Rough solutions of the Einstein-vacuum equations

Transcription

1 Annals of Mathematics, 6 005, Rough solutions of the Einstein-vacuum equations By Sergiu Klainerman and Igor Rodnianski To Y. Choquet-Bruhat in honour of the 50 th anniversary of her fundamental paper [Br] on the Cauchy problem in General Relativity Abstract This is the first in a series of papers in which we initiate the study of very rough solutions to the initial value problem for the Einstein-vacuum equations epressed relative to wave coordinates. By very rough we mean solutions which cannot be constructed by the classical techniques of energy estimates and Sobolev inequalities. Following [Kl-Ro] we develop new analytic methods based on Strichartz-type inequalities which result in a gain of half a derivative relative to the classical result. Our methods blend paradifferential techniques with a geometric approach to the derivation of decay estimates. The latter allows us to take full advantage of the specific structure of the Einstein equations.. Introduction We consider the Einstein-vacuum equations, R αβ g =0 where g is a four-dimensional Lorentz metric and R αβ its Ricci curvature tensor. In wave coordinates α, g α = g μg μν g ν α =0, the Einstein-vacuum equations take the reduced form; see [Br], [H-K-M]. 3 g αβ α β g μν = N μν g, g with N quadratic in the first derivatives g of the metric. We consider the initial value problem along the spacelike hyperplane Σ given by t = 0 =0, 4 g αβ 0 H s Σ, t g αβ 0 H s Σ

2 44 SERGIU KLAINERMAN AND IGOR RODNIANSKI with denoting the gradient with respect to the space coordinates i, i =,, 3 and H s the standard Sobolev spaces. We also assume that g αβ 0 is a continuous Lorentz metric and 5 sup g αβ 0 m αβ 0 as r, =r where = 3 i= i and m αβ is the Minkowski metric. The following local eistence and uniqueness result well-posedness is well known see [H-K-M] and the previous result of Ch. Bruhat [Br] for s 4. Theorem.. Consider the reduced equation 3 subject to the initial conditions 4 and 5 for some s>5/. Then there eists a time interval [0,T] and unique Lorentz metric solution g C 0 [0,T] R 3, g μν C 0 [0,T]; H s with T depending only on the size of the norm g μν 0 H s. In addition, condition 5 remains true on any spacelike hypersurface, i.e. any level hypersurface of the time function t = 0. We establish a significant improvement of this result bearing on the issue of minimal regularity of the initial conditions: Main Theorem. Consider a classical solution of the equations 3 for which also holds. The time T of eistence depends in fact only on the size of the norm g μν 0 H s, for any fied s>. Remark.. Theorem. implies the classical local eistence result of [H-K-M] for asymptotically flat initial data sets Σ,g,k with g, k H s Σ and s > 5, relative to a fied system of coordinates. Uniqueness can be proved for additional regularity s>+ 5. We recall that an initial data set Σ,g,k consists of a three-dimensional complete Riemannian manifold Σ,g, a -covariant symmetric tensor k on Σ verifying the constraint equations: j k ij i trk =0, R k + trk =0, where is the covariant derivative, R the scalar curvature of Σ,g. An initial data set is said to be asymptotically flat AF if there eists a system of In other words for any solution of the reduced equations 3 whose initial data satisfy the constraint equations, see [Br] or [H-K-M]. The fact that our solutions verify plays a fundamental role in our analysis. We assume however that T stays sufficiently small, e.g. T. This a purely technical assumption which one should be able to remove.

3 NONLINEAR WAVE EQUATIONS 45 coordinates,, 3 defined in a neighborhood of infinity 3 on Σ relative to which the metric g approaches the Euclidean metric and k approaches zero. 4 Remark.3. The Main Theorem ought to imply eistence and uniqueness 5 for initial conditions with H s, s>, regularity. To achieve this we only need to approimate a given H s initial data set i.e. g H s Σ, k H s Σ, s> for the Einstein vacuum equations by classical initial data sets, i.e. H s data sets with s > 5, for which Theorem. holds. The Main Theorem allows us to pass to the limit and derive eistence of solutions for the given, rough, initial data set. We do not know however if such an approimation result for the constraint equations eists in the literature. For convenience we shall also write the reduced equations 3 in the form 6 g αβ α β φ = Nφ, φ where φ =g μν, N = N μν and g αβ = g αβ φ. Epressed relative to the wave coordinates α the spacetime metric g takes the form: 7 g = n dt + g ij d i + v i dtd j + v j dt where g ij is a Riemannian metric on the slices, given by the level hypersurfaces of the time function t = 0, n is the lapse function of the time foliation, and v is a vector-valued shift function. The components of the inverse metric g αβ can be found as follows: g 00 = n, g 0i = n v i, g ij = g ij n v i v j. In view of the Lorentzian character of g and the spacelike character of the hypersurfaces, 8 c ξ g ij ξ i ξ j c ξ, c n v g for some c>0. The classical local eistence result for systems of wave equations of type 6 is based on energy estimates and the standard H s L Sobolev inequality. 3 We assume, for simplicity, that Σ has only one end. A neighborhood of infinity means the complement of a sufficiently large compact set on Σ. 4 Because of the constraint equations the asymptotic behavior cannot be arbitrarily prescribed. A precise definition of asymptotic flatness has to involve the ADM mass of Σ,g. Taking the mass into account we write g ij = + M r δij + or as r = According to the positive mass theorem M 0 and M =0 implies that the initial data set is flat. Because of the mass term we cannot assume that g e L Σ, with e the 3D Euclidean metric. 5 Properly speaking uniqueness holds, with s>, only for the reduced equations. Uniqueness for the actual Einstein equations requires one more derivative; see [H-K-M].

4 46 SERGIU KLAINERMAN AND IGOR RODNIANSKI Indeed using energy estimates and simple commutation inequalities one can show that, 9 φt H s E φ0 H s with a constant E, 0 E = ep C t 0 φτ L dτ. By the classical Sobolev inequality, E ep Ct sup 0 τ t φτ H s dτ provided that s> 5. The classical local eistence result follows by combining this last estimate, for a small time interval, with the energy estimates 9. This scheme is very wasteful. To do better one would like to take advantage of the mied L t L norm appearing on the right-hand side of 0. Unfortunately there are no good estimates for such norms even when φ is simply a solution of the standard wave equation φ =0 in Minkowski space. There eist however improved regularity estimates for solutions of in the mied L t L norm. More precisely, if φ is a solution of and ɛ>0 arbitrarily small, φ L t L [0,T ] R 3 CT ɛ φ0 H +ɛ. Based on this fact it was reasonable to hope that one can improve the Sobolev eponent in the classical local eistence theorem from s> 5 to s>. This can be easily done for solutions of semilinear equations; see [Po-Si]. In the quasilinear case, however, the situation is far more difficult. One can no longer rely on the Strichartz inequality for the flat D Alembertian in ; we need instead its etension to the operator g αβ α β appearing in 6. Moreover, since the metric g αβ depends on the solution φ, it can have only as much regularity as φ itself. This means that we have to confront the issue of proving Strichartz estimates for wave operators g αβ α β with very rough coefficients g αβ. This issue was recently addressed in the pioneering works of Smith[Sm], Bahouri-Chemin [Ba-Ch], [Ba-Ch] and Tataru [Ta], [Ta], we refer to the introduction in [Kl] and [Kl-Ro] for a more thorough discussion of their important contributions. The results of Bahouri-Chemin and Tataru are based on establishing a Strichartz type inequality, with a loss, for wave operators with very rough

5 NONLINEAR WAVE EQUATIONS 47 coefficients. 6 The optimal result 7 in this regard, due to Tataru, see [Ta], requires a loss of σ = 6. This leads to a proof of local well-posedness for systems of type 6 with s>+ 6. To do better than that one needs to take into account the nonlinear structure of the equations. In [Kl-Ro] we were able to improve the result of Tataru by taking into account not only the epected regularity properties of the coefficients g αβ in 6 but also the fact that they are themselves solutions to a similar system of equations. This allowed us to improve the eponent s, needed in the proof of well-posedness of equations of type 6, 8 to s>+ 3. Our approach was based on a combination of the paradifferential calculus ideas, initiated in [Ba-Ch] and [Ta], with a geometric treatment of the actual equations introduced in [Kl]. The main improvement was due to a gain of conormal differentiability for solutions to the Eikonal equations 3 H αβ α u β u =0 where the background metric H is a properly microlocalized and rescaled version of the metric g αβ in 6. That gain could be traced to the fact that a certain component of the Ricci curvature of H has a special form. More precisely denoting by L the null geodesic vectorfield associated to u, L = H αβ β u α, and rescaling it in an appropriate fashion, 9 L = bl, we found that the null Ricci component R LL =RicHL, L, verifies the remarkable identity: 4 R LL = Lz Lμ L ν H αβ α β H μν +e where z O H and e O H. Thus, apart from Lz which is to be integrated along the null geodesic flow generated by L, the only terms which depend on the second derivatives of H appear in H αβ α β H and can therefore be eliminated with the help of the equations 6. In this paper we develop the ideas of [Kl-Ro] further by taking full advantage of the Einstein equations in wave coordinates 6. An important aspect of our analysis here is that the term Lz appearing on the right-hand side of 4 vanishes identically. We make use of both the vanishing of the Ricci curvature of g and the wave coordinate condition. The other important new features are the use of energy estimates along the null hypersurfaces 6 The derivatives of the coefficients g are required to be bounded in L t H s and L t L norms, with s compatible with the regularity required on the right-hand side of the Strichartz inequality one wants to prove. 7 Recently Smith-Tataru [Sm-Ta] have shown that the result of Tataru is indeed sharp. 8 The result in [Kl-Ro] applies to general equations of type 6 not necessarily tied to. In [Kl-Ro] we have also made the simplifying assumptions n = and v =0. 9 such that L, T H = where T is the unit normal to the level hypersurfaces associated to the time function t.

6 48 SERGIU KLAINERMAN AND IGOR RODNIANSKI generated by the optical function u and a deeper analysis of the conormal properties of the null structure equations. Our work is divided in three parts. In this paper we give all the details in the proof of the Main Theorem with the eception of those results which concern the asymptotic properties of the Ricci coefficients the Asymptotics Theorem, and the straightforward modifications of the standard isoperimetric and trace inequalities on -surfaces. We give precise statements of these results in Section 4. Our second paper [Kl-Ro] is dedicated to the proof of the Asymptotics Theorem which relies on an important result concerning the Ricci defect RicH. This result is proved in our third paper [Kl-Ro3]. We strongly believe that the result of our main theorem is not sharp. The critical Sobolev eponent for the Einstein equations is s c = 3. A proof of wellposedness for s = s c will provide a much stronger version of the global stability of Minkowski space than that of [Ch-Kl]. This is completely out of reach at the present time. A more reasonable goal now is to prove the L - curvature conjecture, see [Kl], corresponding to the eponent s =.. Reduction to decay estimates The proof of the Main Theorem can be reduced to a microlocal decay estimate. The reduction is standard; 0 we quickly review here the main steps. The precise statements and their proofs are given in Section 8. Energy estimates. Assuming that φ is a solution of 6 on [0,T] R 3 we have the a priori energy estimate: 5 φ L C φ0 Ḣs [0,T ]Ḣs with a constant C depending only on φ L and φ [0,T ] L L. [0,T ] L The Strichartz estimate. To prove our Main Theorem we need, in addition to 5 an estimate of the form: φ L [0,T ] L C φ0 H s for any s>. We accomplish it by establishing a Strichartz type inequality of the form, 6 φ L C φ0 [0,T ] L H +γ with any fied γ > 0. We achieve this with the help of a bootstrap argument. More precisely we make the assumption, 0 See [Kl-Ro] and the references therein. i.e., a classical solution according to Theorem..

7 NONLINEAR WAVE EQUATIONS 49 Bootstrap Assumption. 7 8 φ L + φ [0,T ] H+γ L B 0, [0,T ] L and use it to prove the better estimate: φ L CB 0 T δ [0,T ] L for some δ>0. Thus, for sufficiently small T > 0, we find that 6 holds true. Proof of the Main Theorem. This can be done easily by combining the energy estimates with the Strichartz estimate stated above. The Dyadic Strichartz Estimate. The proof of the Strichartz estimate can be reduced to a dyadic version for each φ λ = P λ φ, λ sufficiently large, where P λ is the Littlewood-Paley projection on the space frequencies of size λ Z, φ λ L CB 0 c [0,T ] L λ T δ φ H +γ, with λ c λ. Dyadic linearization and time restriction. Consider the new metric g <λ = P <λ g = μ M 0λ P μg, for some sufficiently large constant M 0 > 0, restricted to a subinterval I of [0,T] of size I Tλ 8ɛ0 with ɛ 0 > 0 fied such that γ > 5ɛ 0. Without loss of generality 3 we can assume that I =[0, T ], T Tλ 8ɛ 0. Using an appropriate now standard, see [Ba-Ch], [Ta], [Kl], [Kl-Ro] paradifferential linearization together with the Duhamel principle we can reduce the proof of the dyadic Strichartz estimate mentioned above to a homogeneous Strichartz estimate for the equation g αβ <λ α β ψ =0, with initial conditions at t = 0 verifying, 0 λ m m ψ0 L 0 λ m ψ0 L. There eists a sufficiently small δ>0, 5ɛ 0 + δ<γ, such that 9 P λ ψ L I L CB 0 T δ ψ0 Ḣ+δ. Rescaling. Introduce the rescaled metric 4 H λ t, =g <λ λ t, λ The low frequencies are much easier to treat. 3 In view of the translation invariance of our estimates. 4 H λ is a Lorentz metric for λ Λ with Λ sufficiently large. See the discussion following 33 in Section 8.

8 50 SERGIU KLAINERMAN AND IGOR RODNIANSKI and consider the rescaled equation H αβ λ α β ψ =0 in the region [0,t ] R 3 with t λ 8ɛ0. Then, with P = P, would imply the estimate 9. P ψ L I L CB 0 t δ ψ0 L Reduction to an L L decay estimate. The standard way to prove a Strichartz inequality of the type discussed above is to reduce it, by a TT type argument, to an L L dispersive type inequality. The inequality we need, concerning the initial value problem Hλ ψ = α H αβ λ H λ β ψ =0, H λ with data at t = t 0 has the form, m P ψt L CB 0 + dt k ψt + t t 0 δ 0 L for some integer m 0. Final reduction to a localized L L decay estimate. We state this as the following theorem: Theorem.. Let ψ be a solution of the equation, k=0 0 Hλ ψ =0 on the time interval [0,t ] with t λ 8ɛ0. Assume that the initial data are given at t = t 0 [0,t ], supported in the ball B 0 of radius centered at the origin. We fi a large constant Λ > 0 and consider only the frequencies λ Λ. q There eist a function dt, with t d Lq [0,t ] for some q> sufficiently close to, an arbitrarily small δ>0and a sufficiently large integer m>0 such that for all t [0,t ], m P ψt L CB 0 + dt k ψt + t t 0 δ 0 L. Remark.. In view of the proof of the Main Theorem presented above, which relies on the final estimate 8, we can in what follows treat the bootstrap constant B 0 as a universal constant and bury the dependence on it in the notation introduced below. k=0

9 NONLINEAR WAVE EQUATIONS 5 Definition.3. We use the notation A B to epress the inequality A CB with a universal constant, which may depend on B 0 and various other parameters depending only on B 0 introduced in the proof. The proof of Theorem. relies on a generalized Morawetz-type energy estimate which will be presented in the net section. We shall in fact construct a vectorfield, analogous to the Morawetz vectorfield in the Minkowski space, which depends heavily on the background metric H = H λ. In the net proposition we display most of the main properties of the metric H which will be used in the following section. Proposition.4 Background estimates. Fi the region [0,t ] R 3, with t λ 8ɛ0, where the original Einstein metric 5 g = gφ verifies the bootstrap assumption 7. The metric Ht, =H λ t, =P <λ gλ t, λ can be decomposed relative to our spacetime coordinates 3 H = n dt + h ij d i + v i dt d j + v j dt where n and v are related to n, v according to the rule. components n, v, and h satisfy the conditions The metric 4 c ξ h ij ξ i ξ j c ξ, n v h c>0, n, v c. In addition, the derivatives of the metric H verify the following: m H L [0,t ] L λ 8ɛ0, m 0 +m H L [0,t ] L λ 4ɛ0, m 0 +m H L [0,t ] L λ 4ɛ0, m 0 +m H L [0,t ] L λ m, m +4ɛ 0 +m H L [0,t ] L λ 4ɛ0, +4ɛ 0 m m H αβ α β H L [0,t ] L λ 8ɛ0, m 0 m RicH L [0,t ] L λ, m 0 m RicH L [0,t ] L λ 8ɛ0, m 0. 5 Recall that in fact g is φ. Thus, in view of the nondegenerate Lorentzian character of g the bootstrap assumption for φ reads as an assumption for g.

10 5 SERGIU KLAINERMAN AND IGOR RODNIANSKI Proof. This follows from Proposition 8. and a straightforward rescaling argument. Remark.5. Among the long list of estimates above, the main ones reflect that H is controlled in L t L, + H is in L t L, and that RicH H. The remaining estimates follow from these by rescaling, Sobolev, and frequency localization. 3. Generalized energy estimates and the Boundedness Theorem Consider the Lorentz metric H = H λ, as in, verifying, in particular, the properties of Proposition.4 in the region [0,t ] R 3, t λ 8ɛ0. We denote by D the compatible covariant derivative and by the induced covariant differentiation on. We denote by T the future oriented unit normal to and by k the second fundamental form. Associated to H we have the energy momentum tensor of H, 33 Q μν = Q[ψ] μν = μ ψ ν ψ H μνh αβ α ψ β ψ. The energy density associated to an arbitrary timelike vectorfield K is given by QK, T. We consider also the modified energy density, 34 QK, T = Q[ψ]K, T =Q[ψ]K, T+tψT ψ ψ T t, and the total conformal energy, 35 Q[ψ]t = Q[ψ]K, T. We recall below the statement of the main generalized energy estimate upon which we rely; see [Kl-Ro]. Proposition 3.. Let K be an arbitrary vectorfield with deformation tensor K π μν = L K H μν = D μ K ν + D ν K μ and ψ a solution of H ψ =0. Then Q[ψ]t =Q[ψ]t 0 Q αβ K π αβ + 36 ψ H Ω [t 0,t] R 4 3 [t 0,t] R 3 where 37 K π = K π ΩH and Ω is an arbitrary function.

11 NONLINEAR WAVE EQUATIONS 53 Remark 3.. In the particular case of the Minkowski spacetime we can choose K to be the conformal time-like Killing vectorfield K = t + r t + r +t r t r. In this case we can choose Ω = 4t and obtain the total conservation law, Q[ψ]t =Q[ψ]t 0. This conservation law can be used to get the desired decay estimate for the free wave equation; see [Kl]. As in [Kl-Ro] we construct a special vectorfield K whose modified deformation tensor K π is such that we can control the error terms Q αβ K π αβ + ψ H Ω. [t 0,t] R 4 3 [t 0,t] R 3 As in [Kl-Ro] we set 6 38 K = nu L + u L with u, u, L, L defined as follows: 39 Optical function u. This is an outgoing solution of the Eikonal equation H αβ α u β u =0 with initial conditions uγ t =t on the time ais. The time ais is defined as the integral curve, originating from zero on Σ 0, of the forward unit normal T to the hypersurfaces. The point Γ t is the intersection between Γ and. The level surfaces of u, denoted C u are outgoing null cones with vertices on the time ais. Clearly, 40 T u = u h where h is the metric induced by H on, u h = 3 i= e iu relative to an orthonormal frame e i on. 4 Canonical null pair L, L. L = bl = T + N, L =T L = T N with L = H αβ β u α the geodesic null generator of C u, b the lapse of the null foliationor shortly null lapse defined by 4 b = L,T = T u, 6 Observe that this definition of K differs from the one in [Kl-Ro] by an important factor of n.

12 54 SERGIU KLAINERMAN AND IGOR RODNIANSKI and N the eterior unit normal, along, to the surfaces S t,u, i.e. the surfaces of intersection between and C u. We shall also use the notation The function u = u +t. e 3 = L, e 4 = L. The S t,u foliation. The intersection between the level hypersurfaces 7 and u form compact - Riemannian surfaces denoted by S t,u. We define rt, u by the formula AreaS t,u = 4πr. We denote by / the induced covariant derivative on S t,u. A vectorfield X is called S-tangent if it is tangent to S t,u at every point. Given an S-tangent vectorfield X we denote by / N X the projection on S t,u of N X. Remark 3.3. Observe that in Minkowski space u = t r, r =, L = t + r, S t,u are the spheres centered at t, 0 and radius r = t u. With the help of these constructions the proof of the L L decay estimate stated in Theorem. can be reduced to the following: Theorem 3.4 Boundedness Theorem. Consider the Lorentz metric H = H λ as in verifying, in particular, the properties of Proposition.4 in the region [0,t ] R 3, t λ 8ɛ0.Letψ be a solution of the wave equation H ψ = α H αβ 43 H β ψ =0 H with initial data ψ[t 0 ], at t = t 0 >, supported in the geodesic ball B 0. Let D u be the region determined by u>u in the slab [0,t ] R 3. For all t 0 t t, ψt is supported in D t0 D 0 and 44 Proof. See Section 5. Q[ψ]t Q[ψ]t 0. We consider also the auiliary energy type quantity, E[ψ]t =E i [ψ]t+e e [ψ]t where, E i [ψ]t= ζt ψ + ψ Σ t E e [ψ]t= ζ u Lψ + u Lψ + u /ψ + ψ. with ζ a smooth cut-off function equal to in the wave zone region u t. 7 The level hypersurfaces of u are outgoing null cones C u with vertices on the time ais Γ t.

13 NONLINEAR WAVE EQUATIONS 55 In the proof of Theorem 3.4 we need the following comparison between the quantity Qt and the auiliary norm Et = E[ψ]t. Theorem 3.5 The comparison theorem. as in Theorem 3.4, for any t t, Under the same assumptions E[ψ]t Q[ψ]t. Proof. See Section The Asymptotics Theorem and other geometric tools In this section we record the crucial properties of all the important geometric objects associated to our spacetime foliations, C u and S t,u introduced above. Most of the results of this section will be proved only in the second part of this work. We start with some simple facts concerning the parameters of the foliation relative to the spacetime geometry associated to the metric H = H λ. The foliation. Recall, see 3, that the parameters of the foliation are given by n, v, the induced metric h and the second fundamental form k ij, according to the decomposition, 45 H = n dt + h ij d i + v i dt d j + v j dt, with h ij the induced Riemannian metric on, n the lapse and v = v i i the shift of H. Denoting by T the unit, future oriented, normal to and k the second fundamental form k ij = D i T, j we find, 46 t = nt + v, t,v =0, k ij = L T H ij = n t h ij L v h ij with L X denoting the Lie derivative with respect to the vectorfield X. We also have the following see 8, 4, and 35 in Section 8: 47 c ξ h ij ξ i ξ j c ξ, c n v h for some c>0. Also n, v, n + v + h + k H. S t,u - foliation. We define the Ricci coefficients associated to the S t,u foliation and null pair L, L.

14 56 SERGIU KLAINERMAN AND IGOR RODNIANSKI Definition 4.. Using an arbitrary orthonormal frame e A A=, on S t,u we define the following tensors on the surfaces S t,u : 50 χ AB = D A e 4,e B, χ AB = D A e 3,e B, η A = D 3e 4,e A, η A = D 4e 3,e A, ξ A = D 3e 3,e A. Using the parameters n, v, k of the foliation we findsee [Kl-Ro] and [Kl-Ro], χ AB = χ AB k AB, η A = k AN + n / A n, ξ A = k AN η A + n / A n, η A = b / A b + k AN. Thus all the Ricci coefficients can be epressed in terms of k ij, n, the scalar function b and, most important, the Ricci coefficients χ and η. Recall that χ decomposes into its trace tr χ and traceless part ˆχ; see [Kl-Ro]. We shall also denote by θ AB = / A N,e B the second fundamental form of S t,u relative to. It is easy to check that χ AB = k AB + θ AB. We consider the parameters b, trχ, ˆχ and η associated to the S t,u foliation according to 4 and 50. For convenience we shall introduce the quantity: 5 Θ= tr χ r + tr χ + ˆχ + η. nt u Remark 4.. Strictly speaking we need only one of the two quantities tr χ r, tr χ nt u in the epression above. Indeed we show in [Kl-Ro] that these two are comparable. Remark 4.3. Simple calculations based on Definition 4., see also Ricci equations in Section of [Kl-Ro], allow us to derive the following: 5 DL, DL, N r +Θ+ H. Remark 4.4. We shall make use of the following simple commutation estimates; see Lemma 3.5 in [Kl-Ro], 53 / N / / / N f r +Θ+ H f. We state below the crucial theorem which establishes the desired asymptotic behavior of these quantities relative to λ.

15 NONLINEAR WAVE EQUATIONS 57 Theorem 4.5 The asymptotics theorem. In the spacetime region D 0 see Theorem 3.4 the quantities b, Θsatisfy the following estimates: b n λ 4ɛ0, Θ L t L λ 3ɛ0, Θ Lq S t,u λ 3ɛ0, q 4. In addition, in the eterior region u t/, 57 Θ L S t,u t λ ɛ0 + λ ɛ Ht L for an arbitrarily small ɛ>0. The following estimates hold for the derivatives of tr χ: 58 sup L tr χ u r t 59 L S t,u L t sup / tr χ L S t,u L t u t sup /,L u t + sup L tr χ u nt u L S t,u L t λ 3ɛ0, t + sup / tr χ u t In addition, there are weak estimates of the form, 60 tr χ nt u nt u L S t,u L t λ 3ɛ0. L S t,u λ C for some large value of C. There also is the following comparison between the functions r and t u, 6 c r t u c. The proof of the Asymptotics Theorem is truly at the heart of this work and it is quite involved. Our second paper [Kl-Ro] is almost entirely dedicated to it. Remark 4.6. Observe that the estimate 55 holds true also for H. We shall show, see [Kl-Ro, Prop. 7.4], that H also verifies the estimate 56. Thus we can incorporate the term H in the definition 5 of Θ. 6 Θ= tr χ r + tr χ + ˆχ + η + H. nt u For convenience we shall also often use Θ to denote OΘ. We shall do this freely throughout this paper. The proof of the net proposition will be delayed to [Kl-Ro3]; see also [Kl-Ro].

16 58 SERGIU KLAINERMAN AND IGOR RODNIANSKI Proposition 4.7. Let S t,u be a fied surface in D 0. i Isoperimetric inequality. For any smooth function f : S t,u R there is the following isoperimetric inequality: f 63 S t,u /f + trθ f. S t,u ii Sobolev Inequality. For any δ 0, and p from the interval p, ], δp p+δp 64 sup f r ɛp p+δp S t,u /f + r f S t,u [ ] δ /f p + r p f p S t,u p+δp. iii Trace Inequality. For an arbitrary function f : R such that f H +ɛ R 3, 65 f L S t,u +ɛ f L + ɛ f L. More generally, for any q [, 66 f Lq S t,u 3 +ɛ q f L + 3 ɛ q f L. Also, considering the region Et t = {0 u t }, we have the following: 67 f L S t,u Nf L Et t f L Et t + t f L Et t. We shall make use of the following; see Lemma 6.3 in [Kl-Ro]. Proposition 4.8. The following inequality holds for all t [,t ] and <p<: V w t p sup V 68 L u S t,u /w + r w, where p is the eponent dual to p. We shall also make use of the variant, V w t p V p p 69 L sup V L u S t,u /w + r w. In particular, if V L is bounded by some positive power of λ, and we restrict ourselves to the eterior region Et t, we deduce that for every ε>0 and some constant C, V w t λ Cε Et t sup V ε L S E[w]t. t,u 0 u t/ Proof. The proof is straightforward and relies only on the isoperimetric inequality 63; see also 6.. in [Kl-Ro].

17 NONLINEAR WAVE EQUATIONS Proof of the Boundedness Theorem We first calculate the components of the modified 8 deformation tensor π = K π = K π 4tH of our vectorfield K = nu L + u L. Recall that u =t u and L = L +T ;thus L u =4ub, L u =4un, L u =4un b. We proceed as in Section 6. of [Kl-Ro] to calculate the null components of π = K π relative 9 to e 4 = L, e 3 = L and e A A=, an arbitrary orthonormal frame on S t,u, 7 π 44 =u n k NN n e 4 n, π 34 =4unn b +u n knn n e 4 n + u n knn n e 3 n, π 33 = 8u nn b u n knn + n e 3 n, π 3A = u nη A + k AN n / A n+u nξ A, π 4A = u nη A k AN n / A n, π AB =tnt u tr χ δab +4tnt uˆχ AB u nk AB nt u where k NN = k NN n N n. The following proposition concerning the behavior of the null components of π is an immediate consequence of the above formulae and the Asymptotics Theorem stated above. Proposition 5.. u π 44 L t L, λ 3ɛ0 u π 34 L t L, λ 3ɛ0 u π 33 L t L, λ 3ɛ0 u π 3A L t L, λ 3ɛ0 u π 4A L t L, λ 3ɛ0 u π AB L t L. λ 3ɛ0 The proof of the boundedness theorem relies on the generalized energy identity 36 with K = n u L + u L and Ω = 4t. Thus, Q[ψ]t=Q[ψ]t 0 Q αβ K π 7 αβ + ψ H t [t 0,t] R 3 [t 0,t] R 3 = Q[ψ]t 0 J + Y. 8 corresponding to the choice Ω = 4t. 9 We say that e i =,,3,4 forms a null frame.

18 60 SERGIU KLAINERMAN AND IGOR RODNIANSKI Observe that we can decompose: J = Q αβ [ψ] π αβ [t 0,t] R 3 = [t 0,t] R 4 π 33Lψ + 4 π 44Lψ + π 34 /ψ π 4A Lψ / A ψ 3 π 3A Lψ / A ψ + π AB / A ψ / B ψ +tr π LψLψ /ψ. Consider, for eample, I = [t 0,t] R π 3 4A Lψ / A ψ. We can estimate it as follows: I uu π 4A u Lψ + u / A ψ [t 0,t] R 3 t t 0 uu π 4A L E[ψ]τ dτ. Making use of the comparison theorem and the estimate uu π 4A L t L we infer that, λ 3ɛ0 t I uu π 4A L Q[ψ]τ dτ λ 3ɛ0 sup Q[ψ]τ. t 0 [t 0,t] We can proceed in the same manner with all the terms of J with the eception of [t 0,t] R tr πlψlψ. Observe that 0 3 tr π = δ AB π AB =tnt u [t 0,t] R 3 u ntrklψlψ t tr χ u ntrk, nt u [t 0,t] R 3 trk u Lψ + u Lψ H L E[ψ]τ dτ. t 0 Since H L t L, this term can be treated in the same manner as I. λ 4ɛ0 We are thus left with the integral B = tnt u tr χ LψLψ. [t 0,t] R nt u 3 All other terms J Bcan be estimated in precisely the same manner, using the comparison theorem and the estimates of Proposition 5., by 73 J B λ 3ɛ0 sup Q[ψ]τ. [t 0,t] 0 We use tr here to denote the trace relative to the surfaces S t,u. Thustrk = δ AB k AB. We use Trk = h ij k ij to denote the usual trace of k with respect to.

19 NONLINEAR WAVE EQUATIONS 6 To estimate the remaining term B requires a more involved argument. In fact we shall need more information concerning the geometry of the null cones C u and surfaces S t,u. Denote Et t the eterior region Et t = {0 u t/}. Let ζ be a smooth cut-off function with support in Et t. Observe that t ψ + ψ 74 ζ ζ Q[ψ]t. We can split the remaining integral B = B i + B e, B i = tnt u tr χ LψLψ ζ, [t 0,t] R nt u 3 B e = tnt u tr χ LψLψζ. [t 0,t] R nt u 3 With the help of 74 the first integral can be estimated as follows: B i tr χ [t 0,t] R nτ u τ ψ ζ 3 t tr χ nτ u L Q[ψ]τ dτ. t 0 In view of the estimate tr χ nt u L t L Theorem 4.5 we infer that, λ 3ɛ0, given by the Asymptotics B i λ 3ɛ0 sup Q[ψ]τ. [t 0,t] Therefore, it remains to estimate B e. According to the Asymptotics Theorem the quantity z =trχ nt u verifies the following estimates: z L t L λ 3ɛ0, z L S t,u λ ɛ0, sup u t /z L S t,u L t λ 3ɛ0, sup L z L S t,u L t λ 3ɛ0. u t Remark 5.. The same estimates hold true if we replace tr χ nt u by tr χ r. It would therefore suffice to prove the following result. Using the estimates 75, 76 we shall prove that: 77 B e = tnt uzlψlψζ λ ɛ0 sup Q[ψ]τ. [t 0,t] R 3 [t 0,t]

20 6 SERGIU KLAINERMAN AND IGOR RODNIANSKI To prove 77 we need to rely on the fact that ψ is a solution of the wave equation H ψ = 0. We shall also make use of the following standard integration by parts formulae, 78 FNG = NF + trθ + n Nn F G, where N is the unit normal to S t,u. If Y is a vectorfield in T tangent to S t,u then 79 F div / Y = It is also not difficult to verify that /F +b /b+ n /nf Y. 80 FTG = [t 0,t] R 3 T F +Trk+ divvg + [t 0,t] R 3 FG FG. 0 Writing L = T N we integrate by parts and epress the integral B e in the form, 8 Be = I + I + I 3 I 4, I = ζntt uz LLψ ψ, [t 0,t] R 3 I = Lζntt uz [t 0,t] R 3 +trθ + n Nn Tr k divvζntt uz Lψ ψ, I 3 = ζntt uzlψψ, Σ t I 4 = ζntt uzlψψ. 0 We first handle the boundary terms I 3, I 4. With the help of Proposition 4.8 which we can apply in view of the estimates 57 for Θ as well as the estimate 6 for H we have nt uzψ L Et t λ Cɛ sup nz ɛ/ L S E t,u [ψ]t. u t These are simple adaptations of the formulae in Lemma 6., [Kl-Ro].

21 NONLINEAR WAVE EQUATIONS 63 Therefore, I 3 ζntt uzlψψ nt uz tlψψ Et t tlψ L nt uzψ L Et t nt u zψ L Et te [ψ]t λ Cɛ sup nz ɛ/ L S E[ψ]t t,u E[ψ]t. λ ɛ0 s t The last inequality followed from the boundness of n and 75. A similar estimate holds for the second boundary term I 4. To estimate I we observe that, as an immediate consequence of Theorem 4.5, we have Denoting Lt, Lt u, Lζ t. Θt, = tr χ + ˆχ + η + H, nt u we easily find t I τ Lz + τ z + τ Θ z Lψ ψ dτ. t 0 Et τ To treat the term involving Lz we proceed as in the case of I 3 ; we estimate Et τ τ Lz Lψ ψ dτ by Cauchy-Schwartz followed by an application of Proposition 4.8. The space integral of the other two terms can be estimated as follows: τ z + τ Θ z Lψ ψ dτ z L + τ Θ L z L E[ψ]τ. Et τ Consequently, using the inequalities 75, 76 for z as well as the weak estimate 60 and the estimates for Θ from the Asymptotics Theorem 4.5 we obtain as desired. I t t 0 λ Cɛ sup Lz ɛ/ L u τ S t,u E[ψ]τ dτ + z L Σ τ + τ Θ L Σ τ z L Σ τ λ C ɛ sup Lz L S t,u ɛ/ L u τ t + z L t L + λ Θ L t L z L t L sup Q[ψ]τ [t 0,t] λ ɛ0 sup Q[ψ]τ [t 0,t]

22 64 SERGIU KLAINERMAN AND IGOR RODNIANSKI It remains therefore to consider I. We shall make use of the fact that ψ is a solution of the wave equation. This allows us to epress the LLψ in terms of the angular laplacian / and lower order terms. Epressed relative to a null frame the wave operator H ψ takes the form H ψ = H αβ ψ ;αβ = ψ ;43 + ψ ;AA, where ψ ;eie j = e j e i ψ D ei e j ψ. We use the Ricci formulas: D 3 e 4 = η A e A + k NN e 4, and D B e A = / B e A + χ ABe 3 + χ AB e 4 to derive 8 H ψ = LLψ + /ψ+η A / A ψ + tr χlψ + tr χ + k NN Lψ. As a result of this calculation 83 I = ζnττ uzllψ ψ = ζnττ uz /ψψ [t 0,t] R 3 [t 0,t] R 3 + ζnττ uz tr χlψψ [t 0,t] R 3 + ζnττ uz η A / A ψ + [t 0,t] R tr χ + k NN Lψ ψ 3 = I + I + I 3. Consider first I 3. Since t u t, t I 3 τ z Θ /ψ+ τ 84 +ΘLψ ψ t 0 Et τ t τ z L Σ τ Θ L Σ τ + z L Σ τ E[ψ]τ dτ t 0 λ ɛ0 sup Q[ψ]τ [t 0,t] as before. To estimate I we need first to integrate once more by parts. I = Lζnττ uz tr χ 4 [t 0,t] R 3 + trθ + n Nn Tr k divv ζnττ uztr χ ψ + ζnττ uz tr χψ ζnττ uz tr χψ All terms can be treated as above. Take, for eample, the worst term involving Ltr χ. Recall that Ltr χ =L tr χ + L L tr χ + r r r r + r Θ. the Laplace-Beltrami operator on S t,u.

23 NONLINEAR WAVE EQUATIONS 65 Thus ζntt uzltr χψ [t 0,t] R 3 t τ z L tr χ + t 0 Et τ r τ + τ Θ ψ t τ z L tr χ ψ t 0 Et τ r t + z L Σ τ + τ z L Σ τ Θ L Σ τ E[ψ]τ dτ. t 0 The second term has already been treated above; see 83. To estimate the first we apply first Cauchy-Schwartz and then make use of Proposition 4.8, t τ z L tr χ ψ t 0 Et τ r t τ z L tr χ ψ t 0 r L Et τ E [ψ]τ dτ t λ Cɛ sup τ z L tr χ ɛ/ t 0 u τ r L S t,u E[ψ]τ dτ. Taking into account the estimates in 75, 76 and Remark 5. we deduce, t λ Cɛ sup τzl tr χ ɛ/ r L S t,u t 0 u τ λ C ɛ t z L t L sup L tr χ ɛ/ u r L S t,u L t λ ɛ0. t Therefore, 85 I λ ɛ0 sup Q[ψ]τ. [t 0,t] Finally we estimate I = [t 0,t] R ζntt uz /ψψ by integrating once 3 more by parts as follows: I = ζntt uz /ψ [t 0,t] R 3 n b / A bn ζ ntt uz / A ψψ. [t 0,t] R 3 The first integral on the right can easily be estimated t 86 ζntt uz /ψ z L E[ψ]τ dτ [t 0,t] R 3 t 0 z L t L sup Q[ψ]τ [t 0,t] λ 3ɛ0 sup Q[ψ]τ. [t 0,t]

24 66 SERGIU KLAINERMAN AND IGOR RODNIANSKI To estimate the second we write schematically / bn ζtt uz tt u /bz + tt u /z+ tt uzθ = tt u /z+ tt uzθ since / A b = bη A k AN. Thus with the help of Proposition 4.8 using also the weak estimate 60, we have n b / A bn ζττ uz / A ψψ [t 0,t] R 3 t τ /z + τ z Θ τ / A ψ ψ t 0 Et τ t λ Cɛ sup /z ɛ/ L t 0 u τ S + τ z t,u L Σ τ Θ L Σ τ E[ψ]τ dτ. Using 76 once more we have, t λ Cɛ sup /z ɛ/ L t 0 u τ S + τ z t,u L Σ τ Θ L Σ τ dτ λ C ɛ sup /z L S t,u ɛ/ L + t z u t t L t L Θ L t L. λ ɛ0 Therefore, combining this with 86 we infer that, 87 I λ ɛ0 sup Q[ψ]τ. [t 0,t] Recalling also 85 and 84 we conclude that 88 I λ ɛ0 sup Q[ψ]τ. [t 0,t] Since I,I 3,I 4 and B i have already been estimated we finally derive, 89 B λ ɛ0 sup Q[ψ]τ [t 0,t] as desired. This combined with 73 yields, 90 J λ ɛ0 sup Q[ψ]τ. [t 0,t] Going back to the identity 7 we still have to estimate Y. For this we only need to observe that H t depends only on the first derivatives of H. Thus also 9 Y λ ɛ0 sup Q[ψ]τ. [t 0,t] Therefore, sup [t 0,t] which implies the boundedness theorem. Q[ψ]τ Q[ψ]t 0 +λ ɛ0 sup Q[ψ]τ [t 0,t]

25 NONLINEAR WAVE EQUATIONS Proof of the comparison theorem We proceed precisely as in [Kl-Ro, 6.]. The purpose of the calculation below is to estimate the potentially negative term tψt ψ n ψ, appearing in the integral of QK, T[ψ], via integration by parts in terms of the positive terms of the Morawetz energy. 3 Define S and S: 9 S = ul + ul, S = ul ul. Since u = u +t, L = T N, L = T + N, tt = 4 u + ul + L =S u ul L =S t un, 4 tt = tl + L = t t u S t t u N. Therefore, with the help of the identities 78, and Nt =0,Nu = b, ψttψ= ψsψ t unψ = ψsψ+ b +t u trθ + n Nn ψ, ψttψ= ψ t t u Sψ t t u Nψ = ψ t t u Sψ t + Σt t u b +t u trθ + n Nn ψ. Recall that θ AB = χ AB + k AB. Recall also that Θ was defined in 6. Θt, = tr χ r + tr χ + ˆχ + η + H. nt u Thus, ψttψ = ψsψ+ b + n +t uθ ψ, ψttψ = ψ t t u Σt Sψ+ t b t u + n +t uθ ψ. Recall, from the Asymptotics Theorem 4.5, b n λ 4ɛ0. 3 In flat space this was first done in [Kl3]; see also [Kl]. It is also related to the well known Morawetz calculation in [Mor].

26 68 SERGIU KLAINERMAN AND IGOR RODNIANSKI Also, since n is bounded away from zero so is b. Therefore, 3 ψttψ = ψsψ+ +t uθ + λ 4ɛ0 ψ, n ψttψ = ψ t t u Σt Sψ+ t t u +t uθ + λ 4ɛ0 ψ. n Since QK, T[ψ] = n u Lψ + u Lψ +u + u /ψ +tψt ψ n ψ, 4 and 4 u Lψ + u Lψ = Sψ +Sψ we can introduce positive constants A, B : A + B = such that Q[ψ]t= nsψ +AψSψ+ 3 n A n ψ + t uθ + λ 4ɛ0 ψ + nsψ +Bψ t t u Sψ t t + n t u Bψ + t uθ + λ 4ɛ 0 t u ψ + nu + u /ψ. 4 For any values of A, B such that <A< and 0 <B< it is possible to find positive constants c,c such that nsψ +AψSψ+ n 3A ψ c Sψ + ψ, nsψ +Bψ t t u Sψ+ n B t t u ψ c Sψ + Therefore, Q[ψ]t u Lψ + u Lψ +u + u /ψ + + t + t t u uθ + λ 4ɛ0 ψ, Q[ψ]t u Lψ + u Lψ +u + u /ψ + t + t u t uθψ. Therefore it suffices to show that t t u t uθψ λ ɛ0 t /ψ + + t t u ψ. t t u ψ t t u ψ t t u ψ.

27 NONLINEAR WAVE EQUATIONS 69 Consider the worst term t Σt t u Θψ t Θ ψ t Σt 94 t u ψ. According to the estimate 68 of Proposition 4.8, applied to eponent p such that p = q, t Θ ψ t 4 q sup Θ L u q S t,u /ψ + r ψ. t Or, since according to 6, c r t u c, and with the help of the estimate 57 for Θ with q> sufficiently close to, t Θ ψ λ 5ɛ0 t /ψ t + t u ψ. Thus, back to 94, t Σt t u Θψ λ ɛ0 t /ψ t 95 + t u ψ as desired in the proof of 93. The remaining term on the left-hand side of 93 is easier to treat. 7. Proof of the L L decay estimate; Theorem. In this section we rely on the Boundedness Theorem 3.4 to prove the crucial Theorem.. Recall that E[ψ] =E i [ψ]+e e [ψ], where E i [ψ]t= t ψ + ψ ζ, E e [ψ]t= u Lψ + u /ψ +u Lψ + ψ ζ with a cut-off function ζ equal to in the region u t. Estimate for ζpψ. Observe that since the projector P is an averaging operator on the scale of size and ζ is a cut-off function with the scale of size t, we can essentially write that ζpψ P ψ ζ. Thus the Bernstein inequality, followed by the facts ζ ψ L t E [ψ]t and ζ t, imply that 96 P ψt ζ L ψ ζ L t E [ψ]t as desired.

28 70 SERGIU KLAINERMAN AND IGOR RODNIANSKI Estimate for ζpψ. It clearly suffices to establish the estimate for Pψt, at any point t, with 0 u t 4. According to the Sobolev inequality 64, with p = 4, of Proposition 4.7 we have for any positive δ<, sup Pψ S t,u t 4δ 4+δ S t,u /Pψ + t 4δ 4+δ [ Pψ S t,u /Pψ 4 + t 4 Pψ 4 ] δ 4+δ By the isoperimetric inequality 63 applied to Pψ and /Pψ, Pψ 4 /Pψ Pψ + Pψ, S t,u S t,u S t,u t S t,u /Pψ 4 / Pψ /Pψ + /Pψ. S t,u S t,u S t,u t S t,u In addition, by the trace inequality 65, 4 f Nf f + f. S t,u Et t Et t t Et t Here, Et t = {0 u t } and N is the vectorfield of the unit normals to S t,t ρ. Thus, setting ε = 4δ 4+δ, using the fact that t, and applying the Hölder inequality, we obtain 97 sup Pψ t ε / N /Pψ + /Pψ + NPψ S t,u Et t t + Pψ ε I ε, I = / N / Pψ + / Pψ + / N /Pψ + /Pψ + t 4 NPψ + Pψ. Note that we can always replace the outside N derivative with a generic derivative. More precisely, Nf i if. We make the following three observations: The derivatives in the second factor I can be ignored in view of the presence of the projection P. Thus we can crudely bound it by I ψ E[ψ]t. The terms t Et t NPψ + Pψ are easily estimated by t E[ψ]t.. 4 The tensor version of the estimate requires the covariant / N derivative. Recall that / N denotes the projection on S t,u of the covariant derivative N.

29 NONLINEAR WAVE EQUATIONS 7 3 It remains to handle the terms Et t / N /Pψ + /Pψ. Consider first the integral Et t /Pψ. Let ζ be a cut-off function of the eterior region Et t such that ζ Et t = and ζ t. We introduce the angular vectorfields A i = ζ i < i,n > N. Clearly, for any scalar function f, /f 3 i= A if in the eterior region Et t.now, 3 /Pψ A i Pψ Et t i= Et t 3 3 PA i ψ + [P, A i ]ψ i= Et t i= Et t 3 PA i ψ + error i= /ψ + Error. We estimate the error term Et t i [P, A i]ψ with the help of the following: Lemma 7.. Consider a vectorfield X = i Xi i vanishing on the complement of the eterior region Et t of and P the standard Littlewood-Paley projection on frequencies of size. Then, for arbitrary scalar functions f there is the inequality: 5 [P, X]f L Et t sup i X j L Et t f L. i,j Proof. We postpone the proof until the end of Section 8; see Lemma We apply the above lemma to the vectorfields A k = ζ δ j k N kn j j. Observe that the components A j k are bounded and ζ t.thus Error t + N L Et t ψ L Σ. t Recall the epression, see 6, Θ = tr χ r + ˆχ + η + H and the inequality 5 N r + Θ. Observe also that in the eterior region Et t, 5 In fact the eterior region on the right-hand side of the inequality should be somewhat enlarged by size one. Since this enlargement does not affect our arguments we prefer to ignore it.

30 7 SERGIU KLAINERMAN AND IGOR RODNIANSKI r t. Therefore, Error t ψ + Θ LEt t L Σ. t We can finally conclude that /Pψ /ψ + t 98 + Θ L Et t ψ Et t Σt t + Θ L Et t E[ψ]t. We now consider Et t / N /Pψ. In view of the simple commutation estimates 53 we can write: / N /Pψ / NPψ + r +Θ Pψ Et t Et t Et t 3 A i NPψ + r +Θ Pψ. i= Et t Et t Observe that A i NPψ=A i N j j Pψ=N j A i j Pψ+[A i,n j ] j Pψ = NPA i ψ+n j [A i, j P ]ψ +[A i,n j ] j Pψ. Therefore, by Lemma 7., with P replaced by P, as well as the estimates 5 A i NPψ A i ψ + t + Θ L Et t ψ Et t Et t Σt and finally, / N /Pψ t + Θ 99 L Et t E[ψ]t. Et t Substituting 98, 99 back into 97 we infer that in the eterior region, sup Pψ t ε t + Θ εe L Et ε t [ψ]t I ε S t,u t ε t + Θ L Et t εe[ψ]t. Finally, together with the interior estimates 96 this implies that 00 Pψt L + t ε + tε Θ ε L Et t E [ψ]t. Observe that according to 57 of the Asymptotics Theorem, Θ obeys the following estimate in the eterior region: Define Θt L Et t t λ ε0 + λ ε Ht L. dt =t ε λ ε Ht L ε.

31 NONLINEAR WAVE EQUATIONS 73 Therefore, Pψt L + dt E + t ε [ψ]t. To prove the desired L L decay estimate it remains to check that for some q>, 6 q t d L q. [0,t ] Since t λ 4ε0 it clearly suffices to show that d L q λ. In view of [0,t ] the estimates, see Proposition.4, we infer that H L t L λ 4ε0, H L t L λ +ε0, d L q tε [0,t ] λ ε H ε t as desired. L q ε L t ε λ ε H q ε L t L q H L t L λ, 8. Proof of the reduction steps In this section we give precise statements and proofs for the reduction steps discussed in Section. Recall the equation 3, written in the form 6, 0 g αβ α β φ = Nφ, φ where φ =g μν, N = N μν and g αβ = g αβ φ. In fact g αβ =φ. We consider solutions φ of 0 such that the components of both φ and φ are uniformly bounded. Moreover g μν approaches the Minkowski metric m μν at infinity according to 5. To avoid repeating this statement in what follows we introduce the following notation: Definition 8.. We say that f H s = H s R 3 if f H s, f is continuous and tends to zero as. Observe that H s, with s> 3, is the closure of C0 in the norm f H s. Given a solution φ of 0 we say that φ =g μν C[0,T]; m + H s if, for every t [0,T], g μν t m μν H s and t φ H s. Throughout the section we shall use the following notation: Definition 8.. For any function f on = R 3, P λ f = F χλ ξ ˆfξ with χ supported in the unit dyadic region ξ and λ Z an integer power of. Also f = λ P λf. We shall denote f Z λ = P λ f = μ λ f μ. We shall also use the notation f <λ = P <λ f = μ< M 0λ f μ, for a fied, sufficiently large constant M 0, such as 00. We shall often, improperly, refer to 6 We can assume that ε <q<+0 ɛ 0.

32 74 SERGIU KLAINERMAN AND IGOR RODNIANSKI the P λ s as Littlewood-Paley projections. They are in fact only approimate projections or rather cut-off operators. Remark 8.3. Observe that if f is continuous, approaches a constant c at infinity, i.e. sup =r f c 0asr, and f H s, s> 3, then P λ f H s Energy estimates. We start with the following well known statement: Proposition 8.5 Energy estimate. Let φ C[0,T]; m + H s be a solution of 0 on the time interval [0,T] for some s> 3 such that φ, φ L [0,T ] L Λ 0. Then φ verifies the following energy estimate. 0 φ L C φ [0,T L, Λ 0 φ0 Ḣs. ]Ḣs [0,T ] L Remark 8.6. Throughout this section we shall often ignore the dependence on Λ 0 and the constant M 0 involved in the definition of P <λ. Proof. The proof of Proposition 8.5 can be easily reduced to the following lemma. Lemma 8.7. Let φ satisfy the conditions of Proposition 8.5. each dyadic λ Z, φ λ = P λ φ verifies the equation Then for 03 t φ λ +n g 0i <λ φ t i φ λ +n g ij <λ φ i j φ λ = R λ, where for any s> and t [0,T] the right-hand side R λ has Fourier support in {ξ : 4λ ξ 4λ} and obeys the estimate 04 λ R λ t Ḣ s C φt L φt Ḣs with C a constant depending only on Λ 0. Moreover φ λ also satisfies the equation 05 g αβ <λ α β φ λ = R λ with a different R λ which verifies the same estimate 04 and the frequency property. 7 This can be proved easily by a density argument.

33 NONLINEAR WAVE EQUATIONS 75 Proof. The proof is based on the technique of the paradifferential calculus and is now standard. 8 For a detailed treatment see [Ba-Ch], or [Kl-Ro] for notation similar to that used here. Remark 8.8. In the subsequent paper we shall also need the following more general result concerning other dyadic projections of our equation. Lemma 8.9. Under the assumptions of Lemma 8.7, g αβ <λ α β φ <λ = F λ. The function F λ obeys the estimates F λ L t L C φ L t L φ L t L, F λ L t Ḣ C φ L t L φ L t In addition, for any dyadic μ, g αβ <λ α β P λμ φ = F λ,μ, where F λ,μ verifies Ḣ. F λ,μ L t L Cλμ γ λ φ L t L φ L t F λ,μ L t Ḣ Cλ γ μ γ φ L t L φ L t The components of the metric g satisfy similar equations. Ḣ +γ, Ḣ +γ. The proof of Lemma 8.9 proceeds in the same manner as the proof of Lemma 8.7 after we apply the respective projections P <λ and P λμ. To finish the proof of Proposition 8.5 we choose a large parameter Λ in such a way that for any λ Λ the metric n g ij <λ is uniformly elliptic. This is always possible since P <λ is an approimation of the identity and the original metric n g ij is uniformly elliptic in [0,T]. For the values of the dyadic parameter λ Λ rewrite the equation for φ λ in the form t φ λ +n g 0i <λ t i φ λ +n g ij <λ i j φ λ = R λ noting that the change of the metric introduces the error term of type E. For λ Λ we keep the form of the equation as in Lemma 8.7, t φ λ +n g 0i <λ t i φ λ +n g ij <λ i j φ λ = R λ. In either case, the standard H energy estimate for the wave equation yields φ λ L [0,T ] L CΛ 0 φ λ 0 L + R λ L [0,T ] L. 8 The equations discussed in the literature are somewhat different from the one treated here because of the nontriviality of the components g 00 and g 0i of the metric. This adds only minor technical complications.

34 76 SERGIU KLAINERMAN AND IGOR RODNIANSKI Using Lemma 8.7 and the Gronwall inequality we immediately obtain for s>, φ L ep φ [0,T L φ0 ]Ḣs [0,T ] L Ḣ. s The estimate for s = follows by standard energy estimates without the paradifferential decomposition Reduction to the Strichartz-type estimates. As discussed in Section we need to prove the Strichartz type inequality 6. This is achieved by the following: Theorem 8. A. Let φ C[0,T]; m+h +γ be a solution of 0 on the time interval [0,T], T. Assume that 06 φ L + φ [0,T ] H+γ L B 0. [0,T ] L There eists a small positive eponent δ = δb 0 such that φ satisfies the following local in time Strichartz-type estimate, 07 φ L CB 0 T δ. [0,T ] L Remark 8.. In view of Remark. and Definition.3 we shall treat B 0 as a universal constant in what follows and hide the dependence on it in the notation The dyadic version of the Strichartz-type estimate. Fi a large frequency parameter Λ. It easily follows from the triangle inequality that for p [, ], φ L p φ Λ L p + λ>λ φ λ L p. Thus, Theorem 8. follows from the following dyadic version of the Strichartztype estimates for φ λ = P λ φ. Theorem 8.4 A. Let φ be as in Theorem 8.. There eists a small positive eponent δ = δb 0 such that for each λ Λ, the function φ λ satisfies the Strichartz-type estimate 08 φ λ L c [0,T ] L λt δ with constants c λ such that λ c λ. Remark 8.5. The corresponding estimate for small frequencies, i.e. for φ <λ, follows trivially from the Sobolev inequality, φ <λ L [0,T ] L T φ<λ L H Λ 3 +γ T φ L Λ [0,T ] H+γ T. [0,T ] Since Λ is a fied large parameter, which could depend only upon B 0, we have the desired bound for the low frequency part of φ.

35 NONLINEAR WAVE EQUATIONS 77 Remark 8.6. We shall need the following version of the estimate 04 for R λ and any s<+γ: 09 R λ t Ḣs c λ φ L φ H +γ with constants c λ : λ c λ. The estimate 09 can be easily obtained from 04 by making use of the fact that the Fourier support of R λ is localized on the set {ξ : λ ξ 4λ}. As a consequence, using the bootstrap assumption 06, we also have the estimate 0 R λ t L [0,T ]Ḣs c λ T φ L [0,T ] L φ L [0,T ] H+γ c λ Dyadic linearization and time restriction. This step reduces Theorem 8.4 to a Strichartz-type estimate for the linearized equation g αβ <λ α β ψ = 0 on smaller subintervals of [0,T]. We partition [0,T] by the intervals I k = [t k,t k+ ],k=0,.., λ 8ɛ0 with the properties I k Tλ 8ɛ0 and φ L L λ 4ɛ0 B 0. The eistence of such a partition is insured by the bootstrap condition 06. Theorem 8.8 A3. Fi λ Λ and k Z [0,λ 8ɛ0 ] and let ψ be a solution of the linear wave equation g αβ <λ α β ψ =0 on the interval I k =[t k,t k+ ], verifying, 0 λ m ψt k L m ψt k L 0 λ m ψt k L for every m 0. Then there eists a sufficiently small eponent δ>0 such that: P λ ψ L L I k δ ψt k Ḣ+δ. The size of δ depends only on ɛ 0,B 0. In particular, for any ɛ 0 > 0, δ can be chosen such that δ<0 γ. Remark 8.9. The condition implies that, modulo a negligible tail, the Fourier support of ψt k belongs to the set {ξ : 0 λ ξ 0 λ}. In general, we shall say that function f obeys the property 3 M if 3 M λ m f L m f L M λ m f L. 4 Lemma 8.0. Assume f in R 3 is a function whose frequency is localized to the region ξ M0 λ and c f c for some positive number c. Then u = f verifies, m u L M0 λ m.