the main difficulty is that after differentiating theelliptic equations (2.24)–(2.26), we have terms the involve second derivatives of (cid:101) φ k , which by (7.5b) mayappear to have L (Σ t ) norm of size δ − . These terms in particular require a careful analysis using thetransversality of the different waves. We will first control the inhomogeneous terms in the elliptic equationsfor ∂ t g : in Section 9.4.1 , we carry out the more straightforward bounds, and the estimates correspondingto the interaction of the different waves are proven in Section 9.4.2 . The main weighted W , s (cid:48)− s (cid:48)(cid:48) estimateswill then be proven in Section 9.4.3 . Let Ω( g ) be as in (9.8) . Then the following estimate holds for all t ∈ [0 , T B ) : (cid:107) ∂ t [Ω( g )( ∂ α φ reg )( ∂ σ φ reg )] (cid:107) L (Σ t ) (cid:46) (cid:15) . Proof. This follows from the Hölder inequality, (9.8), compact support of φ reg and the bootstrap assumptions(7.4) and (7.8c). Lemma 9.13. Let Ω( g ) be as in (9.8) . Then (cid:107) ∂ t [Ω( g )( ∂ α (cid:101) φ k )( ∂ σ (cid:101) φ k )] − ∂ i [Ω( g )( N X ik + β i )( ∂ α (cid:101) φ k )( ∂ σ (cid:101) φ k )] (cid:107) L (Σ t ) (cid:46) (cid:15) , (9.48) and (cid:107) ∂ t [Ω( g )( ∂ α (cid:101) φ k )( ∂ σ φ reg )] − ∂ i [Ω( g )( N X ik + β i )( ∂ α (cid:101) φ k )( ∂ σ φ reg )] (cid:107) L (Σ t ) (cid:46) (cid:15) , (9.49) Proof. Step 1: Proof of (9.48) . By (2.40) and (2.4), L k = (cid:126)n − X k , (cid:126)n = N ( ∂ t − β i ∂ i ) . Therefore, we candecompose ∂ t as follows: ∂ t = N(cid:126)n + β i ∂ i = N L k + ( N X ik + β i ) ∂ i . (9.50) In fact, one can replace s (cid:48) − s (cid:48)(cid:48) with an arbitrarily large p < ∞ (as long as one takes (cid:15) smaller). (Note, however, that ourargument does not give an estimate for p = ∞ .) The particular estimate we prove here is sufficient for our later applications. ∂ t [Ω( g )( ∂ α (cid:101) φ k )( ∂ σ (cid:101) φ k )] − ∂ i [Ω( g )( N X ik + β i )( ∂ α (cid:101) φ k )( ∂ σ (cid:101) φ k )]= N L k [Ω( g )( ∂ α (cid:101) φ k )( ∂ σ (cid:101) φ k )] (cid:124) (cid:123)(cid:122) (cid:125) =: I − { ∂ i [( N X ik + β i )] } Ω( g )( ∂ α (cid:101) φ k )( ∂ σ (cid:101) φ k ) (cid:124) (cid:123)(cid:122) (cid:125) =: II . (9.51)To estimate term I in (9.51), we use supp( (cid:101) φ k ) ⊆ B (0 , R ) , (9.8), Lemma 8.2, and the bootstrap assump-tions (7.2b), (7.5a) and (7.8c) to obtain (cid:107) I (cid:107) L (Σ t ) (cid:46) (cid:107) N L k (Ω( g )) (cid:107) L ∞ (Σ t ∩ B (0 ,R )) (cid:107) ∂ (cid:101) φ k (cid:107) L ∞ (Σ t ) + (cid:107) N Ω( g ) (cid:107) L ∞ (Σ t ∩ B (0 ,R ) (cid:107) L k ∂ (cid:101) φ k (cid:107) L (Σ t ) (cid:107) ∂ (cid:101) φ k (cid:107) L ∞ (Σ t ) (cid:46) (cid:15) . (9.52)The term II can be treated similarly. Using supp( (cid:101) φ k ) ⊆ B (0 , R ) , (9.8), Lemmas 8.2, 8.4 and the bootstrapassumptions (7.2b) and (7.8c), we obtain (cid:107) II (cid:107) L (Σ t ) (cid:46) (cid:107) Ω( g ) ∂ i [( N X ik + β i )] }(cid:107) L ∞ (Σ t ∩ B (0 ,R )) (cid:107) ∂ (cid:101) φ k (cid:107) L ∞ (Σ t ) (cid:46) (cid:15) . (9.53)Combining (9.51)–(9.53) yields (9.48). Step 2: Proof of (9.49) . The estimate (9.49) can be proven in a similar manner. The main only difference isthat (cid:107) N L k [Ω( g )( ∂ α (cid:101) φ k )( ∂ σ φ reg )] (cid:107) L (Σ t ) (cid:46) (cid:15) (c.f. term I in (9.51)) has to be proved slightly differently andwe use additionally the bootstrap assumption (7.4) for φ reg . The rest of the argument proceeds similarly. Lemma 9.14. The following estimate holds for all t ∈ [0 , T B ) : (cid:107) ∂ t { e γ N | L β | }(cid:107) L s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α +2 (Σ t ) (cid:46) (cid:15) + (cid:15) (cid:88) (cid:101) g ∈{ (cid:101) γ,β i , (cid:101) N } (cid:107) ∂ t (cid:101) g (cid:107) W , s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) , (9.54) (cid:107) ∂ t { ∂ i (log( N e − γ ))( L β ) jl }(cid:107) L s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α +2 (Σ t ) (cid:46) (cid:15) + (cid:15) (cid:88) (cid:101) g ∈{ (cid:101) γ,β i , (cid:101) N } (cid:107) ∂ t (cid:101) g (cid:107) W , s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) . (9.55) Proof. Step 1: Preliminaries. Note that ∂ t γ asymp = 0 (since γ asymp is a constant; see (7.1)), and thus both ∂ t γ and ∂ t β i does not have a logarithmic growing contribution. Hence, (cid:88) g ∈{ γ,β i } (cid:107) ∂ t ∂ j g (cid:107) L s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α +1 (Σ t ) (cid:46) (cid:88) (cid:101) g ∈{ (cid:101) γ,β i , (cid:101) N } (cid:107) ∂ t (cid:101) g (cid:107) W , s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) . (9.56)For ∂ t N , the logarithmic terms give a worse decay for large (cid:104) x (cid:105) , but after using (7.2a), we still have (cid:107) ∂ t ∂ i N (cid:107) L s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) (cid:46) (cid:107) ∂ t ∂ i (cid:101) N (cid:107) L s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) + | ∂ t N asymp | ( t ) (cid:107)(cid:104) x (cid:105) − (cid:107) L s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) (cid:46) (cid:107) ∂ t (cid:101) N (cid:107) W , s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) + (cid:15) . (9.57)Additionally, Proposition 9.1, (7.2a) and (7.2b) imply that (cid:88) g ∈{ γ,β i ,N } (cid:107) ∂ t g (cid:107) L s (cid:48)− s (cid:48)(cid:48)− − s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) (cid:46) (cid:88) g ∈{ (cid:101) γ,β i , (cid:101) N } (cid:107) ∂ t (cid:101) g (cid:107) L ∞ (Σ t ) + (cid:15) (cid:107) log(2 + | x | ) (cid:107) L s (cid:48)− s (cid:48)(cid:48)− − s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) (cid:46) (cid:15) . (9.58) Step 2: Proof of (9.54) . Note that because of the signs of γ asymp and N asymp , the factors e γ and N are54avorable in terms of the (cid:104) x (cid:105) weights. Therefore, by Hölder’s inequality, (9.56), (9.58) and (7.2b), we obtain (cid:107) ∂ t { e γ N | L β | }(cid:107) L s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α +2 (Σ t ) (cid:46) (cid:107) ∂ t L β (cid:107) L s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α +1 (Σ t ) (cid:107) L β (cid:107) L ∞ (Σ t ) + (cid:107) ∂ t ( e γ N ) (cid:107) L s (cid:48)− s (cid:48)(cid:48)− − s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) (cid:107) L β (cid:107) L ∞ (Σ t ) (cid:46) (cid:107) ∂ t β (cid:107) W , s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) (cid:107) β (cid:107) W , ∞ (Σ t ) + max g ∈{ γ,N } (cid:107) ∂ t g (cid:107) L s (cid:48)− s (cid:48)(cid:48)− − s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) (cid:107) β (cid:107) W , ∞ (Σ t ) (cid:46) (cid:15) + (cid:15) (cid:88) (cid:101) g ∈{ (cid:101) γ,β i , (cid:101) N } (cid:107) ∂ t (cid:101) g (cid:107) W , s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) . Step 3: Proof of (9.55) . This is similar to (9.54), except we use also (9.57) and have less room in the weightsin the ( ∂ i ∂ t N )( L β ) jl and ( ∂ i N )( ∂ t N )( L β ) jl terms. More precisely, (cid:107) ∂ t { ∂ i (log( N e − γ ))( L β ) jl }(cid:107) L s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α +2 (Σ t ) (cid:46) ( (cid:107) ∂ i N (cid:107) L ∞ + (cid:107) ∂ i γ (cid:107) L ∞ (Σ t ) ) (cid:107) ∂ t L β (cid:107) L s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α +1 (Σ t ) + (cid:107) ∂ i ∂ t N (cid:107) L s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) (cid:107) L β (cid:107) L ∞ − α (Σ t ) + (cid:107) ∂ t N (cid:107) L s (cid:48)− s (cid:48)(cid:48)− − s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) (cid:107) ∂ i N (cid:107) L ∞ (Σ t ) (cid:107) L β (cid:107) L ∞ − α (Σ t ) + (cid:107) ∂ i ∂ t γ (cid:107) L s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α +1 (Σ t ) (cid:107) L β (cid:107) L ∞ (Σ t ) (cid:46) (cid:15) + (cid:15) (cid:88) (cid:101) g ∈{ (cid:101) γ,β i , (cid:101) N } (cid:107) ∂ t (cid:101) g (cid:107) W , s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) . We now analyze the contribution coming from two different waves, say (cid:101) φ k and (cid:101) φ j (for k (cid:54) = j ). Before we proveour main estimate (Lemma 9.18), we need some preliminary observations making use of the transversalityof the two singular zones (see Lemmas 9.15 and 9.16).Given k (cid:54) = j , we now construct a polar coordinate system. Let p ∈ Σ t be the point correspondingto u k = u j = 0 , and let z = ( z , z ) be the elliptic gauge coordinates of the point p . Introduce the polarcoordinates ( r, ϑ ) be the polar coordinates corresponding to the elliptic gauge coordinate system centered at ( z , z ) , with (recall our convention ( c ⊥ k , c ⊥ k ) = ( − c k , c k ) ) x − z = r (cos ϑ (cid:20) c ⊥ k c ⊥ k (cid:21) + sin ϑ (cid:20) c k c k (cid:21) ) . (9.59)In particular, ( r = 1 , ϑ = 0) corresponds to x = z + (cid:20) c ⊥ k c ⊥ k (cid:21) in elliptic gauge coordinates. Using moreover(8.12), one sees that { ( r, ϑ ) : ϑ = 0 } is an approximation of the curve { x : u k ( t, x ) = 0 } .Define ϑ ∈ ( − π, π ) so that ( r = 1 , ϑ = ϑ ) corresponds to z + (cid:20) c ⊥ j c ⊥ j (cid:21) in elliptic gauge coordinates (recallthe (cid:20) c ⊥ j c ⊥ j (cid:21) has unit length by (2.30)). In other words, we impose (cid:20) c ⊥ j c ⊥ j (cid:21) = cos ϑ (cid:20) c ⊥ k c ⊥ k (cid:21) + sin ϑ (cid:20) c k c k (cid:21) . (9.60)Note that (2.30) implies (cid:12)(cid:12)(cid:12)(cid:12) det (cid:20) c k c ⊥ k c k c ⊥ k (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) = 1 . Hence, combining this with (9.60) and using (2.31), we obtain | sin ϑ | = (cid:12)(cid:12)(cid:12)(cid:12) det (cid:20) ϑ ϑ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) det (cid:20) c k c ⊥ k c k c ⊥ k (cid:21) (cid:20) ϑ ϑ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) det (cid:20) − c k − c j c k c j (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≥ κ . (9.61) Note that such a point is indeed uniquely defined since ( u k , u j ) forms a coordinate system. (cid:101) φ k is localized in the regionwhere sin ϑ for the above polar coordinates system. Lemma 9.15. For (cid:15) > sufficiently small, u k / ∈ ( − δ, δ ) in the set { ( r, ϑ ) : r ≥ κ − δ, | sin ϑ | ≥ κ } .Proof. Take a point y ∈ Σ t such that its ( r, ϑ ) coordinates satisfy r ≥ κ − δ and | sin ϑ | ≥ κ . We want toshow that | u k ( y ) | ≥ δ . To this end, we integrate along the radial line γ : [0 , → Σ t (connecting z and y )given by γ ( s ) = z + sr (cos ϑ (cid:20) c ⊥ k c ⊥ k (cid:21) + sin ϑ (cid:20) c k c k (cid:21) ) . To proceed, we use the fundamental theorem of calculus and (8.12) to obtain u k ( y ) = u k ( γ (1)) = u k ( γ (1)) − u k ( γ (0)) = (cid:90) dds u k ( γ ( s )) ds = (cid:90) { r [ − (cos ϑ ) c k + (sin ϑ ) c k ]( ∂ u k )( γ ( s )) + r [(cos ϑ ) c k + (sin ϑ ) c k ]( ∂ u k )( γ ( s )) } d s = r (cid:90) (sin ϑ )( c k + c k ) d s + O ( (cid:15) r ) = r sin ϑ + O ( (cid:15) r ) , (9.62)since c k + c k = 1 by (2.30). By (9.62), it is clear that since r ≥ κ − δ and | sin ϑ | ≥ κ , if we choose (cid:15) tobe sufficiently small, then | u k ( y ) | ≥ r | sin ϑ | ≥ · κ − δ · κ = δ , as desired.The following lemma is related to Lemma 9.15, but adapted for u j . Lemma 9.16. For (cid:15) > sufficiently small, u j / ∈ ( − δ, δ ) in the set { ( r, ϑ ) : r ≥ κ − δ, | sin ϑ | ≤ κ } .Proof. In an entirely analogous manner as Lemma 9.15, we can show thatif r ≥ κ − δ and | sin( ϑ − ϑ ) | ≥ κ , then u j / ∈ ( − δ, δ ) . (9.63)Now, given a point ( r, ϑ ) ∈ { ( r, ϑ ) : r ≥ κ − δ, | sin ϑ | ≤ κ } , we know that• | sin ϑ cos ϑ | ≤ | sin ϑ | ≤ κ , and• | sin ϑ cos ϑ | ≥ κ (using (9.61) and the fact | sin ϑ | ≤ κ = ⇒ | sin ϑ | ≤ = ⇒ | cos ϑ | ≥ ).Therefore, | sin( ϑ − ϑ ) | = | sin ϑ cos ϑ − sin ϑ cos ϑ | ≥ κ . Consequently, it follows from (9.63) that u j / ∈ ( − δ, δ ) at the given point.Before we control the interaction terms, we need one more simple lemma. Lemma 9.17. For any k and any k (cid:48) (cid:54) = k , the following estimate holds for all t ∈ [0 , T B ) : (cid:107) ∂ (cid:101) φ k (cid:107) L uk L ∞ uk (cid:48) (Σ t ) (cid:46) (cid:15) · δ − . Proof. Using the wave equation if necessary, we only need to estimate ∂∂ x (cid:101) φ k . Using the -dimensionalSobolev embedding, we have (cid:107) ∂∂ x (cid:101) φ k (cid:107) L uk L ∞ uk (cid:48) (Σ t ) (cid:46) (cid:107) /∂ u k (cid:48) ∂∂ x (cid:101) φ k (cid:107) L (Σ t ) (cid:46) (cid:107) ∂E k ∂ x (cid:101) φ k (cid:107) L (Σ t ) (cid:46) (cid:15) · δ − , where we have used (2.61), (7.3d), (7.2a), (7.2b), Lemma 8.2 and (8.13) to compare /∂ u k (cid:48) and E k , and usedLemma 8.4 to commute [ ∂, E k ] . Finally, we apply the bootstrap assumption (7.5d).We are now ready to prove the main estimate for the interaction terms. Here, z is as defined above before the lemma, which corresponds to the center for the polar coordinates. emma 9.18. Let Ω( g ) be as in (9.8) . Then, for k (cid:54) = j , there exist t -independent functions (cid:101) ζ int and (cid:101) ζ ang with L ∞ norms (cid:46) (defined precisely in the proof ) such that for any p ∈ [1 , , the following estimate holdsfor all t ∈ [0 , T B ) : (cid:107) ∂ t [Ω( g )( ∂ α (cid:101) φ k )( ∂ σ (cid:101) φ j )] − ∂ i [(1 − (cid:101) ζ int ) (cid:101) ζ ang Ω( g )( N X ik + β i )( ∂ α (cid:101) φ k )( ∂ σ (cid:101) φ j )] − ∂ i [(1 − (cid:101) ζ int )(1 − (cid:101) ζ ang )Ω( g )( N X ij + β i )( ∂ α (cid:101) φ k )( ∂ σ (cid:101) φ j )] (cid:107) L p (Σ t ) (cid:46) (cid:15) (2 − p ) p . Proof. Step 1: Defining the decomposition. Recall the polar coordinates ( r, ϑ ) in (9.59). We introduce twocut-off functions. First, define a radial cut-off function (cid:101) ζ int = (cid:101) ζ int ( r ) be a non-negative function which = 1 when r ≤ κ − δ and = 0 when r ≥ κ − δ . (cid:101) ζ int can chosen so that | (cid:101) ζ int | (cid:46) , | ∂ i (cid:101) ζ int | (cid:46) δ − . (9.64)Second, define an angular cut-off function (cid:101) ζ ang = (cid:101) ζ ang ( ϑ ) to be a non-negative function, smooth in ϑ , which = 1 when | sin ϑ | ≤ κ and = 0 when | sin ϑ | ≥ κ . Note that while the derivatives of (cid:101) ζ ang with respect to ϑ are δ -independent, the derivative ∂ i ϑ is unbounded and obeys only | ∂ i ϑ | (cid:46) r . As a result, (cid:101) ζ ang can only bechosen to obey the following bounds: | (cid:101) ζ ang | (cid:46) , | ∂ i (cid:101) ζ ang | ( x ) (cid:46) | x − z | . (9.65)Using the above cutoffs and (9.50), for I := Ω( g )( ∂ α (cid:101) φ k )( ∂ σ (cid:101) φ j ) , we can rewrite ∂ t I = (cid:101) ζ int ∂ t I + (1 − (cid:101) ζ int ) (cid:101) ζ ang ∂ t I + (1 − (cid:101) ζ int )(1 − (cid:101) ζ ang ) ∂ t I = (cid:101) ζ int ∂ t I (cid:124) (cid:123)(cid:122) (cid:125) =: I + (1 − (cid:101) ζ int ) (cid:101) ζ ang ( N L k + ( N X ik + β i ) ∂ i ) I (cid:124) (cid:123)(cid:122) (cid:125) =: II + (1 − (cid:101) ζ int )(1 − (cid:101) ζ ang )( N L j + ( N X ij + β i ) ∂ i ) I (cid:124) (cid:123)(cid:122) (cid:125) =: III . (9.66)In the following steps, we consider each of terms I , II and III . Step 2: The region near the interaction zone (Term I in (9.66) ). The key here is to use the smallness of theinteraction zone. We have (cid:107) I (cid:107) L (Σ t ) = (cid:107) (cid:101) ζ int ∂ t [Ω( g )( ∂ α (cid:101) φ k )( ∂ σ (cid:101) φ j )] (cid:107) L (Σ t ) (cid:46) (cid:107) ∂ t Ω( g ) (cid:107) L ∞ (Σ t ∩ B (0 ,R )) (cid:107) ∂ (cid:101) φ k (cid:107) L ∞ (Σ t ) (cid:107) ∂ (cid:101) φ j (cid:107) L ∞ (Σ t ) + (cid:107) ∂ (cid:101) φ k (cid:107) L (Σ t ∩{ x : | x − z | (cid:46) δ } ) (cid:107) ∂ (cid:101) φ j (cid:107) L ∞ (Σ t ) + (cid:107) ∂ (cid:101) φ k (cid:107) L ∞ (Σ t ) (cid:107) ∂ (cid:101) φ j (cid:107) L (Σ t ∩{ x : | x − z | (cid:46) δ } ) , (9.67)where we have used that supp( (cid:101) ζ int ) ⊆ { x : | x − z | (cid:46) δ } .The first term in (9.67) is obviously (cid:46) (cid:15) using (9.8) and the bootstrap assumption (7.8c).For the second term in (9.67), we start by noting that by Corollary 8.6, ( sup y, y (cid:48) ∈{ x : | x − z | (cid:46) δ } | u k ( y ) − u k ( y (cid:48) ) | ) + ( sup y, y (cid:48) ∈{ x : | x − z | (cid:46) δ } | u j ( y ) − u j ( y (cid:48) ) | ) (cid:46) δ. As a result, by Corollary 8.6, Hölder’s inequality and Lemma 9.17, we have (cid:107) ∂ (cid:101) φ k (cid:107) L (Σ t ∩{ x : | x − z | (cid:46) δ } ) (cid:46) (cid:107) ∂ (cid:101) φ k (cid:107) L uk L ∞ uj (Σ t ) (cid:107) (cid:107) L ∞ uk L uj ( { x : | x − z | (cid:46) δ } ) (cid:46) ( (cid:15) δ − ) δ = (cid:15) . In particular, using also the bootstrap assumption (7.8c), we obtain (cid:107) ∂ (cid:101) φ k (cid:107) L (Σ t ∩{ x : | x − z | (cid:46) δ } ) (cid:107) ∂ (cid:101) φ j (cid:107) L ∞ (Σ t ) (cid:46) (cid:15) . The third term in (9.67) can be treated similarly as the second term so that altogether we have (cid:107) I (cid:107) L (Σ t ) = (cid:107) (cid:101) ζ int ∂ t [Ω( g )( ∂ α (cid:101) φ k )( ∂ σ (cid:101) φ j )] (cid:107) L (Σ t ) (cid:46) (cid:15) . (9.68)57 tep 3: The remaining region (Terms II and III in (9.66) ). We first consider term II of (9.66). Thekey observation is that by Lemma 9.16, on the support of (1 − (cid:101) ζ int ) (cid:101) ζ ang , u j / ∈ [ − δ, δ ] . As a result, we have (cid:107) (1 − (cid:101) ζ int ) (cid:101) ζ ang ∂ (cid:101) φ j (cid:107) L (Σ t ) (cid:46) (cid:15) by (7.6b).We now move onto the details. We write II = (1 − (cid:101) ζ int ) (cid:101) ζ ang ( N L k + ( N X ik + β i ) ∂ i )[Ω( g )( ∂ α (cid:101) φ k )( ∂ σ (cid:101) φ j )]= (1 − (cid:101) ζ int ) (cid:101) ζ ang N L k [ · · · ] (cid:124) (cid:123)(cid:122) (cid:125) =: II + (1 − (cid:101) ζ int ) (cid:101) ζ ang ( N X ik + β i ) ∂ i [ · · · ] (cid:124) (cid:123)(cid:122) (cid:125) =: II . (9.69)For II in (9.69), we compute II = (1 − (cid:101) ζ int ) (cid:101) ζ ang N { ( L k Ω( g ))( ∂ α (cid:101) φ k )( ∂ σ (cid:101) φ j ) + Ω( g )( L k ∂ α (cid:101) φ k )( ∂ σ (cid:101) φ j ) } (cid:124) (cid:123)(cid:122) (cid:125) =: II , + (1 − (cid:101) ζ int ) (cid:101) ζ ang N Ω( g )( ∂ α (cid:101) φ k )( L k ∂ σ (cid:101) φ j ) (cid:124) (cid:123)(cid:122) (cid:125) =: II , . (9.70)The term II , is easy, particularly because L k is a regular vector field for (cid:101) φ k . More precisely, using (9.8),(7.5a) and (7.8c), we obtain (cid:107) II , (cid:107) L (Σ t ) (cid:46) (cid:15) .For II , in (9.70), the key is that Lemma 9.16 implies that supp( II , ) ⊆ Σ t \ S jδ . Therefore, we use(9.8), (7.6b) and (7.8c) to obtain (cid:107) II , (cid:107) L (Σ t ) (cid:46) (cid:15) .For II in (9.69), we compute II = ∂ i [(1 − (cid:101) ζ int ) (cid:101) ζ ang ( N X ik + β i )Ω( g )( ∂ α (cid:101) φ k )( ∂ σ (cid:101) φ j )] (cid:124) (cid:123)(cid:122) (cid:125) =: II , − (1 − (cid:101) ζ int ) (cid:101) ζ ang [ ∂ i ( N X ik + β i )]Ω( g )( ∂ α (cid:101) φ k )( ∂ σ (cid:101) φ j ) (cid:124) (cid:123)(cid:122) (cid:125) =: II , + ( ∂ i (cid:101) ζ int ) (cid:101) ζ ang ( N X ik + β i )Ω( g )( ∂ α (cid:101) φ k )( ∂ σ (cid:101) φ j ) (cid:124) (cid:123)(cid:122) (cid:125) =: II , − (1 − (cid:101) ζ int )( ∂ i (cid:101) ζ ang )( N X ik + β i )Ω( g )( ∂ α (cid:101) φ k )( ∂ σ (cid:101) φ j ) (cid:124) (cid:123)(cid:122) (cid:125) =: II , . (9.71) II , is one of the main terms we have in the statement of the lemma. II , can be handled just asterm II in (9.51) so that (cid:107) II , (cid:107) L (Σ t ) (cid:46) (cid:15) by (9.53). The term II , has L ∞ norm (cid:46) (cid:15) δ − (using (7.2b),(7.8c), Lemma 8.2, (9.8), (9.64) and (9.65)), but ∂ i (cid:101) ζ int is supported in {| x − z | (cid:46) δ } . Thus, using Hölder’sinequality, (cid:107) II , (cid:107) L (Σ t ) (cid:46) (cid:107) II , (cid:107) L ∞ (Σ t ) (cid:107) (cid:107) L (Σ t ∩{| x − z | (cid:46) δ } ) (cid:46) (cid:15) δ − ( δ ) (cid:46) (cid:15) . Now II , is compactly supported in B (0 , R ) , and is bounded in L ∞ above by (cid:46) (cid:15) | x − z | (by (7.2b), (7.8c),Lemma 8.2, (9.8), (9.64) and (9.65)). It follows that for p ∈ [1 , , (cid:107) II , (cid:107) L p (Σ t ) (cid:46) (cid:15) ( (cid:90) B (0 ,R ) d x | x − z | p ) p (cid:46) (cid:15) (2 − p ) p . Putting all the estimates above (using also that the L norm controls the L p norm since the support ofeach term ⊆ B (0 , R ) ), we obtain that for every p ∈ [1 . , (cid:107) II − ∂ i [(1 − (cid:101) ζ int ) (cid:101) ζ ang ( N X ik + β i )Ω( g )( ∂ α (cid:101) φ k )( ∂ σ (cid:101) φ j )] (cid:107) L p (Σ t ) (cid:46) (cid:15) (2 − p ) p . (9.72)Finally, term III in (9.66) can be treated in a similar way as term II , except we use Lemma 9.15 insteadof Lemma 9.16. Hence, we have (cid:107) III − ∂ i [(1 − (cid:101) ζ int )(1 − (cid:101) ζ ang )Ω( g )( N X ij + β i )( ∂ α (cid:101) φ k )( ∂ σ (cid:101) φ j )] (cid:107) L p (Σ t ) (cid:46) (cid:15) (2 − p ) p ; (9.73)we omit the details.Combining (9.66), (9.68), (9.72) and (9.73) yields the lemma. Note that | x − z | / ∈ L loc in two dimensions. .4.3 The main weighted W , s (cid:48)− s (cid:48)(cid:48) estimates for ∂ t g Proposition 9.19. Decomposing γ, β i , N as in (7.1) , the following estimates hold for all t ∈ [0 , T B ) : | ∂ t N asymp | ( t ) + (cid:88) (cid:101) g ∈{ (cid:101) γ,β i , (cid:101) N } (cid:107) ∂ t (cid:101) g (cid:107) W , s (cid:48)− s (cid:48)(cid:48) − s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) (cid:46) (cid:15) . (9.74) Using also Proposition 9.1, it follows moreover that | ∂ t N asymp | ( t ) + (cid:88) (cid:101) g ∈{ (cid:101) γ,β i , (cid:101) N } (cid:107) ∂ t (cid:101) g (cid:107) L ∞ − α (Σ t ) (cid:46) (cid:15) . (9.75) Proof. The fact that γ, β i , N admit the decomposition (7.1), and that γ asymp being a constant (and hence ∂ t γ asymp = 0 ), is again a consequence of the local existence result in [35, Theorem 5.4]. From now on, wefocus on deriving the estimates using (2.24)–(2.26). Step 1: Decomposition of ∆ ∂ t g . Differentiating (2.24)–(2.26) by ∂ t , we obtain, for g ∈ { γ, β i , N } , ∆ ∂ t g = G g + T g , (9.76)where G g are the metric terms, given explicitly by G γ := − ∂ t [ e γ N | L β | ] , G N := ∂ t [ e γ N | L β | ] , G β j := 2 ∂ t [ δ ik δ jl ∂ k (log( N e − γ ))( L β ) il ]; (9.77)and, for any g , T g takes the schematic form T g = ∂ t { Ω( g ) ∂ α φ∂ σ φ } .By (9.7) and (9.76), ∂ t g = ∆ − ( G g + T g ) . (9.78) Step 1(a): The metric term G g in (9.78) . By Lemma 9.14, the terms in (9.77) can be bounded as follows: (cid:88) g ∈{ γ,β i ,N } (cid:107) G g (cid:107) L s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α +2 (Σ t ) (cid:46) (cid:15) + (cid:15) (cid:88) (cid:101) g ∈{ (cid:101) γ,β i , (cid:101) N } (cid:107) ∂ t (cid:101) g (cid:107) W , s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) . (9.79) Step 1(b): The scalar field term T g in (9.78) . Expanding T g = ∂ t { Ω( g ) ∂ α φ∂ σ φ } = ∂ t { Ω( g )( ∂ α φ reg + (cid:88) k =1 ∂ α (cid:101) φ k )( ∂ σ φ reg + (cid:88) (cid:96) =1 ∂ σ (cid:101) φ (cid:96) } , and using Lemmas 9.12, 9.13 and 9.18, we obtain a decomposition T g = F g + ∂ i H i g , (9.80)where F g and H i g ( i = 1 , ) are smooth and compactly supported in B (0 , R ) (for each t ) and (cid:107) F g (cid:107) L s (cid:48)− s (cid:48)(cid:48) (Σ t ) (cid:46) (cid:15) , (cid:107) H i g (cid:107) L ∞ (Σ t ) (cid:46) (cid:15) . (9.81) Step 2: Bounding ∂ t N asymp . By (9.78) and Proposition 9.4, ∂ t N asymp = π (cid:82) Σ t ( F N + G N + ∂ i H iN ) dx = π (cid:82) Σ t ( F N + G N ) dx (since supp( H iN ) ⊆ B (0 , R ) ). Hence, by part 2 of Proposition 9.1, (9.79) and (9.81), | ∂ t N asymp | ( t ) = (cid:12)(cid:12)(cid:12)(cid:12) π (cid:90) Σ t ( F N + G N ) dx (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:107) F N (cid:107) L s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48) + α +1 (Σ t ) + (cid:107) G N (cid:107) L s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48) + α +1 (Σ t ) (cid:46) (cid:15) + (cid:15) (cid:88) (cid:101) g ∈{ (cid:101) γ,β i , (cid:101) N } (cid:107) ∂ t (cid:101) g (cid:107) W , s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) . (9.82)59 tep 3: Bounding ∆ − ( F g + G g ) . Using the obvious notation ∂ t g asymp = ∂ t N asymp for g = N , and ∂ t g asymp =0 for g ∈ { γ, β i } . By Proposition 9.4, (cid:88) g ∈{ γ,β i ,N } (cid:107) ∆ − ( F g + G g ) − ∂ t g asymp ( t ) ω ( | x | ) log | x |(cid:107) W , s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) (cid:46) (cid:107) F g + G g (cid:107) L s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α +2 (Σ t ) (cid:46) (cid:15) + (cid:15) (cid:88) (cid:101) g ∈{ (cid:101) γ,β i , (cid:101) N } (cid:107) ∂ t (cid:101) g (cid:107) W , s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) , (9.83)where we have used (9.81), the support properties of F g , and (9.79).Sobolev embedding (1(b) of Proposition 9.1) applied to (9.83) gives additionally that (cid:107) ∆ − ( F g + G g ) − ∂ t g asymp ( t ) ω ( | x | ) log | x |(cid:107) W , s (cid:48)− s (cid:48)(cid:48) − s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) (cid:46) (cid:15) + (cid:15) (cid:88) (cid:101) g ∈{ (cid:101) γ,β i , (cid:101) N } (cid:107) ∂ t (cid:101) g (cid:107) W , s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) . (9.84) Step 3: Bounding ∆ − ∂ t H i g . Using Proposition 9.5, (9.81), supp( H iγ ) ⊆ B (0 , R ) , as well as part 2 ofProposition 9.1, we have (cid:88) g ∈{ γ,β j ,N } (cid:107) ∂ i ∆ − H i g (cid:107) W , s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) (cid:46) (cid:88) g ∈{ γ,β j ,N } (cid:107) ∂ i ∆ − H i g (cid:107) W , s (cid:48)− s (cid:48)(cid:48) − s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) (cid:46) (cid:15) . (9.85) Step 4: Obtaining the W , s (cid:48)− s (cid:48)(cid:48) − s (cid:48) + s (cid:48)(cid:48) − α (Σ t ) estimates. We now combine (9.78), (9.80), (9.82), (9.83) and (firstterm in) (9.85) to obtain | ∂ t N asymp | ( t ) + (cid:88) (cid:101) g ∈{ (cid:101) γ,β i , (cid:101) N } (cid:107) ∂ t (cid:101) g (cid:107) W , s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) (cid:46) (cid:15) + (cid:15) (cid:88) (cid:101) g ∈{ (cid:101) γ,β i , (cid:101) N } (cid:107) ∂ t (cid:101) g (cid:107) W , s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) . (9.86)Choosing (cid:15) smaller if necessary, we can absorb the term (cid:15) (cid:80) (cid:101) g ∈{ (cid:101) γ,β i , (cid:101) N } (cid:107) ∂ t (cid:101) g (cid:107) W , s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) on the RHS of(9.86) by the corresponding term on the LHS, giving | ∂ t N asymp | ( t ) + (cid:88) (cid:101) g ∈{ (cid:101) γ,β i , (cid:101) N } (cid:107) ∂ t (cid:101) g (cid:107) W , s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α (Σ t ) (cid:46) (cid:15) . (9.87) Step 5: Obtaining the W , s (cid:48)− s (cid:48)(cid:48) − s (cid:48) + s (cid:48)(cid:48) − α (Σ t ) estimates. Plugging (9.87) into (9.84), and combining it with (9.78),(9.80), and (second term in) (9.85), we thus obtain the desired estimate (9.74). (cid:104) D x (cid:105) s (cid:48) ∂ i derivatives of ∂ t g Proposition 9.20. Let P be a cutoff in frequency (corresponding to the elliptic gauge coordinates) to | ξ | (cid:46) . Then, for every t ∈ [0 , T B ) , (cid:107)(cid:104) D x (cid:105) s (cid:48) [ ∂ i ∂ t N − ( ∂ t N asymp )( t ) P ∂ i ( ω ( | x | ) log | x | )] (cid:107) L (Σ t ) (cid:46) (cid:15) , (9.88) (cid:107)(cid:104) D x (cid:105) s (cid:48) ∂ i ∂ t γ (cid:107) L (Σ t ) + (cid:107)(cid:104) D x (cid:105) s (cid:48) ∂ i ∂ t β (cid:107) L (Σ t ) (cid:46) (cid:15) . (9.89) Proof. We only prove (9.88) since it features a low-frequency correction which is not in L (coming from ∂ t N asymp potentially non-vanishing). The estimate (9.89) is similar but slightly simpler; we omit the details. The reader may have noted that we have not used the estimate (9.84) proven above. It will be used in Step 5 below. 60y (9.78) and (9.80), we write (cid:104) D x (cid:105) s (cid:48) ∂ i ∂ t N − (cid:104) D x (cid:105) s (cid:48) P ( ∂ t N asymp ) ∂ i ( ω ( | x | ) log | x | )= (cid:104) D x (cid:105) s (cid:48) P ∂ i ∂ t N − (cid:104) D x (cid:105) s (cid:48) P ( ∂ t N asymp ) ∂ i ( ω ( | x | ) log | x | ) + (cid:104) D x (cid:105) s (cid:48) ( I − P ) ∂ i ∂ t N = (cid:104) D x (cid:105) s (cid:48) P [ ∂ i ∆ − ( F N + G N + ∂ j H jN ) − ( ∂ t N asymp ) ∂ i ( ω ( | x | ) log | x | )] (cid:124) (cid:123)(cid:122) (cid:125) =: I + (cid:104) D x (cid:105) s (cid:48) ( I − P ) ∂ i ∂ t N (cid:124) (cid:123)(cid:122) (cid:125) =: II . (9.90)For I , we use bounded frequency, i.e. the fact (cid:104) D x (cid:105) s (cid:48) P : L (Σ t ) → L (Σ t ) is bounded, Hölder’s inequalityand (9.74) to obtain (cid:107) I (cid:107) L (Σ t ) (cid:46) (cid:107) ∂ i ∆ − ( F N + G N + ∂ j H jN ) − ( ∂ t N asymp ) ∂ i ( ω ( | x | ) log | x | ) (cid:107) L (Σ t ) (cid:46) (cid:107) ∂ i ∂ t (cid:101) N (cid:107) L (Σ t ) (cid:46) (cid:107)(cid:104) x (cid:105) − s (cid:48) + s (cid:48)(cid:48) − α ∂ i ∂ t (cid:101) N (cid:107) L s (cid:48)− s (cid:48)(cid:48) (Σ t ) (cid:107)(cid:104) x (cid:105) − s (cid:48) − s (cid:48)(cid:48) + α (cid:107) L − s (cid:48) + s (cid:48)(cid:48) (Σ t ) (cid:46) (cid:15) . (9.91)For II in (9.90), we use that the frequency is bounded away from so that by Plancherel’s theorem, (cid:107) II (cid:107) L (Σ t ) = (cid:107)(cid:104) D x (cid:105) s (cid:48) ( I − P ) ∂ i ∂ t N (cid:107) L (Σ t ) (cid:46) (cid:107)(cid:104) D x (cid:105) − s (cid:48) ∆ ∂ t N (cid:107) L (Σ t ) . (9.92)The remaining of the proof concerns bounding (9.92). First, by (9.76) and (9.80), Sobolev embedding( (cid:104) D x (cid:105) − s (cid:48) : L (Σ t ) → L (Σ t ) is bounded) and Plancherel’s theorem, (cid:107)(cid:104) D x (cid:105) − s (cid:48) ∆ ∂ t N (cid:107) L (Σ t ) (cid:46) (cid:107) F N (cid:107) L (Σ t ) + (cid:107) G N (cid:107) L (Σ t ) + max i (cid:107)(cid:104) D x (cid:105) s (cid:48) H iN (cid:107) L (Σ t ) . (9.93)The F N and G N terms are easier. Since supp( F N ) ⊆ B (0 , R ) , by Hölder’s inequality and (9.81), (cid:107) F N (cid:107) L (Σ t ) (cid:46) (cid:15) . (9.94)Using Hölder’s inequality, s (cid:48) − s (cid:48)(cid:48) < , (9.79) and (9.87), we also have (cid:107) G N (cid:107) L (Σ t ) (cid:46) (cid:107) G N (cid:107) L s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α +2 (cid:107)(cid:104) x (cid:105) s (cid:48) − s (cid:48)(cid:48) +2 α − (cid:107) L − s (cid:48) +3 s (cid:48)(cid:48) (cid:46) (cid:107) G N (cid:107) L s (cid:48)− s (cid:48)(cid:48)− s (cid:48) + s (cid:48)(cid:48)− α +2 (cid:46) (cid:15) . (9.95)To handle H iN , we need a more explicit form of H iN . Going back to Lemmas 9.12, 9.13 and 9.18, we seethat schematically H iN takes one of the following four forms Ω( g )( N X ik + β i )( ∂ α (cid:101) φ k )( ∂ σ (cid:101) φ k ) , Ω( g )( N X ik + β i )( ∂ α (cid:101) φ k )( ∂ σ φ reg ) , (1 − (cid:101) ζ int ) (cid:101) ζ ang Ω( g )( N X ik + β i )( ∂ α (cid:101) φ k )( ∂ σ (cid:101) φ j ) (cid:124) (cid:123)(cid:122) (cid:125) =: ∗ , (1 − (cid:101) ζ int )(1 − (cid:101) ζ ang )Ω( g )( N X ij + β i )( ∂ α (cid:101) φ k )( ∂ σ (cid:101) φ j ) . (9.96)They can all be handled similarly; with the last two terms being slightly harder due to the cutoffs (cid:101) ζ int and (cid:101) ζ ang . We will take the ∗ term as an example. We first handle the fractional derivatives of the cutoffs.First, by interpolation, (9.64) and the support of ∂ x (cid:101) ζ , (cid:107)(cid:104) D x (cid:105) s (cid:48) [ (cid:36) (1 − (cid:101) ζ int )] (cid:107) L (Σ t ) (cid:46) (cid:107) (cid:36) (1 − (cid:101) ζ int ) (cid:107) H (Σ t ) (cid:46) . (9.97)Also, by Sobolev embedding and (9.65), (cid:107)(cid:104) D x (cid:105) s (cid:48) ( (cid:36) (cid:101) ζ ang ) (cid:107) L (Σ t ) (cid:46) (cid:107) (cid:36) (cid:101) ζ ang (cid:107) W , (Σ t ) (cid:46) . (9.98)Notice now that since supp( φ ) ⊆ B (0 , R ) , we have ∗ = (cid:36) ∗ . Therefore, by repeated applications ofLemma 9.2, we have (cid:107)(cid:104) D x (cid:105) s (cid:48) ( ∗ ) (cid:107) L (Σ t ) (cid:46) (cid:107)(cid:104) D x (cid:105) s (cid:48) [ (cid:36) (1 − (cid:101) ζ int )] (cid:107) L (Σ t ) (cid:107) (cid:36) (cid:101) ζ ang (cid:107) L ∞ (Σ t ) (cid:107) ∂ (cid:101) φ k (cid:107) L ∞ (Σ t ) (cid:107) ∂ (cid:101) φ j (cid:107) L ∞ (Σ t ) + (cid:107) (cid:36) (1 − (cid:101) ζ int ) (cid:107) L ∞ (Σ t ) (cid:107)(cid:104) D x (cid:105) s (cid:48) ( (cid:36) (cid:101) ζ ang ) (cid:107) L (Σ t ) (cid:107) ∂ (cid:101) φ k (cid:107) L ∞ (Σ t ) (cid:107) ∂ (cid:101) φ j (cid:107) L ∞ (Σ t ) + (cid:107) (cid:36) (1 − (cid:101) ζ int ) (cid:107) L ∞ (Σ t ) (cid:107) (cid:36) (cid:101) ζ ang (cid:107) L ∞ (Σ t ) (cid:107) ∂ (cid:104) D x (cid:105) s (cid:48) (cid:101) φ k (cid:107) L (Σ t ) (cid:107) ∂ (cid:101) φ j (cid:107) L ∞ (Σ t ) + (cid:107) (cid:36) (1 − (cid:101) ζ int ) (cid:107) L ∞ (Σ t ) (cid:107) (cid:36) (cid:101) ζ ang (cid:107) L ∞ (Σ t ) (cid:107) ∂ (cid:101) φ k (cid:107) L ∞ (Σ t ) (cid:107) ∂ (cid:104) D x (cid:105) s (cid:48) (cid:101) φ j (cid:107) L (Σ t ) (cid:46) (cid:15) , (9.99)61here in the last estimate we have used (9.97), (9.98), and the bootstrap assumptions (7.5c) and (7.8c).We can handle the other terms in (9.96) in a similar manner as (9.99), so that when combined with(9.92), (9.93), (9.94) and (9.95), we obtain the following bound for II in (9.90): (cid:107) II (cid:107) L (Σ t ) (cid:46) (cid:15) . (9.100)Finally, combining (9.90), (9.91) and (9.100), we obtain (9.88). Our final elliptic estimate concerns third derivatives of the metric; see Proposition 9.21. Note that1. the estimate allows for at most one ∂ t derivative, and2. the bound blows up as δ → . Proposition 9.21. The following estimate holds for all t ∈ [0 , T B ) : (cid:88) g ∈{ γ, β i , N } (cid:107) ∂∂ x g (cid:107) L (Σ t ) (cid:46) (cid:15) · δ − . (9.101) Proof. By the L -boundedness of ∂ ij ∆ − , it suffices to show that (cid:107) ∂ ∆ g (cid:107) L (Σ t ) (cid:46) (cid:15) · δ − .Differentiating (2.24)–(2.26) by ∂ , it follows from (7.2a), (7.2b) and (7.8c) that (cid:88) g ∈{ γ,β i ,N } | ∂ ∆ g | (cid:46) (cid:15) (cid:104) x (cid:105) − − α (cid:124) (cid:123)(cid:122) (cid:125) =: I + (cid:15) (cid:88) g ∈{ γ,β i ,N } (cid:104) x (cid:105) − | ∂∂ x g | (cid:124) (cid:123)(cid:122) (cid:125) =: II + (cid:15) | ∂ φ | (cid:124) (cid:123)(cid:122) (cid:125) =: III . We control the L (Σ t ) norm of each term. Obviously, (cid:107) I (cid:107) L (Σ t ) (cid:46) (cid:15) . By (7.2c) (with Proposition 9.1) andProposition 9.19, (cid:107) II (cid:107) L (Σ t ) (cid:46) (cid:15) . Finally, by (7.4) and (7.5b), (cid:107) III (cid:107) L (Σ t ) (cid:46) (cid:15) · δ − . K Proposition 9.22. The following estimate holds for all t ∈ [0 , T B ) : (cid:107) K (cid:107) L ∞ − α (Σ t ) + (cid:107) ∂ x K (cid:107) L ∞ − α (Σ t ) + (cid:107) ∂ t K (cid:107) L s (cid:48)− s (cid:48)(cid:48) − s (cid:48) + s (cid:48)(cid:48) + α (Σ t ) (cid:46) (cid:15) . (9.102) Proof. We use the formula (2.14) to write K in terms γ , β and N . Notice that e γ N is favorable in terms ofthe (cid:104) x (cid:105) weights. Hence, the estimates follow from Propositions 9.7, 9.11 and 9.19. 10 Estimates for the Ricci coefficients and related geometric quan-tities We continue to work under the assumptions of Theorem 7.1.Our goal in this section is to control the remaining geometric quantities, particularly those related tothe eikonal functions u k . In Section 10.1 , we bound the Ricci coefficients χ k , η k and their derivatives.In Section 10.2 , we bound the metric coefficients µ k and Θ k (in the ( u k , θ k , t k ) coordinates. Finally, in Section 10.3 , we estimate the second derivatives of the commutation fields.62 In this subsection we bound the Ricci coefficients and their derivatives. estimates, which require a treatmentof the quadratic interaction between two impulsive waves: Proposition 10.1. The following estimates hold for all t ∈ [0 , T B ) and all u k ∈ R : (cid:107) χ k (cid:107) L ∞ − α (Σ t ) + (cid:107) η k (cid:107) L ∞ − α (Σ t ) (cid:46) (cid:15) , (10.1) (cid:107) ∂ x χ k (cid:107) L θk (Σ t ∩ C kuk ) (cid:46) (cid:15) , (10.2) (cid:107) E k η k (cid:107) L θk (Σ t ∩ C kuk ) (cid:46) (cid:15) , (10.3) (cid:107) ∂ x η k (cid:107) L (Σ t ∩ B (0 , R )) (cid:46) (cid:15) . (10.4) Proof. In this proof, we prove estimates by solving transport equations and integrating along the integralcurves of L k . Recall in particular that in the coordinate system ( u k , θ k , t k ) , ∂ t k = N · L k by (2.45). Step 1: Controlling χ k and η k (Proof of (10.1) ). Using the transport equations (2.94), (2.93), the bootstrapassumptions (7.3a), (7.3b), (7.8c), and the estimates in (8.1), we obtain | L k χ k | + | L k η k | (cid:46) (cid:15) · (cid:104) x (cid:105) − α (cid:46) (cid:15) · (cid:104) x (cid:105) − α . (10.5)Note that• the initial χ k and η k are bounded by (cid:104) x (cid:105) − α (see point 5 in the proof of Lemma 7.6), and• that (cid:104) x (cid:105) are comparable between any two points on the integral curve of L k (see Step 0 of Proposi-tion 8.5).Hence, integrating (10.5) along the integral curve of L k , we obtain (10.1). Step 2: Controlling derivatives of χ k (Proof of (10.2) ). Step 2(a): Preliminary reductions. First, wecommute (2.94) with ∂ i , and rewrite L k = N − · ∂ t k (using (2.45)): ∂ t k ∂ i χ k = N [ L k , ∂ i ] χ k (cid:124) (cid:123)(cid:122) (cid:125) =: A − N ([ ∂ i , L k ] φ )( L k φ ) (cid:124) (cid:123)(cid:122) (cid:125) =: B − N ( L k ∂ i φ )( L k φ ) (cid:124) (cid:123)(cid:122) (cid:125) =: D − N ∂ i ( χ k − ( K ( X k , X k ) − X k log( N )) · χ k ) (cid:124) (cid:123)(cid:122) (cid:125) =: E . (10.6)We first control A , B and E of (10.6) in the L θ k (Σ t ∩ C ku k ) norm (see Definition 3.12).Using Lemma 8.4, (7.2a)–(7.2b), (8.3), (8.6) and (10.5) in order, we obtain | A | (cid:46) (cid:15) (cid:104) x (cid:105) − α + (cid:15) | ∂χ k | (cid:46) (cid:15) (cid:104) x (cid:105) − α +2 (cid:15) ( | L k χ k | + | ∂ x χ k | ) (cid:46) (cid:15) (cid:104) x (cid:105) − α +2 (cid:15) + (cid:15) (cid:104) x (cid:105) − α +2 (cid:15) | ∂ x χ k | . Note that using L k θ k = 0 , | ( θ k ) | Σ | (cid:46) (cid:104) x (cid:105) (by (2.44)), and the comparability of (cid:104) x (cid:105) along integral curves of L k , we have (cid:104) x (cid:105) − (cid:46) (cid:104) θ k (cid:105) − .Hence, using also (7.3c), (cid:107) A (cid:107) L θk (Σ t ∩ C kuk ) (cid:46) (cid:15) + (cid:15) (cid:107)(cid:104) x (cid:105) − − α ∂ x χ k (cid:107) L θk (Σ t ∩ C kuk ) (cid:46) (cid:15) . Using Lemmas 8.2 and 8.4, supp( φ ) ⊆ B (0 , R ) and (7.8c), it follows easily that | B | (cid:46) (cid:15) (cid:104) x (cid:105) − α . UsingLemma 8.2, (7.3a)–(7.3c), (7.2a)–(7.2b), we have | E | (cid:46) (cid:15) (cid:104) x (cid:105) − α . Arguing as for term A , both B and E can be controlled in L θ k (Σ t ∩ C ku k ) by (cid:46) (cid:15) .Combining all the above estimates, it follows that (with D as in (10.6)) (cid:107) ∂ t k ∂ i χ k + D (cid:107) L θk (Σ t ∩ C kuk ) (cid:46) (cid:15) . (10.7)We now turn to the term D in (10.6) (and (10.7)). Using the decomposition φ = (cid:80) q =1 (cid:101) φ q + φ reg , the L ∞ bootstrap assumption (7.8c) for ∂φ , and Lemma 8.2, we obtain the following pointwise bounds for D : | D | (cid:46) | L k ∂ i φ reg | + | L k ∂ i (cid:101) φ k | + (cid:88) q (cid:54) = k | L k ∂ i φ q | . (10.8)63e now bound ∂ i χ k using (10.7), by first integrating along the integral curve of t k for every fixed ( u k , θ k ) ,and then taking the (cid:107)(cid:104) θ k (cid:105) − − α · (cid:107) L θk norm. Writing in the ( u k , θ k , t k ) coordinate system, (10.7), (10.8) andthe initial data bound (obtained in part 5 of the proof of Lemma 7.6) imply that (cid:107) ∂ i χ k (cid:107) L (Σ t ∩ C kuk ) (cid:46) (cid:107) ∂ i χ k (cid:107) L (Σ ∩ C kuk ) + (cid:107) (cid:90) t | D | ( u k , · , t (cid:48) ) dt (cid:48) (cid:107) L (Σ ∩ C kuk ) (cid:46) (cid:15) + (cid:15) (cid:107) (cid:90) t | L k ∂ i φ reg | ( u k , · , t (cid:48) ) dt (cid:48) (cid:107) L ((Σ t ∩ C kuk )) (cid:124) (cid:123)(cid:122) (cid:125) =: I + (cid:15) (cid:107) (cid:90) t | L k ∂ i φ k | ( u k , · , t (cid:48) ) dt (cid:48) (cid:107) L (Σ t ∩ C kuk ) (cid:124) (cid:123)(cid:122) (cid:125) =: II + (cid:15) (cid:88) q (cid:54) = k (cid:107) (cid:90) t | L k ∂ i φ q | ( u k , · , t (cid:48) ) dt (cid:48) (cid:107) L (Σ t ∩ C kuk ) (cid:124) (cid:123)(cid:122) (cid:125) =: III . (10.9)We will bound the terms I , II and III in the following substeps. Step 2(b): The easy terms I and II in (10.9) . To handle the terms I and II in (10.9), we first use Minkowski’sinequality in the θ k variable, and then use the Cauchy–Schwarz inequality in t k to obtain that I (cid:46) (cid:15) (cid:107) L k ∂ i φ reg (cid:107) L ( C kuk ([0 ,T B ))) , II (cid:46) (cid:15) (cid:107) L k ∂ i (cid:101) φ k (cid:107) L ( C kuk ([0 ,T B ))) . (10.10)The terms in (10.10) are bounded above by (cid:15) by (7.7a) and (7.7c) respectively. Step 2(c): The main term III in (10.9) . We now turn to term III in (10.9), which is more delicate andrequires the transversality of the different waves.Fix q (cid:54) = k . Take a constant- ( u k , θ k ) curve (parametrized by t k ) which passes through the support of φ for some t k ∈ [0 , . Using the bootstrap assumptions (7.2a)–(7.2b) on the metric, and the fact that supp( φ ) ⊆ B (0 , R ) , it is easy to check that for t ∈ [0 , T B ) ⊆ [0 , , the whole curve is contained in B (0 , R ) .Define T ∓ k,q (depending on the chosen constant- ( u k , θ k ) curve) by T − k,q ( u k , θ k ) := inf { t ≥ , ( u k , θ k , t ) ∈ S qδ } , T + k,q ( u k , θ k ) := sup { t ≥ , ( u k , θ k , t ) ∈ S qδ } (recall the definition (2.35)).Let us consider only the case that < T − k,q ( u k , θ k ) < T + k,q ( u k , θ k ) < t (if not the proof is even easier). Inview of the fact that ∂ t k u q ∈ ( κ , on B (0 , R ) by (8.14), and that (by definition) (cid:82) T + k,q ( u k ,θ k ) T − k,q ( u k ,θ k ) ∂ t k u q dt k = 2 δ ,we get that δ ≤ T + k,q ( u k , θ k ) − T − k,q ( u k , θ k ) ≤ δ κ . (10.11)We split the integral (cid:82) t term III in (10.9) into an integral in ( S kδ ) c , i.e. (cid:82) T − k,q ( u k ,θ k )0 + (cid:82) tT + k,q ( u k ,θ k ) , andanother integral in S kδ , i.e. (cid:82) T + k,q ( u k ,θ k ) T − k,q ( u k ,θ k ) .Note that, since (cid:101) φ q ≡ on C q ≤− δ (by Lemma 8.1), the (cid:82) T − k,q ( u k ,θ k )0 integral is trivial. Using the Cauchy–Schwarz inequality and the bootstrap assumption (7.7b), we obtain (cid:107) (cid:90) tT + k,q ( u k ,θ k ) | L k ∂ i (cid:101) φ q | ( u k , · , t (cid:48) ) dt (cid:48) (cid:107) L θk (Σ t ∩ C kuk ) (cid:46) (cid:15) · (cid:107) L k ∂ i ˜ φ q (cid:107) L ( C kuk ([0 ,T B )) ∩ S qδ ) (cid:46) (cid:15) . It therefore follows that (cid:107) (cid:90) T − k,q ( u k ,θ k )0 | L k ∂ i (cid:101) φ q | ( u k , · , t (cid:48) ) dt (cid:48) (cid:107) L θk (Σ t ∩ C kuk ) + (cid:107) (cid:90) tT + k,q ( u k ,θ k ) | L k ∂ i (cid:101) φ q | ( u k , · , t (cid:48) ) dt (cid:48) (cid:107) L θk (Σ t ∩ C kuk ) (cid:46) (cid:15) . (10.12)64hen we turn to the integral on the singular zone, whose smallness we will exploit. This time, Cauchy–Schwarz gives, in view of (10.11): (cid:107) (cid:90) T + k,q ( u k ,θ k ) T − k,q ( u k ,θ k ) | L k ∂ i ˜ φ q | ( u k , · , t (cid:48) ) dt (cid:48) (cid:107) L θk (Σ t ∩ C kuk ) (cid:46) δ · (cid:107) L k ∂ i ˜ φ q (cid:107) L ( C kuk ∩ [0 ,T B ]) (cid:46) δ · ( (cid:15) δ − ) (cid:46) (cid:15) , (10.13)where we used (7.7d).Combining (10.12) and (10.13), term III in (10.9) is bounded by III (cid:46) (cid:15) . Step 2(d): Putting everything together. Combining the estimates in Steps 2(b) and 2(c), we have thus shownthat the terms I , II and III in (10.9) are bounded above by (cid:46) (cid:15) (for all t ∈ [0 , T B ) and u k ∈ R ). Thus,using (10.9), we obtain the desired estimate (10.2). Step 3: Controlling E k η k (Proof of (10.3) ). The proof is broadly similar to that of (10.2) so we only explainthe difference. By (2.93) and using similar arguments as in Step 2, we get sup ≤ t The following estimates hold for all t ∈ [0 , T B ) : (cid:107) L k χ k (cid:107) L ∞ − α (Σ t ) + (cid:107) L k η k (cid:107) L ∞ − α (Σ t ) (cid:46) (cid:15) , (10.23) (cid:107) L k χ k (cid:107) L (Σ t ∩ B (0 ,R )) + (cid:107) L k η k (cid:107) L (Σ t ∩ B (0 ,R )) + (cid:107) L k ∂ x χ k (cid:107) L (Σ t ∩ B (0 ,R )) + (cid:107) L k E k η k (cid:107) L (Σ t ∩ B (0 ,R )) (cid:46) (cid:15) . (10.24) Proof. The estimates for L k χ k and L k η k follows from (10.5). The estimates for L k ∂ x χ k and L k E k η k followfrom combining (10.7), (10.14) with (7.5a) and (7.8c). Finally, the estimates for L k η k and L k χ k followfrom differentiating (2.93) and (2.94) by L k , and then controlling the resulting terms using (7.5a), (7.8c),Lemmas 8.2 and 8.4, Propositions 9.11, 9.19 and (9.22), and (10.23). µ k and Θ k We next consider the estimates for µ k and Θ k . Proposition 10.3. The following estimates hold for all t ∈ [0 , T B ) and all u k ∈ R : (cid:107) log µ k − γ asymp ω ( | x | ) log | x |(cid:107) L ∞ − α (Σ t ) + (cid:107) ∂ x µ k (cid:107) L ∞ − α (Σ t ) (cid:46) (cid:15) , (10.25) (cid:107) log( Θ k ) − γ asymp ω ( | x | ) log | x |(cid:107) L ∞ − α (Σ t ) + (cid:107)(cid:104) x (cid:105) − α ∂ x log Θ k (cid:107) L θk (Σ t ∩ C kuk ) (cid:46) (cid:15) . (10.26) Proof. By initial condition (2.98), and the bounds in Proposition 9.7, (cid:107) log µ k − γ asymp ω ( | x | ) log | x |(cid:107) L ∞ − α (Σ ) (cid:46) (cid:15) . Integrating the transport equation (2.91), and using the estimates in Lemma 8.2 regarding | X ik | togetherwith (9.9), (9.10), (9.102), and the comparability of (cid:104) x (cid:105) along integral curves of ∂ t k , we get sup ≤ t For every sufficiently regular function f , (cid:107) ∂∂ x f (cid:107) L (Σ t ∩ B (0 , R )) (cid:46) (cid:88) Y k ∈{ L k ,X k ,E k } (cid:88) Z k ∈{ X k ,E k } (cid:107) Y k Z k f (cid:107) L (Σ t ∩ B (0 , R )) + (cid:107) ∂ x f (cid:107) L (Σ t ∩ B (0 , R )) , (10.27) (cid:107) ∂ f (cid:107) L (Σ t ∩ B (0 , R )) (cid:46) (cid:88) Y k ,Z k ∈{ L k ,X k ,E k } (cid:107) Y k Z k f (cid:107) L (Σ t ∩ B (0 , R )) + (cid:107) ∂f (cid:107) L (Σ t ∩ B (0 , R )) . (10.28) Proof. Acting the vector fields in (2.53) on f , and further differentiating by ∂ , and using the estimates inLemmas 8.2 and 8.4 and (9.9)–(9.10), we obtain the pointwise bound in B (0 , R ) . | ∂∂ i f | (cid:46) | ∂E k f | + | ∂X k f | + | E k f | + | X k f | , which implies (10.27) after also using Lemma 8.3.To prove (10.28), we need to additionally control ∂∂ t f , which can be done with a similar argument exceptfor starting with (8.8); we omit the details. Proposition 10.5. The following estimates for the second derivatives of the coefficients of the vector fields L k , E k and X k hold for all t ∈ [0 , T B ) : (cid:107) ∂ E ik (cid:107) L (Σ t ∩ B (0 , R )) + (cid:107) ∂ X ik (cid:107) L (Σ t ∩ B (0 , R )) + (cid:107) ∂∂ x L αk (cid:107) L (Σ t ∩ B (0 , R )) (cid:46) (cid:15) . Proof. Step 1: Estimates for E ik and X ik . By Lemma 10.4, to obtain the estimates for E ik and X ik , it sufficesto bound (cid:107) ∂Z k E ik (cid:107) L (Σ t ∩ B (0 , R )) and (cid:107) ∂Z k X ik (cid:107) L (Σ t ∩ B (0 , R )) for Z k ∈ { L k , X k , E k } . Moreover, in view of(2.84), it in fact suffices to bound only (cid:107) ∂Z k E ik (cid:107) L (Σ t ∩ B (0 , R )) .To control (cid:107) ∂Z k E ik (cid:107) L (Σ t ∩ B (0 , R )) , we differentiate the equations (2.83a)–(2.83c) in Proposition 2.20. Totreat the resulting terms, note that we only need estimates in B (0 , R ) . We then use Lemmas 8.2 and 8.4to bound L αk , X ik , E ik and their first derivatives in L ∞ , use (9.9), (9.10) and (9.75) to control the metricand its first derivatives in L ∞ , and use (9.102) and (10.1) to bound K , χ k and η k in L ∞ . Thus, on the set Σ t ∩ B (0 , R ) , we have the pointwise bound (cid:88) Z k ∈{ L k ,X k ,E k } | ∂Z k E ik | (cid:46) (cid:15) + (cid:88) g ∈{ γ,β i ,N } | ∂∂ x g | + | ∂K | + | ∂χ k | + | ∂η k | . (10.29)We take the L (Σ t ∩ B (0 , R )) norm (10.29): the metric terms are bounded by (9.26), and (9.74) (and thefact that s (cid:48) − s (cid:48)(cid:48) < ), | ∂K | is bounded by (9.102), and | ∂χ k | and | ∂η k | are bounded by Lemma 8.3, (10.23),(10.2) and (10.4). We thus obtain (cid:80) Z k ∈{ L k ,X k ,E k } (cid:107) ∂Z k E ik (cid:107) L (Σ t ∩ B (0 , R )) (cid:46) (cid:15) , as desired. Step 2: Estimates for L αk and L ik . Using Lemma 10.4, it suffices to bound (cid:80) Z k ∈{ X k ,E k } (cid:107) ∂Z k L αk (cid:107) L (Σ t ∩ B (0 , R )) (note in particular that Z k (cid:54) = L k ). Differentiating (2.85) and (2.86), and using the bound (10.29) above to-gether with (9.9), (9.10) and (9.75), we obtain that on Σ t ∩ B (0 , R ) , (cid:88) Z k ∈{ X k ,E k } | ∂Z k L αk | (cid:46) (cid:15) + (cid:88) g ∈{ γ,β i ,N } | ∂∂ x g | + | ∂K | + | ∂χ k | + | ∂η k | . (10.30)Notice that all the terms have appeared in (10.29), and we then proceed as in Step 1. We remark that some components of the second derivatives are in fact better. For instance, the derivatives of L k E ik and E k E ik obey better bounds (in terms of the L p space) than X k E ik . Such improvements will not be made precise, nor will theybe useful. We continue to work under the assumptions of Theorem 7.1. We now conclude the proof of Theorem 7.1. Proof of Theorem 7.1. To conclude the proof of Theorem 7.1, we only need to collect already proven facts:• (7.2a)–(7.2c) all hold with (cid:15) replaced by C(cid:15) thanks to Proposition 9.7 and (9.75).• (7.3a) is improved by (9.102), (7.3b) and (7.3c) are improved by estimates in Proposition 10.1, and(7.3d) and (7.3e) are improved by estimates in Proposition 10.3. All these estimates are improved from (cid:15) to C(cid:15) .• The estimates in (5.13) for u k follow from Corollary 8.6 and Proposition 8.7.• The estimates in (5.14) for L βk , E ik and X ik follow from Lemmas 8.2 and 8.4.• The estimates in (5.16) for the metric components follow from Propositions 9.7, 9.11, and 9.19. A Solving the constraint equations This appendix concerns the constraint equations for the initial data. In our setting, we cannot directly usethe result in [34] (or [35]) to solve the constraints. We will instead need a modification which we sketch inthis appendix.We first explain why [34] is not applicable in our setting. In [34, 35], one prescribes φ and ˙ φ := e γ (cid:126)nφ sothat one can directly impose the integrability condition (cid:90) Σ e γ ( (cid:126)nφ ) ( ∂ j φ ) dx dx = (cid:90) Σ ˙ φ ( ∂ j φ ) dx dx = 0 , j = 1 , . (A.1)Using this condition, one can solve for γ and K as a coupled system of nonlinear elliptic equations. However,in our case, we need to impose initial data with the additional condition that (cid:126)n (cid:101) φ k − X k (cid:101) φ k is better thangeneric first derivatives of (cid:101) φ k . In terms of (cid:101) φ k and ˙ (cid:101) φ k , this corresponds e − γ ˙ (cid:101) φ k − e − γ · δ iq · c kq · ∂ i (cid:101) φ k beingbetter. This, however, cannot be imposed with the scheme in [34] since γ is not known a priori.Instead, we prescribe φ and φ := e γ (cid:126)nφ . In order to impose the condition (A.1), we need to introduce atwo-parameter family of data and show that there exists a choice of parameters such that (A.1) holds.In Section A.1 , we prove a general lemma for solving the constraint equations. In Section A.2 , wethen expand on the non-degeneracy condition (4.8), in preparation of solving the constraint equations in oursetting. In Section A.3 , we then solve the constraint equations to construct examples of initial data setssatisfying the assumptions of Definition 4.3. Finally, in Section A.4 , we solve the constraint equations inorder to construct δ -impulsive wave data that approximate impulsive wave data, hence proving Lemma 6.1. A.1 A general lemma for solving the constraint equations Let K ⊆ R be a compact, convex set and consider a two parameter family φ λ parametrized by λ =( λ , λ ) ∈ K such that the following holds:1. For any λ ∈ K , ( φ, φ λ ) satisfies supp( φ ) , supp( φ λ ) ⊆ B (0 , R ) and obeys the estimate (cid:107) φ (cid:107) W , ( R ) + (cid:107) φ λ (cid:107) L ( R ) ≤ (cid:15). (A.2)2. λ (cid:55)→ φ λ is a continuous map K → L ( R ) .3. For any γ = − γ asymp ω ( | x | ) log | x | + (cid:101) γ with | γ asymp | ≤ (cid:15) and (cid:107) (cid:101) γ (cid:107) W , (Σ ) ≤ (cid:15) , there exists λ ∈ K suchthat (cid:90) Σ e γ φ λ ( ∂ j φ ) dx dx = 0 , j = 1 , . (A.3)68 emma A.1. For any R > , there exists (cid:15) = (cid:15) ( R ) > such that the following holds.Suppose ( φ, φ λ ) satisfies the conditions 1–3 above. Then, if (cid:15) ∈ (0 , (cid:15) ] , there exist λ ∈ K and functions ( γ, K ) such that ( φ, φ (cid:48) = e − γ φ λ , γ, K ) is an admissible initial data set (see Definition 4.2).Proof. Denote by B Y (0 , (cid:15) ) the closed ball of radius (cid:15) around in a Banach space Y . Let Φ : K × [0 , (cid:15) ] × B W , (Σ ) (0 , (cid:15) ) × B L (Σ ) (0 , (cid:15) ) → K × [0 , (cid:15) ] × B W , (Σ ) (0 , (cid:15) ) × B L (Σ ) (0 , (cid:15) ) be given by ( λ , γ asymp , (cid:101) γ, K ) (cid:55)→ ( λ ∗ , γ ∗ asymp , (cid:101) γ ∗ , K ∗ ) , where1. λ ∗ is chosen so that (A.3) holds.2. γ ∗ = − γ ∗ asymp ω ( | x | ) log | x | + (cid:101) γ ∗ is given by γ ∗ = ∆ − ( − δ il ( ∂ i φ )( ∂ l φ ) − e − γ | K | − ( φ λ ∗ ) ) , (A.4)where ∆ − is as in Definition 9.3.3. Define σ i := δ ij · e γ · φ λ ∗ · ∂ j φ . Impose K ∗ to be K ∗ ij = 2[ L ∆ − σ ] ij , (A.5)with λ ∗ as in 1 above, ∆ − as in Definition 9.3, and L the conformal Killing operator as in (2.14).The equations (A.4) and (A.5) are easy to solve: for (cid:15) > sufficiently small and (cid:15) ∈ (0 , (cid:15) ] , if ( γ asymp , (cid:101) γ, K ) ∈ [0 , (cid:15) ] × B W , (Σ ) (0 , (cid:15) ) × B L (Σ ) (0 , (cid:15) ) , it follows from Proposition 9.4, (A.3) and Proposi-tion 9.1 that | γ asymp | , (cid:107) (cid:101) γ ∗ (cid:107) H − (Σ ) , (cid:107) K ∗ (cid:107) H (Σ ) (cid:46) (cid:15) (A.6)for some implicit constants depending only on R . (This can for instance be proven as [35, Lemma 7.1], with δ = − in the notation there, and noting that there is in fact extra room in the weights.) In particular,using 1(b) of Proposition 9.1, it follows, after choosing (cid:15) smaller if necessary, that Φ indeed maps into K × [0 , (cid:15) ] × B W , (Σ ) (0 , (cid:15) ) × B L (Σ ) (0 , (cid:15) ) (as stated above), and moreover, Φ is compact.By Schauder’s fixed point theorem, Φ has a fixed point ( λ , γ asymp , (cid:101) γ, K ) . As a result, ( φ, φ (cid:48) := e − γ φ λ , γ := − γ asymp ω ( | x | ) log | x | + (cid:101) γ, K ) constitutes an admissible initial data set. A.2 Lemmas on the non-degeneracy assumption In this subsection, we prove two lemmas related to the non-degeneracy condition (4.8). First, in Lemma A.2,we prove the assertion in Remark 4.7 that LHS of (4.8) is non-zero for non-identically zero, compactlysupported φ . Then, in Lemma A.3 we deduce a consequence of the condition (4.8) which will be used whensolving the constraint equations for the impulsive and δ -impulsive gravitational waves. Lemma A.2. Let φ ∈ H (Σ ) be compactly supported and non-identically vanishing. Then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ φ − (cid:104) ∂ φ, ∂ φ (cid:105) L (Σ ,dx ) (cid:107) ∂ φ (cid:107) L (Σ ) ∂ φ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H − (Σ ) (cid:54) = 0 , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ φ − (cid:104) ∂ φ, ∂ φ (cid:105) L (Σ ,dx ) (cid:107) ∂ φ (cid:107) L (Σ ) ∂ φ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H − (Σ ) (cid:54) = 0 . (A.7) Proof. Take φ as in the assumption of the lemma. By the compact support assumption, both ∂ φ and ∂ φ arenot identically . The same argument shows that it is impossible to have ∂ φ + a∂ φ = 0 or a∂ φ + ∂ φ = 0 for some constant a ∈ R . It thus follows from the Cauchy–Schwarz inequality that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ φ − (cid:104) ∂ φ, ∂ φ (cid:105) L (Σ ,dx ) (cid:107) ∂ φ (cid:107) L (Σ ) ∂ φ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L (Σ ) (cid:54) = 0 , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ φ − (cid:104) ∂ φ, ∂ φ (cid:105) L (Σ ,dx ) (cid:107) ∂ φ (cid:107) L (Σ ) ∂ φ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L (Σ ) (cid:54) = 0 . Clearly, since a function with non-zero L norm must be non-vanishing (and hence has non-zero H − norm),we obtain (A.7). 69 emma A.3. Suppose that supp( φ ) ⊆ B (0 , R ) , and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ φ − (cid:104) ∂ φ, ∂ φ (cid:105) L (Σ ,dx ) (cid:107) ∂ φ (cid:107) L (Σ ) ∂ φ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H − (Σ ) × (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ φ − (cid:104) ∂ φ, ∂ φ (cid:105) L (Σ ,dx ) (cid:107) ∂ φ (cid:107) L (Σ ) ∂ φ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H − (Σ ) ≥ (cid:15) . (A.8) Then there exist smooth functions ψ , ψ compactly supported in B (0 , R ) such that max j =1 , (cid:107) ψ j (cid:107) H (Σ ) ≤ , (cid:12)(cid:12)(cid:12)(cid:12) det (cid:20)(cid:82) Σ ψ ∂ φ dx dx (cid:82) Σ ψ ∂ φ dx dx (cid:82) Σ ψ ∂ φ dx dx (cid:82) Σ ψ ∂ φ dx dx (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) (cid:38) (cid:15) . (A.9) Proof. For this proof, let (cid:107) · (cid:107) H (Σ ) := (cid:107)(cid:104) D x (cid:105) ( · ) (cid:107) L (Σ t ) (which is equivalent to that in Definition 3.4). Given ∂ φ and ∂ φ , notice that sup ψ ∈ C ∞ c (Σ ) \{ } , (cid:82) Σ0 ψ∂ φ dx dx =0 (cid:82) Σ ψ∂ φ dx dx (cid:107) ψ (cid:107) H (Σ ) = sup ψ ∈ C ∞ c (Σ ) \{ } , (cid:82) Σ0 ψ∂ φ dx dx =0 (cid:82) Σ ψ ( ∂ φ − (cid:104) ∂ φ, ∂ φ (cid:105) L ,dx ) (cid:107) ∂ φ (cid:107) L ∂ φ ) dx dx (cid:107) ψ (cid:107) H (Σ ) . It is thus easy to see that the supremum is achieved by ψ = (cid:104) D x (cid:105) − ( ∂ φ − (cid:104) ∂ φ, ∂ φ (cid:105) L ,dx ) (cid:107) ∂ φ (cid:107) L ∂ φ ) , and thatthe supremum is sup ψ ∈ C ∞ c (Σ ) \{ } , (cid:82) Σ0 ψ∂ φ dx dx =0 (cid:82) Σ ψ∂ φ dx dx (cid:107) ψ (cid:107) H (Σ ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ φ − (cid:104) ∂ φ, ∂ φ (cid:105) L (Σ ,dx ) (cid:107) ∂ φ (cid:107) L (Σ ) ∂ φ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H − (Σ ) . A similar statement holds after switching ∂ φ and ∂ φ . Therefore, using also that supp( φ ) ⊆ B (0 , R ) , wededuce that there are smooth functions ψ , ψ compactly supported in B (0 , R ) with max j =1 , (cid:107) ψ j (cid:107) H (Σ ) ≤ such that (cid:104) ψ , ∂ φ (cid:105) L (Σ ,dx ) = 0 = (cid:104) ψ , ∂ φ (cid:105) L (Σ ,dx ) and for i, j = 1 , , i (cid:54) = j , without summation , (cid:90) Σ ψ j ∂ j φ dx dx ≥ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ j φ − (cid:104) ∂ j φ, ∂ i φ (cid:105) L (Σ ,dx ) (cid:107) ∂ i φ (cid:107) L (Σ ) ∂ i φ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H − (Σ ) . Assuming also that (A.8) holds, this implies that (A.9) holds for this choice of ψ j . A.3 Construction of impulsive wave data It is now straightforward to apply Lemma A.1 to construct initial data set satisfying conditions in Defini-tion 4.3. We will simply be content with constructing some — instead of classifying all — such initial datasets. To simplify the exposition, let us construct special examples such that (cid:101) φ k and (cid:101) φ (cid:48) k are of size O ( (cid:15) ) , φ reg is of size O ( (cid:15) ) , and φ (cid:48) reg is of size O ( (cid:15) ) . Lemma A.4. There exist a large class of admissible initial data sets ( γ, K, φ, φ (cid:48) ) satisfying the assumptionsof Definition 4.3.Proof. Step 1: Prescribing φ reg , (cid:101) φ k and (cid:101) φ k . Impose φ reg , (cid:101) φ k and (cid:101) φ k satisfying the following conditions:• The transversality condition in 2 and the support properties in 3 of Definition 4.3 hold. Moreover, φ reg and (cid:101) φ k satisfies the stronger support assumptions supp( φ reg ) , supp( (cid:101) φ k ) ⊆ B (0 , R ) .• The following estimates hold for k = 1 , , : (cid:107) φ reg (cid:107) H s (cid:48) (Σ ) ≤ . (cid:15), (A.10a) (cid:107) (cid:101) φ k (cid:107) W , ∞ (Σ ) + (cid:107) (cid:101) φ k (cid:107) H s (cid:48) (Σ ) + (cid:107) (cid:101) φ k (cid:107) L ∞ (Σ ) + (cid:107) (cid:101) φ k (cid:107) H s (cid:48) (Σ ) ≤ (cid:15) , (A.10b) (cid:107) δ iq · c ⊥ kq · ∂ i (cid:101) φ k (cid:107) H s (cid:48)(cid:48) (Σ ) + (cid:107) δ iq · c ⊥ kq · ∂ i (cid:101) φ k (cid:107) H s (cid:48)(cid:48) (Σ ) + (cid:107) (cid:101) φ k − δ iq · c kq · ∂ i (cid:101) φ k (cid:107) H s (cid:48)(cid:48) (Σ ) ≤ (cid:15) . (A.10c) This particular choice we make here allows (A.3) to be checked more easily. 70 For k = 1 , , , there exist signed Radon measures T ij,k , T i,k , T ijE,k , T iE,k and T ijL,k such that (cid:107) ∂ ij (cid:101) φ k − T ij,k (cid:107) L (Σ ) + (cid:107) ∂ i (cid:101) φ k − T i,k (cid:107) L (Σ ) + (cid:107) ∂ ij ( δ lq · c ⊥ kq · ∂ l (cid:101) φ k ) − T ijE,k (cid:107) L (Σ ) + (cid:107) ∂ i ( δ lq · c ⊥ kq · ∂ l (cid:101) φ (cid:48) k ) − T iE,k (cid:107) L (Σ ) + (cid:107) ∂ ij ( (cid:101) φ k − δ lq · c kq · ∂ l (cid:101) φ k ) − T ijL,k (cid:107) L (Σ ) ≤ (cid:15) , (A.11) supp( T ij,k ) ∪ supp( T (cid:48) i,k ) ∪ supp( T ijE,k ) ∪ supp( T (cid:48) iE,k ) ∪ supp( T ijL,k ) ⊆ { u k = 0 } , (A.12)and T.V. ( T ij,k ) + T.V. ( T (cid:48) i,k ) + T.V. ( T ijE,k ) + T.V. ( T (cid:48) iE,k ) + T.V. ( T ijL,k ) ≤ (cid:15) . (A.13)• For φ := φ reg + (cid:80) k =1 (cid:101) φ k , the non-degeneracy condition 7 in Definition 4.3 holds. Step 2: Prescribing φ reg and using Lemma A.1. By the non-degeneracy assumption and Lemma A.3 (with R instead of R ), we can now fix smooth functions ψ , ψ compactly supported in B (0 , R ) satisfying (A.9).For λ = ( λ , λ ) ∈ [ − (cid:15) , (cid:15) ] × [ − (cid:15) , (cid:15) ] =: K , define φ := φ reg + (cid:88) k =1 (cid:101) φ k , φ λ reg := (cid:88) j =1 λ j ψ j , φ λ := φ λ reg + (cid:88) k =1 (cid:101) φ k . We now check that ( φ, φ λ ) obey conditions 1–3 preceding Lemma A.1. The only non-trivial condition tocheck is condition 3, which translates to finding λ = ( λ , λ ) ∈ K such that for j = 1 , , (cid:88) i =1 λ i (cid:90) Σ e γ ψ i ∂ j φ dx dx = − (cid:90) Σ e γ ( (cid:88) k =1 (cid:101) φ k ) ∂ j φ dx dx . (A.14)To see that (A.14) holds, note that given γ as in condition 3, we have | e γ − | ≤ max { e γ , e − γ }| γ | (cid:46) (cid:15) on B (0 , R ) . Hence, using (A.9), (A.10a) and (A.10b), we obtain that for (cid:15) sufficiently small, (cid:12)(cid:12)(cid:12)(cid:12) det (cid:20)(cid:82) Σ e γ ψ ∂ φ dx dx (cid:82) Σ e γ ψ ∂ φ dx dx (cid:82) Σ e γ ψ ∂ φ dx dx (cid:82) Σ e γ ψ ∂ φ dx dx (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:12) det (cid:20)(cid:82) Σ ψ ∂ φ dx dx (cid:82) Σ ψ ∂ φ dx dx (cid:82) Σ ψ ∂ φ dx dx (cid:82) Σ ψ ∂ φ dx dx (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) − C(cid:15) (cid:38) (cid:15) − C(cid:15) (cid:38) (cid:15) , (A.15)which, after using again (A.10a), (A.10b) and the upper bound (cid:107) ψ i (cid:107) H (Σ t ) ≤ , imply the entry-wise bound (cid:20)(cid:82) Σ e γ ψ ∂ φ dx dx (cid:82) Σ e γ ψ ∂ φ dx dx (cid:82) Σ e γ ψ ∂ φ dx dx (cid:82) Σ e γ ψ ∂ φ dx dx (cid:21) − = O ( (cid:15) − ) . (A.16)On the other hand, using (A.10a), (A.10b), we see that the RHS of (A.14) obeys (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Σ e γ ( (cid:88) k =1 (cid:101) φ k ) ∂ j φ dx dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) max k (cid:107) (cid:101) φ k (cid:107) L (Σ t ) (cid:107) ∂ j φ (cid:107) L (Σ t ) (cid:46) (cid:15) . (A.17)Combining (A.16) and (A.17), we see that (A.14) can be solved with λ such that | λ | , | λ | (cid:46) (cid:15) ; in particular, λ ∈ K .Therefore, Lemma A.1 shows that there exist λ ∈ K and an admissible initial data set ( φ, φ (cid:48) := e − γ φ λ , γ, K ) with the prescribed data. Returning to the assumptions in Step 1 above, it is easy to checkthat the data set satisfies all the conditions in Definition 4.3.71 .4 Construction of approximate data (Proof of Lemma 6.1) Proof of Lemma 6.1. Let ( φ, φ (cid:48) , γ, K ) be as in Definition 4.3. In particular, we have decompositions φ = φ reg + (cid:80) k =1 (cid:101) φ k and φ (cid:48) = φ (cid:48) reg + (cid:80) k =1 (cid:101) φ (cid:48) k . Step 1: Definitions of (cid:101) φ ( δ ) , ∗ k and (cid:101) φ ( δ ) , ∗ k : one-dimensional mollifications. As a first step towards prescribing (cid:101) φ ( δ ) k and (cid:101) φ ( δ ) k for the initial data of the δ -impulsive waves, we first define approximations of them, denotedrespectively by (cid:101) φ ( δ ) , ∗ k and (cid:101) φ ( δ ) , ∗ k , which are non-smooth but already satisfy the desired estimates.Let κ : R → [0 , be smooth and such that κ ( τ ) = 1 for τ ≤ , κ ( τ ) = 0 for τ ≥ , and (cid:82) R κ = 1 .Define (cid:101) φ ( δ ) , ∗ k and (cid:101) φ ( δ ) , ∗ k by performing -dimensional mollifications and translating by δ in the directionparallel to ∂ u k : (cid:101) φ ( δ ) , ∗ k ( x ) := 8 δ (cid:90) R κ ( 8 sδ ) (cid:101) φ k ( x − c k s + 12 c k δ, x − c k s + 12 c k δ ) ds, (A.18) (cid:101) φ ( δ ) , ∗ k ( x ) := 8 δ (cid:90) R κ ( 8 sδ )( e γ (cid:101) φ (cid:48) k )( x − c k s + 12 c k δ, x − c k s + 12 c k δ ) ds. (A.19)Using the constraint equation (4.1), we can prove a bound (see e.g., (A.6)) | e γ | , | e − γ | ≤ (cid:15) (cid:104) x (cid:105) (cid:15) (for (cid:15) sufficiently small). This allows us to pass between bounds for ˜ φ (cid:48) k and e γ ˜ φ (cid:48) k with only a small error.In particular, the following are easy to check.1. (cid:101) φ ( δ ) , ∗ k and (cid:101) φ ( δ ) , ∗ k are supported in { x ∈ Σ : u k (0 , x ) ≥ − δ } ∩ B (0 , R ) .• To see this, note that the support being in B (0 , R ) is obvious by (A.18), (A.19) since (cid:101) φ k , (cid:101) φ (cid:48) k aresupported in B (0 , R ) , and δ ≤ , R ≥ . Now since – (cid:101) φ k ( y ) is supported in u k = a k + c ki y i ≥ , – κ ( sδ ) is supported in s ∈ [ − δ , δ ] ,we deduce that (cid:101) φ ( δ ) , ∗ k is non-vanishing only when a k + c ki ( x i + c ki δ + c ki δ ) ≥ . Since (cid:80) i =1 c ki =1 , this means that supp( (cid:101) φ ( δ ) , ∗ k ) ⊆ { x ∈ Σ : u k ≥ − δ } . Similarly for (cid:101) φ ( δ ) , ∗ k .2. There exists a decreasing function b : [0 , → [0 , with lim δ → b ( δ ) = 0 such that (cid:107) (cid:101) φ ( δ ) , ∗ k − (cid:101) φ k (cid:107) H s (cid:48) (Σ ) + (cid:107) (cid:101) φ ( δ ) , ∗ k − e γ (cid:101) φ (cid:48) k (cid:107) H s (cid:48) (Σ ) ≤ b ( δ ) . (A.20)3. The following more quantitative convergences hold for lower norms: (cid:107) (cid:101) φ ( δ ) , ∗ k − (cid:101) φ k (cid:107) H (Σ ) + (cid:107) (cid:101) φ ( δ ) , ∗ k − e γ (cid:101) φ (cid:48) k (cid:107) L (Σ ) (cid:46) (cid:15)δ s (cid:48) . (A.21)4. (cid:101) φ ( δ ) , ∗ k and (cid:101) φ ( δ ) , ∗ k obey the following estimates (which follow from the given estimates (4.4a) and (4.4b)): (cid:107) (cid:101) φ ( δ ) , ∗ k (cid:107) W , ∞ (Σ ) + (cid:107) (cid:101) φ ( δ ) , ∗ k (cid:107) H s (cid:48) (Σ ) + (cid:107) (cid:101) φ ( δ ) , ∗ k (cid:107) L ∞ (Σ ) + (cid:107) (cid:101) φ ( δ ) , ∗ k (cid:107) H s (cid:48) (Σ ) ≤ (cid:15) , (A.22) (cid:107) δ iq c ⊥ kq ∂ i (cid:101) φ ( δ ) , ∗ k (cid:107) H s (cid:48)(cid:48) (Σ ) + (cid:107) δ iq c ⊥ kq ∂ i (cid:101) φ ( δ ) , ∗ k (cid:107) H s (cid:48)(cid:48) (Σ ) + (cid:107) (cid:101) φ ( δ ) , ∗ k − δ iq c kq ∂ i (cid:101) φ ( δ ) , ∗ k (cid:107) H s (cid:48)(cid:48) (Σ ) ≤ (cid:15) . (A.23)5. The following holds (which follow from the estimates and support properties in (4.5)–(4.7)): (cid:107) (cid:101) φ ( δ ) , ∗ k (cid:107) H (Σ ) + (cid:107) (cid:101) φ ( δ ) , ∗ k (cid:107) H (Σ ) + (cid:107) E k (cid:101) φ ( δ ) , ∗ k (cid:107) H (Σ ) + (cid:107) E k (cid:101) φ ( δ ) , ∗ k (cid:107) H (Σ ) + (cid:107) (cid:101) φ ( δ ) , ∗ k − δ iq c kq ∂ i (cid:101) φ ( δ ) , ∗ k (cid:107) H (Σ ) ≤ (cid:15) · δ − , (A.24) (cid:107) (cid:101) φ ( δ ) , ∗ k (cid:107) H (Σ \ S k ( − δ , − δ )) + (cid:107) (cid:101) φ ( δ ) , ∗ k (cid:107) H (Σ \ S k ( − δ , − δ )) ≤ (cid:15) . (A.25)72 To see (A.25), we in particular use a support argument like in point 1 above to show that thesingular part does not contribute outside S k ( − δ , − δ ) . Step 2: Defining φ ( δ ) reg , (cid:101) φ ( δ ) k , φ λ , ( δ ) reg and (cid:101) φ ( δ ) k . We now define φ ( δ ) and φ λ , ( δ ) , in anticipation of using Lemma A.1to obtain an admissible initial data set. We define φ ( δ ) := φ ( δ ) reg + (cid:88) k =1 (cid:101) φ ( δ ) k , φ λ , ( δ ) := φ λ , ( δ ) reg + (cid:88) k =1 (cid:101) φ ( δ ) k , (A.26)where φ ( δ ) reg , (cid:101) φ ( δ ) k , φ λ , ( δ ) reg and (cid:101) φ ( δ ) k are defined as follows:1. For φ reg as in Definition 4.3, define φ ( δ ) reg to be a smooth approximation of φ reg , supported in B (0 , R ) ,and such that (cid:107) φ ( δ ) reg − φ reg (cid:107) H s (cid:48) (Σ ) ≤ (cid:15)δ s (cid:48) . (A.27)2. For (cid:101) φ ( δ ) , ∗ k , (cid:101) φ ( δ ) , ∗ k as in Step 1, define (cid:101) φ ( δ ) k and (cid:101) φ ( δ ) k to respectively be smooth approximations of (cid:101) φ ( δ ) , ∗ k and (cid:101) φ ( δ ) , ∗ k , with support in B (0 , R ) , such that the following holds:(a) (cid:101) φ ( δ ) k and (cid:101) φ ( δ ) k are supported in { x ∈ Σ : u k (0 , x ) ≥ − δ } ∩ B (0 , R ) .(b) ( (cid:101) φ ( δ ) k , (cid:101) φ ( δ ) k ) is close to ( (cid:101) φ ( δ ) , ∗ k , (cid:101) φ ( δ ) , ∗ k ) in the following sense: (cid:107) (cid:101) φ ( δ ) k − (cid:101) φ ( δ ) , ∗ k (cid:107) H s (cid:48) (Σ ) + (cid:107) (cid:101) φ ( δ ) k − (cid:101) φ ( δ ) , ∗ k (cid:107) H s (cid:48) (Σ ) ≤ (cid:15)δ s (cid:48) . (A.28)(c) The estimates (A.22)–(A.25) hold, with ( (cid:101) φ ( δ ) , ∗ k , (cid:101) φ ( δ ) , ∗ k ) replaced by ( (cid:101) φ ( δ ) k , (cid:101) φ ( δ ) k ) , with (cid:15) replaced by (cid:15) , and (in the case of (A.25)) with S k ( − δ , − δ ) replaced by S k ( − δ, .3. For φ λ , ( δ ) reg , first define φ (cid:48) ( δ ) , ∗ reg be a smooth approximation of φ (cid:48) reg , supported in B (0 , R ) , and such that (cid:107) φ (cid:48) ( δ ) , ∗ reg − φ (cid:48) reg (cid:107) H s (cid:48) (Σ ) ≤ (cid:15)δ s (cid:48) . (A.29)Next, using (4.8), we can fix ψ , ψ as in the conclusion of Lemma A.3. Define then φ λ , ( δ ) reg := e γ φ (cid:48) ( δ ) , ∗ reg + λ ψ + λ ψ . (A.30) Step 3: Verifying (A.3) . In order to apply Lemma A.1, we first need to verify (A.3).Suppose γ = − γ asymp ω ( | x | ) log | x | + (cid:101) γ is given such that | γ asymp | ≤ (cid:15) and (cid:107) (cid:101) γ (cid:107) W , (Σ ) ≤ (cid:15) . Let K =[ − (cid:15), (cid:15) ] × [ − (cid:15), (cid:15) ] and our goal is to find λ = ( λ , λ ) ∈ K such that (A.3) is satisfied. Given the definitionsin Step 2, this means that need to solve for λ j (with j = 1 , ) which satisfies (cid:88) i =1 λ i (cid:90) Σ e γ ψ i ∂ j φ ( δ ) dx dx = − (cid:90) Σ ( e γ (cid:101) φ (cid:48) ( δ ) , ∗ reg + e γ (cid:88) k =1 (cid:101) φ ( δ ) k ) ∂ j φ ( δ ) dx dx . (A.31)73 tep 3(a): Controlling the RHS of (A.31) . We compute ( e γ φ (cid:48) ( δ ) , ∗ reg + e γ (cid:88) k =1 (cid:101) φ ( δ ) k ) ∂ j φ ( δ ) = ( e γ φ (cid:48) reg + e γ (cid:88) k =1 (cid:101) φ (cid:48) k ) ∂ j φ (cid:124) (cid:123)(cid:122) (cid:125) =: I + [( e γ − e γ ) φ (cid:48) ( δ ) , ∗ reg + ( e γ − e γ ) (cid:88) k =1 (cid:101) φ ( δ ) k ] ∂ j φ ( δ ) (cid:124) (cid:123)(cid:122) (cid:125) =: II + e γ ( φ (cid:48) ( δ ) , ∗ reg ∂ j φ ( δ ) − (cid:101) φ (cid:48) reg ∂ j φ ) + e γ ( ∂ j φ ( δ ) 3 (cid:88) k =1 (cid:101) φ ( δ ) k − ∂ j φ (cid:88) k =1 e γ (cid:101) φ (cid:48) k ) (cid:124) (cid:123)(cid:122) (cid:125) =: III . (A.32)Term I in (A.32) integrates to since ( γ, K, φ, φ (cid:48) ) is a given admissible initial data set and thus obeys (4.2).For the term II in (A.32), note that γ and γ can be bounded respectively by (A.6) and the assumptionson γ (see beginning of Step 3). Hence, using | e x − | ≤ max { e x , e − x }| x | , we obtain the following bound on B (0 , R ) , | e γ − e γ | ≤ | e γ − | + | e γ − | ≤ max { e γ , e γ , e − γ , e − γ }| ( γ | + | γ | ) (cid:46) (cid:15). (A.33)Using also the bounds for φ (cid:48) ( δ ) , ∗ reg , (cid:101) φ ( δ ) k and φ ( δ ) from Step 2, we thus have (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Σ (Term II in (A.32)) dx dx (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) . (A.34)Finally, for term III , we use (A.21), (A.26)–(A.29) together with (4.4a) and the bounds for γ as for term II to obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Σ (Term III in (A.32)) dx dx (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) δ s (cid:48) . (A.35) Step 3(b): Solving (A.31) . Using (4.3), (4.4a), (A.26), (A.21), (A.27), (A.28), and the fact that | e γ − | (cid:46) (cid:15) on the supports of ψ j and φ , it follows that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) det (cid:34)(cid:82) Σ e γ ψ ∂ φ ( δ ) dx dx (cid:82) Σ e γ ψ ∂ φ ( δ ) dx dx (cid:82) Σ e γ ψ ∂ φ ( δ ) dx dx (cid:82) Σ e γ ψ ∂ φ ( δ ) dx dx (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:12) det (cid:20)(cid:82) Σ ψ ∂ φ dx dx (cid:82) Σ ψ ∂ φ dx dx (cid:82) Σ ψ ∂ φ dx dx (cid:82) Σ ψ ∂ φ dx dx (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) − C ( (cid:15)δ s (cid:48) + (cid:15) ) (cid:38) (cid:15) − C ( (cid:15)δ s (cid:48) + (cid:15) ) (cid:38) (cid:15) , (A.36)for (cid:15) (and hence (cid:15) ) sufficiently small.Using the lower bound on the determinant in (A.36), as well as the upper bounds in (4.4a) and (A.9), itfollows that we have the entry-wise bound (cid:20)(cid:82) Σ e γ ψ ∂ φ dx dx (cid:82) Σ e γ ψ ∂ φ dx dx (cid:82) Σ e γ ψ ∂ φ dx dx (cid:82) Σ e γ ψ ∂ φ dx dx (cid:21) − = O ( (cid:15) − ) . (A.37)Using (A.37) and the estimates in Step 3(a), and recalling also that δ s (cid:48) ≤ (cid:15) , we then invert the linearmatrix to solve (A.31) with some ( λ , λ ) satisfying | λ | + | λ | (cid:46) (cid:15) − ( (cid:15) + (cid:15) δ s (cid:48) ) (cid:46) (cid:15) . (A.38)In particular, λ = ( λ , λ ) ∈ K . We have thus verified (A.3). Step 4: Application of Lemma A.1. By Step 3 and Lemma A.1, we know that there exist λ and functions ( γ ( δ ) , K ( δ ) ) such that ( φ ( δ ) , ( φ (cid:48) ) ( δ ) = e − γ ( δ ) φ λ , ( δ ) , γ ( δ ) , K ( δ ) ) is an admissible initial data set. Step 5: Checking the conclusions of Lemma 6.1. First, we prove point 1 of Lemma 6.1, i.e. we checkthat ( φ ( δ ) , ( φ (cid:48) ) ( δ ) = e − γ ( δ ) φ λ , ( δ ) , γ ( δ ) , K ( δ ) ) (given by Step 4) forms an admissible initial data set for three δ -impulsive waves with parameters (3 (cid:15), s (cid:48) , s (cid:48)(cid:48) , R, κ ) in Definition 4.8.74 The transversality condition holds trivially since c ki is defined as for the given ( φ, φ (cid:48) , γ, K ) .• The required support properties follow from points 1, 2(a), and 3 in Step 2.• The estimates in points 4–5 of Definition 4.3 and in (4.9)–(4.10) follow easily from (A.26), the conditionson φ ( δ ) reg , (cid:101) φ ( δ ) k , φ λ , ( δ ) reg and (cid:101) φ ( δ ) k in Step 2, together with (cid:107) ψ j (cid:107) H (Σ ) ≤ and the bound (A.38).We finally need to check the desired convergence (point 2 of Lemma 6.1). By the definition (A.26), andthe estimates (A.20), (A.27), (A.28) and (A.29), we have (cid:107) φ ( δ ) reg − φ reg (cid:107) H s (cid:48) (Σ ) + (cid:107) ( φ (cid:48) reg ) ( δ ) − φ (cid:48) reg (cid:107) H s (cid:48) (Σ ) + max k ( (cid:107) (cid:101) φ ( δ ) k − (cid:101) φ k (cid:107) H s (cid:48) (Σ ) + (cid:107) ( (cid:101) φ (cid:48) k ) ( δ ) − (cid:101) φ (cid:48) k (cid:107) H s (cid:48) (Σ ) ) (cid:46) b ( δ ) + | λ | + | λ | . (A.39)To proceed, we need to bound | λ | , | λ | with an estimate better than (A.38). For this, we need a betterbound compared to (A.34). Instead of (A.33), we use Sobolev embedding (part 1 of Proposition 9.1) toobtain | e γ ( δ ) − e γ | (cid:46) | γ asymp − γ ( δ ) asymp | + (cid:107) γ − γ ( δ ) (cid:107) H − (Σ ) on the support of φ . This in turn implies that (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Σ (Term II in (A.32)) dx dx (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) ( | γ asymp − γ ( δ ) asymp | + (cid:107) γ − γ ( δ ) (cid:107) H − (Σ ) ) . Hence, combining this with (A.37), (A.35) and the fact that I in (A.32) integrates to , we obtain | λ | + | λ | (cid:46) (cid:15) δ s (cid:48) + (cid:15) ( | γ asymp − γ ( δ ) asymp | + (cid:107) γ − γ ( δ ) (cid:107) H − (Σ ) ) . (A.40)On the other hand, taking the difference of the elliptic equations for ( γ, K ) (which hold because of Defini-tion 4.3) and those for ( γ ( δ ) , K ( δ ) ) (which hold because of Step 4), we have | γ asymp − γ ( δ ) asymp | + (cid:107) γ − γ ( δ ) (cid:107) H − (Σ ) + (cid:107) K − K ( δ ) (cid:107) H (Σ ) (cid:46) (cid:15) ( δ s (cid:48) + | λ | + | λ | + | γ asymp − γ ( δ ) asymp | + (cid:107) γ − γ ( δ ) (cid:107) H − (Σ ) + (cid:107) K − K ( δ ) (cid:107) H (Σ ) ) . 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