The Cauchy problem for wave equations with non Lipschitz coefficients; Application to continuation of solutions of some nonlinear wave equations

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(1)ISSN 0012-9593. ASENAH. quatrième série - tome 41. fascicule 2. mars-avril 2008. ANNALES SCIENTIFIQUES de L ÉCOLE NORMALE SUPÉRIEURE Ferruccio COLOMBINI & Guy MÉTIVIER. The Cauchy Problem for Wave Equations with non Lipschitz Coefficients. SOCIÉTÉ MATHÉMATIQUE DE FRANCE.

(2) Ann. Scient. Éc. Norm. Sup.. 4 e série, t. 41, 2008, p. 177 à 220. THE CAUCHY PROBLEM FOR WAVE EQUATIONS WITH NON LIPSCHITZ COEFFICIENTS; APPLICATION TO CONTINUATION OF SOLUTIONS OF SOME NONLINEAR WAVE EQUATIONS  F COLOMBINI  G MÉTIVIER A. – In this paper we study the Cauchy problem for second order strictly hyperbolic operators of the form Lu :=. Pn. . j,k=0. ∂yj aj,k ∂yk u +. Pn. j=0 {bj ∂yj u. + ∂yj (cj u)} + du = f,. when the coefficients of the principal part are not Lipschitz continuous, but only “Log-Lipschitz” with respect to all the variables. This class of equation is invariant under changes of variables and therefore suitable for a local analysis. In particular, we show local existence, local uniqueness and finite speed of propagation for the noncharacteristic Cauchy problem. This provides an invariant version of a previous paper of the first author with N. Lerner [6]. We also give an application of the method to a continuation theorem for nonlinear wave equations where the coefficients above depend on u: the smooth solution can be extended as long as it remains Log-Lipschitz. R. – On considère le problème de Cauchy pour des équations d’onde strictement hyperboliques : Lu :=. Pn. . j,k=0. ∂yj aj,k ∂yk u +. Pn. j=0 {bj ∂yj u. + ∂yj (cj u)} + du = f,. quand les coefficients de la partie principale sont seulement “Log-Lipschitz” en toutes les variables. Cette classe d’équation est invariante par changement de variables et est donc une classe naturelle pour une étude locale intrinsèque. En particulier, on montre l’existence locale, l’unicité locale et la vitesse finie de propagation pour le problème de Cauchy non caractéristique, donnant une version invariante d’un résultat antérieur du premier auteur avec N. Lerner [6]. Pour les équations non linéaires où les coefficients ci-dessus dépendent de u, la méthode d’estimations permet de montrer que les solutions régulières se prolongent en solutions régulières aussi longtemps qu’elles restent Log-Lipschitz.. 1. Introduction In this paper we study the well-posedness of the Cauchy problem for second order strictly hyperbolic equations whose coefficients are not Lipschitz continuous: (1.1). Lu :=. n X j,k=0. n X ∂yj aj,k ∂yk u + {bj ∂yj u + ∂yj (cj u)} + du = f. j=0. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 0012-9593/02/© 2008 Société Mathématique de France. Tous droits réservés.

(3) 178. F. COLOMBINI AND G. MÉTIVIER. In Section 6, we will present an application of the methods developed for the analysis of the Cauchy problem to nonlinear wave equations, where the various coefficients above depend on u. It is known that the smooth solution can be extended as long as they remain Lipschitz continuous. We prove that this condition can be weakened, and that smooth solutions remain smooth as long as they remain Log-Lipschitz. We refer to Section 6 for a precise result and focus now on the analysis of the Cauchy problem. The question of the well-posedness of the Cauchy problem for the wave equation with nonsmooth coefficients has already been studied in the case that the second order part has the special form, in coordinates y = (t, x): ∂t2 −. (1.2). n X. ∂xj aj,k ∂xk u. j,k=1. and the Cauchy data are given on the space-like hyperplane {t = 0}. In this case, when the coefficients depend only on the time variable t, F. Colombini, E. De Giorgi and S. Spagnolo ([5]) have proved that the Cauchy problem is in general ill-posed in C ∞ when the coefficients are only Hölder continuous of order α < 1, but is well-posed in appropriate Gevrey spaces. This has been extended to the case where the coefficients are Hölder in time and Gevrey in x ([14, 8]). Moreover, it is also proved in [5] that the Cauchy problem is well posed in C ∞ when the coefficients, which depend only on time, are “Log-Lipschitz” (in short LL) : recall that a function a of variables y is said to be LL on a domain Ω if there is a constant C such that (1.3) |a(y) − a(y 0 )| ≤ C|y − y 0 | 1 + Log|y − y 0 | for all y and y 0 in Ω. In [5], it is proved that for LL coefficients depending only on t and for initial data in the Sobolev spaces H s × H s−1 , the solution satisfies (1.4). u(t, ·) ∈ H s−λt ,. ∂t u(t, ·) ∈ H s−1−λt. with λ depending only on the LL norms of the coefficients and the constants of hyperbolicity. In particular, there is a loss of smoothness as time evolves and this loss does occur in general when the coefficients are not Lipschitz continuous, and is sharp, as shown in [3]. The analysis of the C ∞ well-posedness has been extended by F. Colombini and N. Lerner ([6]) to the case of equations, still with principal part (1.2), whose coefficients also depend on the space variables x. They show that the Cauchy problem is well-posed if the coefficients are LL in time and C ∞ in x. They also study the problem under the natural assumption of isotropic LL smoothness in (t, x). In this case one has to multiply LL functions with distributions in H s . This is well defined only when |s| < 1. Therefore, one considers initial data in H s × H s−1 with 0 < s < 1, noticing that further smoothness would not help. Next, the loss of smoothness (1.4) forces us to limit t to an interval where 0 < s − λt, yielding only local in time existence theorems. We also refer to [6] for further discussions on the sharpness of LL smoothness. However, the local uniqueness of the Cauchy problem and the finite speed of propagation for local solutions are not proved in [6]. The main goal of this paper is to address these questions. Classical methods, such as convexification, lead one to consider general equations (1.1) with LL coefficients in all variables. However, the meaning of the Cauchy problem for such equations is not completely obvious: as mentioned above, the maximal expected smoothness 4 e SÉRIE – TOME 41 – 2008 – No 2.

(4) THE CAUCHY PROBLEM FOR WAVE EQUATIONS. 179. of the solutions is H s with s < 1 and their traces on the initial manifold are not immediately defined. More importantly, in the general theory of smooth operators, the traces are defined using partial regularity results in the normal direction; in our case, the limited smoothness of the coefficients is a source of difficulties. It turns out that when s ≤ 21 , one cannot in general define the traces of all the first order derivatives of u, but only the Neumann trace relative to the operator, using a weak formulation of the traces. A 1.1. – L is a second order operator of the form (1.1) on a neighborhood Ω of y, with coefficients aj,k ∈ LL(Ω), bj and cj in C α (Ω), for some α ∈ ] 21 , 1[ and d ∈ L∞ (Ω). Σ is a smooth hypersurface through y and L is strictly hyperbolic in the direction conormal to Σ. Shrinking Ω if necessary, we assume that Σ is defined by the equation {ϕ = 0} with ϕ smooth and dϕ 6= 0. We consider the one-sided Cauchy problem, say on the component Ω+ = Ω ∩ {ϕ > 0}. We use the Sobolev spaces H s (Ω ∩ {ϕ > 0}) for s ∈ R. As usual, we say s that u ∈ Hloc (Ω ∩ {ϕ ≥ 0}), if for any relatively compact open subset Ω1 of Ω, the restriction s of u to Ω1 ∩ {ϕ > 0} belongs to H s (Ω ∩ {ϕ > 0}). Similarly, u ∈ Hcomp (Ω ∩ {ϕ ≥ 0}) if s u ∈ H (Ω ∩ {ϕ > 0}) has compact support in Ω ∩ {ϕ ≥ 0}. The adjoint operator L∗ v :=. (1.5). n X. n X {cj ∂yj v + ∂yj (bj v)} + dv ∂yk aj,k ∂yj v − j=0. j,k=0. has the same form as L. For u and v smooth, v compactly supported in Ω ∩ {ϕ ≥ 0}, one has the (formal) identity (1.6) Lu, v L2 (Ω ) − u, L∗ v L2 (Ω ) = Nν u, v L2 (Σ) − u, Nν0 v L2 (Σ) +. +. where Nν u =. X. νk (aj,k ∂j u)|Σ ,. j,k. (1.7). Nν0 v =. X j,k. νj (aj,k ∂k v)|Σ −. X. νj (bj + cj )v. |Σ. j. and ν = (ν0 , . . . , νd ) 6= 0 is conormal to Σ and the d-integration form on Σ is chosen accordingly. s L 1.2. – i) For all s ∈ ]1 − α, 1 + α[ and u ∈ Hloc (Ω ∩ {ϕ ≥ 0}), all the terms entering in the definition of Lu and L∗ u are well defined as distributions in s−2 Hloc (Ω ∩ {ϕ ≥ 0}). s ii) For all s ∈ ] 32 , 1+α[ and u ∈ Hloc (Ω∩{ϕ ≥ 0}), the traces Nν u and Nν0 u are well defined s− 3. in Hloc 2 (Σ ∩ Ω). Proof. – This is due to multiplicative properties (see [6] and Corollary 3.6): σ σ – If σ ∈ ] − 1, 1[, a ∈ LL(Ω) and v ∈ Hloc (Ω ∩ {ϕ ≥ 0}), then av ∈ Hloc (Ω ∩ {ϕ ≥ 0}). α σ σ – If σ ∈ ] − α, α[, a ∈ C (Ω) and v ∈ Hloc (Ω ∩ {ϕ ≥ 0}), then av ∈ Hloc (Ω ∩ {ϕ ≥ 0}).. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(5) 180. F. COLOMBINI AND G. MÉTIVIER. Next, we recall that the subspace of functions with compact support in Ω+ is dense in H σ (Ω+ ) when |σ| < 21 ; moreover, for 0 ≤ σ < 12 and for u ∈ H σ (Ω+ ) the pairing (u, v)L2 (Ω) for v ∈ L2 extends as the duality hu, viH σ ×H −σ . With this remark in mind, the identity (1.6) holds for smooth functions: s s L 1.3. – For s ∈ ] 23 , 1 + α[, u ∈ Hloc (Ω ∩ {ϕ ≥ 0}) and v ∈ Hcomp (Ω ∩ {ϕ ≥ 0}), there holds (1.8) Lu, v H −σ ×H σ − u, L∗ v H σ ×H −σ = Nν u, DΣ v L2 (Σ) − DΣ u, Nν0 v L2 (Σ). with σ = s −. 3 2. ∈ ]0, 12 [ and DΣ u = u|Σ .. Proof. – It is sufficient to remark that for σ ∈ [0, 12 [, the Green’s formula ∂j u, v H −σ ×H σ = − u, ∂j v H σ ×H −σ + νj DΣ u, DΣ v L2 (Σ) 1−σ 1−σ is satisfied for u ∈ Hloc (Ω ∩ {ϕ ≥ 0}) and v ∈ Hcomp (Ω ∩ {ϕ ≥ 0}). s P 1.4. – Let D(L; H s ) = {u ∈ Hloc (Ω ∩ {ϕ ≥ 0}) : Lu ∈ L2loc (Ω ∩ S {ϕ ≥ 0})}. The operator NΣ and DΣ have unique extensions to s>1−α D(L; H s ) such that s− 3. i) For all s ∈ ]1 − α, α[, NΣ (resp. DΣ ) is continuous from D(L; H s ) into Hloc 2 (Σ ∩ Ω) s− 1. (resp. Hloc 2 (Σ ∩ Ω)). 2−s0 ii) for all s0 ∈ ]1 − α, 12 [ such that s0 ≤ s and all v ∈ Hcomp (Ω ∩ {ϕ ≥ 0}) there holds (1.9) 0 3 −s − DΣ u, N v 1 −s . Lu, v L2 − u, L∗ v H s0 ×H −s0 = Nν u, DΣ v s− 23 s− 1 ν 2 2 2 H. ×H. H. ×H 0. 2−s , L∗ v ∈ This proposition is proved in Section 5. Note that by Lemma 1.2, for v ∈ Hcomp 0. 0 3 2 −s. 0. 3 2 −s. 1. −s0. −s s 2 Hcomp and that u ∈ Hloc if s0 ≤ s. Moreover, DΣ v ∈ Hcomp ⊂ Hcomp and NΣ0 v ∈ Hcomp ⊂ 1. −s. 2 . Hcomp With this proposition, the Cauchy problem with source term in L2 and solution in H s , s > 1 − α, makes sense.. T 1.5 (Local existence). – Consider s > 1 − α and a neigborhood ω of y in Σ. Then there are s0 ∈ ]1 − α, α[ and a neighborhood Ω0 of y in R1+n such that for all Cauchy data (u0 , u1 ) in H s (ω) × H s−1 (ω) near y and all f ∈ L2 (Ω0 ∩ {ϕ > 0}) the Cauchy problem (1.10). Lu = f,. DΣ u = u0 ,. NΣ u = u1 ,. 0. has a solution u ∈ H s (Ω0 ∩ {ϕ > 0}). T 1.6 (Local uniqueness). – If s > 1 − α and u ∈ H s (Ω ∩ {ϕ > 0}) satisfies (1.11). Lu = 0,. DΣ u = 0,. NΣ u = 0,. then u = 0 on a neighborhood of y in Ω ∩ {ϕ ≥ 0}. R 1.7. – If the coefficients of the first order term L1 (see (2.3)) are also LL, the statements above are true with α = 1 since the coefficients are then C α for all α < 1. If the bj are C α and the cj are C α̃ , the conditions are 1 − α̃ < α and the limitation on s is 1 − α̃ < s. 4 e SÉRIE – TOME 41 – 2008 – No 2.

(6) THE CAUCHY PROBLEM FOR WAVE EQUATIONS. 181. R 1.8. – Theorem 1.6 implies that if u is in H s and satisfies Lu = 0 near y and if u vanishes on {ϕ < 0}, then u vanishes on a neighborhood of y (see Section 5.2). Moreover, this local propagation of zero across any space-like manifold implies finite speed of propagation by classical arguments which we do not repeat here. In particular, if Ω0 ∩ {ϕ ≥ 0} is contained in the domain of dependence of ω, there is existence and uniqueness for the Cauchy problem (1.10) in Ω0 ∩ {ϕ ≥ 0}. The proof of these results is given in Section 5 below. Because all the hypotheses are invariant under smooth changes of coordinates, we can assume that in the coordinates y = (t, x), the initial surface is {t = 0}, and in these coordinates, we prove the existence and uniqueness theorems. We deduce them from similar results on strips ]0, T [×Rn and there, the main part of the work is to prove good energy estimates for (weak) solutions. In this framework, the results of Theorem 1.5 are improved by using non isotropic spaces, and by making a detailed account of the loss of spatial smoothness as time evolves, as in [5, 6]. The precise results are stated in section 2 below and are proved in section 4 using the paradifferential calculus of J.-M. Bony, whose LL-version is presented in section 3.. 2. The global in space problem In this section we denote by (t, x) the space-time variables. On Ω = [0, T0 ] × Rn consider a second order hyperbolic differential operator (2.1). Lu = L2 u + L1 u + du. with (2.2). L2 = ∂t a0 ∂t +. n X. (∂t aj ∂xj + ∂xj aj ∂t ) −. j=1. (2.3). L1 = b0 ∂t + ∂t c0 +. n X. ∂xj aj,k ∂xk ,. j,k=1 n X. (bj ∂xj + ∂xj cj ).. j=1. The coefficients satisfy on Ω = [0, T0 ] × Rn (2.4). aj,k = ak,j ,. a0 , aj , aj,k ∈ L∞ (Ω) ∩ LL(Ω),. (2.5). b0 , c0 , bj , cj ∈ L∞ (Ω) ∩ C α (Ω),. (2.6). d ∈ L∞ (Ω),. for some α ∈ ] 21 , 1[. Recall that the space LL is defined by (1.3), the semi-norm kakLL being the best constant C in (1.3). In addition, for α ∈ ]0, 1[, C α denotes the usual Hölder space, equipped with the norm (2.7). kakC α = kakL∞ + sup. y6=y 0. |a(y) − a(y 0 )| . |y − y 0 |α. When α = 1, this defines the norm kakLip in the space of Lipschitz functions. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(7) 182. F. COLOMBINI AND G. MÉTIVIER. We assume that L is hyperbolic in the direction dt, which means that there are δ0 > 0 and δ1 > 0 such that for all (t, x, ξ) ∈ [0, T0 ] × Rn × Rn ã X Å aj ak (2.8) a0 (t, x) ≥ δ0 , aj,k + ξj ξk ≥ δ1 |ξ|2 . a0 1≤j,k≤n. We denote by AL∞ , ALL and B constants such that for all indices (2.9). ka0 , aj , aj,k kL∞ (Ω) ≤ AL∞ ,. (2.10). kb0 , c0 , bj , cj kC α (Ω) ≤ B,. ka0 , aj , aj,k kLL(Ω) ≤ ALL , kdkL∞ (Ω) ≤ B.. 2.1. Giving sense to the Cauchy problem Consider the vector fields (2.11). X = a0 ∂t +. n X. aj ∂xj = a0 Y.. j=1. Formal computations immediately show that the second order part of L can be written L2 u = ZXu − L̃2 u. (2.12) with (2.13). Zv = ∂t v +. n X. ∂xj (ãj v),. L̃2 u =. j=1. n X. ∂xj ãj,k ∂xk u ,. j,k=1. ãj,k = aj,k + aj ak /a0 , and ãj = aj /a0 . Consequently, it follows that (2.14). ˜ Lu = (Z + b̃0 )(X + c0 )u − L̃2 u + L̃1 u + du. with (2.15). L̃1 u =. n X. b̃j ∂xj u +. j=1. n X. ∂xj (c̃j u). j=1. and b̃0 = b0 /a0 ,. b̃j = bj − b̃0 aj ,. c̃j = cj − ãj c0 ,. d˜ = d − c0 c̃0 .. The next lemma shows that these identities are rigorous under minimal smoothness assumption on u. L 2.1. – Suppose that u ∈ H ρ (]0, T [×Rn ) for some ρ ∈ ]1 − α, α[. Then cu, Xu and L1 u belong to H ρ−1 (]0, T [×Rn ). Moreover L2 u is well defined as a distribution in H ρ−2 (]0, T [×Rn ). Proof. – Both u and its space-time derivatives (∂t u, ∂xj u) belong to H ρ−1 . Following [6], their multiplication by a bounded LL function belong to the same space (see also Corollary 3.6). This shows that all the individual terms present in the definition of Xu belong to H ρ−1 and those occurring in L2 u and ZXu are well defined in H ρ−2 in the sense of distributions. Next we recall that the multiplication (b, u) 7→ bu is continuous from C α × H s to H s when |s| < α. This implies that the terms b∂u and ∂(cu) that occur in L1 u and L̃1 u belong to H ρ−1 since ρ ∈ ]1 − α, α[. The last term du is in L2 , thus in H ρ−1 , since c ∈ L∞ and u ∈ L2 . 4 e SÉRIE – TOME 41 – 2008 – No 2.

(8) THE CAUCHY PROBLEM FOR WAVE EQUATIONS. 183. The identity (2.12) is straightforward from (2.2) since all the algebraic computations make sense by the preceding remarks. Next we need partial regularity results in time, showing that the traces of u and Xu at t = 0 are well defined, as distributions, for solutions of Lu = f . This is based on the remark that this equation is equivalent to the system ( ˜ + f, Zv + b̃0 v = L̃2 u − L̃1 u − du (2.16) Y u + c̃0 u = v/a0 with c̃0 = c0 /a0 . The important remark is that, for this system, the coefficients of ∂t , both ‹, Z = ∂t + Z, e the for u and v, are equal to 1, thus smooth. Using the notation Y = ∂t + Y system reads ( ˜ + f, ∂t v = −Z̃v − b̃0 v − L̃2 u − L̃1 u − du (2.17) ∂t u = −Ỹ u + v/a0 . L 2.2. – Suppose that ρ ∈ ]1 − α, α[ and u ∈ H ρ (]0, T [×Rn ) is such that Lu ∈ L1 ([0, T ]; H ρ−1 (Rn )). Then u ∈ L2 ([0, T ]; H ρ (Rn )) and ∂t u ∈ L2 ([0, T ]; H ρ−1 (Rn )). 1 Therefore, u ∈ C 0 ([0, T ]; H ρ− 2 (Rn )). 3 Moreover, Xu ∈ L2 ([0, T ]; H ρ−1 (Rn )) and Xu ∈ C 0 ([0, T ]; H ρ− 2 (Rn )). 1 3 In particular, the traces u|t=0 and Xu|t=0 are well defined in H ρ− 2 (Rn ) and H ρ− 2 (Rn ), respectively. 0. Proof. – a) We use the spaces H s,s of Hörmander ([7], chapter 2), which are defined on R1+n as the spaces of temperate distributions such that their Fourier transform û satisfies 0 (1 + τ 2 + |ξ|2 )s/2 (1 + |ξ|2 )s /2 û ∈ L2 . The spaces on [0, T ] × Rn are defined by restriction. In 0 0 0 0 particular, H 0,s ([0, T ] × Rn ) = L2 ([0, T ]; H s (Rn )). Recall that ∂xj maps H s,s to H s,s −1 and that (2.18). 0. 0. u ∈ H s,s , ∂t u ∈ H s,s −1. ⇒. 0. u ∈ H s+1,s −1 .. ˜ as well as L̃1 u, Xu and v belong to H ρ−1 = For u ∈ H ρ , the first derivatives of u, du, H , as well as their multiplication by a LL or C α coefficient. Thus L̃2 u and Z̃v belong ρ−1,−1 to H and b). ρ−1,0. (2.19). ∂t v = f + g,. f = Lu ∈ L1 (]0, T [; H ρ−1 ), g ∈ H ρ−1,−1 .. Let Z v0 (t) =. t. f (t0 ) dt0 ∈ C 0 (H ρ−1 ).. 0. In particular, v ∈ L2 (]0, T [; H ρ−1 ) = H 0,ρ−1 ⊂ H ρ−1,0 , since ρ − 1 ≤ 0. Thus, v − v0 ∈ H ρ−1,0 and ∂t (v − v0 ) = g ∈ H ρ−1,−1 . By (2.18) v − v0 ∈ H ρ,−1 ⊂ H 0,ρ−1 since ρ ≥ 0. Next, reasoning for fixed time and then taking L2 norms we note that the multiplication by a LL or C α function maps L2 (]0, T [; H ρ−1 ) = H 0,ρ−1 into itself. Thus, by the second equation of (2.17), ∂t u = −Ỹ u + v/a0 ∈ H 0,ρ−1 . This finishes the proof of the first part of the lemma. c) In particular, it implies that v = Xu + b0 u ∈ H 0,ρ−1 . Thus, Z̃v and L̃2 u which involve multiplication by C α or LL function, followed by a spatial derivative, belong to H 0,ρ−2 . Therefore, the equation implies that in (2.19) g ∈ H 0,ρ−2 . Thus applying (2.18) to v − v0 ∈ ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(9) 184. F. COLOMBINI AND G. MÉTIVIER 3. H 0,ρ−1 implies that v − v0 ∈ H 1,ρ−2 ⊂ C 0 ([0, T ]; H ρ− 2 (Rn )). Since |ρ − 12 | < α and 1 1 u ∈ C 0 ([0, T ]; H ρ− 2 (Rn )), the product b̃0 u belongs to C 0 ([0, T ]; H ρ− 2 (Rn )). Since v0 is 3 also in this space, we conclude that Xu ∈ C 0 ([0, T ]; H ρ− 2 (Rn )). 3. R 2.3. – If ρ > 21 , then the multiplication by LL functions maps H ρ− 2 into itself 3 and we can conclude that ∂t u ∈ C 0 ([0, T ]; H ρ− 2 (Rn )), as well as all the first derivatives of u, so that their traces at t = 0 are well defined. When ρ ≤ 12 , the continuity of ∂t u is not clear. However, the trace of Xu has an intrinsic meaning, as a consequence of Proposition 1.4 (see Section 5). Lemma 2.2 allows us to consider the Cauchy problem (2.20). Lu = f,. u|t=0 = u0 , Xu|t=0 = u1 , S S when f ∈ ρ>−α L1 ([0, T ]; H ρ (Rn )) and u ∈ ρ>1−α H ρ (]0, T [×Rn ). 2.2. The main results We first state uniqueness for the Cauchy problem: T 2.4. – If u ∈ (2.21). S. ρ>1−α. Lu = 0,. H ρ (]0, T [×Rn ) satisfies u|t=0 = 0,. Xu|t=0 = 0. then u = 0. As in [5, 6], we prove existence of solutions in Sobolev spaces having orders decreasing in time. The proper definition is given as follows. The operators |D| and. (2.22). Λ := Log(2 + |D|). are defined by Fourier transform, associated to the Fourier multipliers |ξ| and Log(2 + |ξ|) respectively. 1. D 2.5. – i) H s (Rn ) or H s denotes the usual Sobolev space on Rn . H s+ 2 log 1 1 s− 21 log denote the spaces Λ− 2 H s and Λ 2 H s respectively. and H ii) Given parameters σ and λ, we denote by Cσ,λ (T ) the space of functions u such that for all t0 ∈ [0, T ], u ∈ C 0 ([0, t0 ], H σ−λt0 ). iii) Hσ± 12 log,λ (T ) denotes the spaces of functions u on [0, T ] with values in the space of temperate distributions in Rn such that (2.23). 1. (1 + |D|)σ−λt Λ± 2 u(t, ·) ∈ L2 ([0, T ]; L2 (Rn )).. iv) Lσ,λ (T ) denotes the space of functions u on [0, T ] with values in the space of temperate distributions in Rn such that (2.24). (1 + |D|)σ−λt u(t, ·) ∈ L1 ([0, T ]; L2 (Rn )).. 4 e SÉRIE – TOME 41 – 2008 – No 2.

(10) 185. THE CAUCHY PROBLEM FOR WAVE EQUATIONS. Cσ,λ (T ) is equipped with the norm sup ku(t)kH σ−λt .. (2.25). t ∈ [0,T ]. The norms in Hσ± 12 log,λ (T ) and Lσ,λ (T ) are given by (2.23) and (2.24). Equivalently, Hσ± 12 log,λ (T ) and Lσ,λ (T ) are the completions of C0∞ ([0, T ] × Rn ) for the norms Z T 12 ku(t)k2 σ−λt± 1 log dt . (2.26) kukHσ± 1 log,λ (T ) = 2. H. 0. 2. and Z kukLσ,λ (T ) =. (2.27). T. ku(t)kH σ−λt dt. 0. T 2.6. – Fix θ < θ1 in ]1 − α, α[. Then there are λ > 0 and K > 0, which depend only on the constants AL∞ , ALL , B, δ0 , δ1 , θ and θ1 , given by (2.8), (2.9) and (2.10), such that for ß ™ θ1 − θ (2.28) T = min T0 , λ u0 ∈ H 1−θ (Rn ), u1 ∈ H −θ (Rn ) and f = f1 +f2 with f1 ∈ L−θ,λ (T ) and f2 ∈ H−θ− 12 log,λ (T ), the Cauchy problem (2.20), has a unique solution u ∈ C1−θ,λ (T ) ∩ H1−θ+ 21 log,λ (T ) with ∂t u ∈ C−θ,λ (T ) ∩ H−θ+ 21 log,λ (T ). Moreover, it satisfies sup ku(t0 )k2H 1−θ−λt0 + sup k∂t u(t0 )k2H −θ−λt0. 0≤t0 ≤t. +. 0≤t0 ≤t. Z t. ku(t0 )k2. 0 1 H 1−θ−λt + 2 log. 0. . + k∂t u(t0 )k2. 0 1 H −θ−λt + 2 log. dt0. (2.29) n ≤ K ku0 k2H 1−θ + ku1 k2H −θ Z t 2 Z t 0 0 + kf1 (t )kH −θ−λt0 dt + kf2 (t0 )k2 0. 0. 0 1 H −θ−λt − 2 log. o dt0 .. Note that for t ∈ [0, T ], 1 − θ − λt ≥ 1 − θ1 > 1 − α, so that f ∈ L1 ([0, T ]; H −θ2 ) with θ1 < θ2 < α. Similarly, u ∈ L2 ([0, T ]; H 1−θ1 ) and ∂t u ∈ L2 ([0, T ]; H −θ1 ) implying that u ∈ H 1−θ1 ([0, T ] × Rn ). Therefore, we are in a situation where we have given sense to the Cauchy problem. R 2.7. – This is a local in time existence theorem since the life span (2.28) is limited by the choice of λ. Thus the dependence of λ0 on the coefficient is of crucial importance. In case of Lipschitz coefficients, there is no loss of derivatives; this would correspond to λ = 0. Using the notations in (2.9) (2.10) and (2.8), the analysis of the proof below shows that there is a function K0 (·) such that one can choose Å ã ALL AL∞ (2.30) λ= K0 , min{δ0 , δ1 } δ0 revealing the importance of the LL-norms of the coefficients and the role of the hyperbolicity constant δ1 /δ0 . In particular, it depends only on the second order part of operator L. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(11) 186. F. COLOMBINI AND G. MÉTIVIER. R 2.8. – A closer inspection of the proof also shows that if the coefficients of the principal part of L are (a0 , aj , aj,k ) = (a00 + a000 , a0j + a00j , a0j,k + a00j,k ) with (a00 , a0j , a0j,k ) Lipschitz continous and (a000 , a00j , a00j,k ) Log Lipschitz, with LL norm bounded by A00LL , one can replace ALL by A00LL in the definition of λ. In particular if instead of (1.3) the coefficients satisfy |a(y) − a(y 0 )| ≤ Cω(|y − y 0 |). (2.31). with a modulus of continuity ω such that (2.32). lim. ε→0+. ω(ε) = 0, ε|Logε|. they can be approximated by Lipschitz functions with errors arbitrarily small in the LL norm. This can be done by usual mollifications, which will preserve the L∞ bounds AL∞ and keep uniform hyperbolicity constants δ0 and δ1 . As a consequence, λ can be taken arbitrarily small, yielding global in time existence with arbitrarily small loss of regularity (see Theorem 2.1 in [3] when the coefficients depend only on time). 3. Paradifferential calculus with LL coefficients In this section we review several known results on paradifferential calculus and give the needed extensions to the case of Log-Lipschitz coefficients. 3.1. The Paley-Littlewood analysis Introduce χ ∈ C0∞ (R), real valued, even and such that 0 ≤ χ ≤ 1 and for |ξ| ≤ 1.1 , χ(ξ) = 0 for |ξ| ≥ 1.9 . For k ∈ Z, introduce χk (ξ) := χ 2−k ξ , χ ek (x) its inverse Fourier transform with respect to ξ and the operators (3.1). (3.2). χ(ξ) = 1. Sk u := χ ek ∗ u = χk (Dx )u , ∆0 = S0 ,. and for k ≥ 1. ∆k = Sk − Sk−1 .. We note that ∆k and Sk are self adjoint. Moreover, by evenness, χ ek is real, so that ∆k and Sk preserve reality. For all temperate distributions u one has X (3.3) u= ∆k u . k≥0. The next propositions immediately follow from the definitions. P 3.1. – Consider s ∈ R. A temperate distribution u belongs to H s (Rn ) (resp. ) if and only if H s± 12 log. i) for all k ∈ N, ∆k u ∈ L2 (Rd ). 1 ii) the sequence δk = 2ks k∆k ukL2 (Rd ) (resp. δk = (k + 1)± 2 2ks k∆k ukL2 (Rd ) ) belongs to `2 (N). Moreover, the norm of the sequence δk in `2 is equivalent to the norm of u in the given space. P 3.2. – Consider s ∈ R and R > 0. Suppose that {uk }k ∈ N is a sequence of functions in L2 (Rd )such that: 4 e SÉRIE – TOME 41 – 2008 – No 2.

(12) THE CAUCHY PROBLEM FOR WAVE EQUATIONS. 187. i) the spectrum of u0 is contained in {|ξ| ≤ R} and for k ≥ 1 the spectrum of uk is contained in R1 2k ≤ |ξ| ≤ R 2k . 1 ii) the sequence δk = 2ks kuk kL2 (Rd ) (resp. δk = (k + 1)± 2 2ks k∆k ukL2 (Rd ) ) belongs to `2 (N). P 1 Then u = uk belongs to H s (Rd ) (resp. H s± 2 log ). Moreover, the norm of the sequence δk in `2 is equivalent to the norm of u in the given space. When s > 0, it is sufficient to assume that the spectrum of uk is contained in |ξ| ≤ R 2k . Next we collect several results about the dyadic analysis of LL spaces. P 3.3. – There is a constant C such that for all a ∈ LL(Rn ) and all integers k>0 k∆k akL∞ ≤ Ck2−k kakLL .. (3.4) Moreover, for all k ≥ 0. ka − Sk akL∞ ≤ C(k + 1)kakLL kSk akLip ≤ C kakL∞ + (k + 1)kakLL .. (3.5) (3.6). If α ∈ ]0, 1[ and a ∈ C α (Rn ), then k∆k akL∞ ≤ C2−αk kakC α .. (3.7). Proof. – Sk is a convolution operator with χ ek which is uniformly bounded in L1 . Thus kSk akL∞ ≤ CkakL∞ .. (3.8). Moreover, since the integral of ∂j χ ek vanishes Z ∂j (Sk a)(x) = ∂j χ ek (y) a(x − y) − a(x) dy. Using the LL smoothness of a yields k∇Sk akL∞ ≤ C(k + 1)kakLL .. (3.9). This implies (3.6). The proof of (3.4) is similar (cf [6]). The third estimate is classical. 3.2. Paraproducts Following J.-M. Bony ([2]), for N ≥ 3 one defines the para-product of a and u as ∞ X (3.10) TaN u = Sk−N a ∆k u. k=N. The remainder (3.11). RaN u. is defined as RaN u = au − TaN u.. The next proposition extends classical results (see [2, 13]) to the case of LL coefficients and Log Sobolev spaces. P 3.4. – i) For a ∈ L∞ and s ∈ R, TaN continuously maps H s to H s and s± 12 log s± 21 log H to H . Moreover, the operator norms are uniformly bounded for s in a compact set. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(13) 188. F. COLOMBINI AND G. MÉTIVIER 0. 1. 1. ii) If a ∈ L∞ ∩ LL and N 0 ≥ N ≥ 3, TaN − TaN maps H s+ 2 log into H s+1− 2 log , for all s ∈ R. 1 1 iii) If a ∈ L∞ ∩ LL, N ≥ 3 and s ∈ ]0, 1[, RaN maps H −s+ 2 log into H 1−s− 2 log , and kRaN uk. (3.12). 1 log. H 1−s− 2. ≤ CkakLL kuk. 1 log. H −s+ 2. with C uniformly bounded for s in a compact subset of ]0, 1[. Proof. – The first statement is an immediate consequence of (3.8) and Propositions 3.1 and 3.2. P 0 Next, TaN u − TaN u = k vk with vk = (Sk−N a − Sk−N 0 a) ∆k u. By Proposition 3.3 kvk kL2 ≤ C(k + 1)2−k k∆k ukL2 . With Proposition 3.2, this implies ii). To prove iii) we can assume that N = 3. Then X X X (3.13) Ra u = ∆k a Sk−3 u + ∆j a∆k u. k |k−j|≤2. k≥3. If u ∈ H. −s+ 21 log. , then k∆j ukL2 ≤ √. 2js εj j+1. with {εj } ∈ `2 . We note that the sequence √ X k+1 √ (3.14) εek = 2(j−k)s εj j + 1 j≤k is also in `2 with ke εk k`2 ≤ Ckεj k`2 with C uniformly bounded when s in a compact subset of ]0, +∞]. Thus kSk−3 ukL2 ≤ √. 2ks ε0k k+1. with {ε0k } ∈ `2 . Therefore,. √ k∆k a Sk−3 ukL2 ≤ C k + 1 2(s−1)k ε0k . 1. Proposition 3.2 implies that the first sum in (3.13) belongs to H 1−s− 2 log . Similarly, X √ ≤ C k + 1 2(s−1)k ε00k , ∆j a∆k u |k−j|≤2. with {ε00k }. L2. 2. ∈ ` . Now the spectrum of ∆j a∆k u is contained in the ball {|ξ| ≤ 2k+3 }; because 1 1 − s > 0, Proposition 3.2 implies that the second sum in (3.13) also belongs to H 1−s− 2 log , and the norm is uniformly bounded when s remains in a compact subset of [0, 1[. R 3.5. – By ii) we see that the choice of N ≥ 3 is essentially irrelevant in our analysis, as in [2]. To simplify notation, we make a definite choice of N , for instance N = 3, and use the notation Ta and Ra for TaN and RaN . 4 e SÉRIE – TOME 41 – 2008 – No 2.

(14) 189. THE CAUCHY PROBLEM FOR WAVE EQUATIONS. C 3.6. – The multiplication (a, u) 7→ au is continuous from (L∞ ∩ LL) × H to H s+δ log for s ∈ ] − 1, 1[ and δ ∈ {− 21 , 0, 12 }. s+δ log. Proof. – (See [6].) Property iii) says that Ra is smoothing by almost one derivative in neg0 ative spaces, and therefore, for all σ ∈ ] − 1, 1[ it maps H σ to H σ for all σ 0 > max {σ, 0} such that σ 0 < min {σ + 1, 1}. Combining this observation with i), the corollary follows. In particular, we note the following estimate (3.15). kauk. 1 log. H s+ 2. ≤ C kakL∞ kuk. 1 log. H s+ 2. + kakLL kukH s .. p 2 P of degree m on Rn . Dep 3.7. – Consider q = (1 + |ξ| ) and ψ(ξ) a symbol ∞ note by Q = (1 − ∆) and Ψ the associated operators. If a ∈ L ∩ LL, then the commutator 1 1 [Q−s Ψ, Ta ] maps H −s+ 2 log into H 1−m− 2 log and (3.16). k[Q−s Ψ, Ta ]uk. 1 log. H 1−m− 2. ≤ CkakLL kuk. 1 log. H −s+ 2. with C uniformly bounded for s ∈ [0, 1] and ψ in a bounded set. Proof. – We use Theorem 35 of [4], which states that if H is a Fourier multiplier with symbol h of degree 0 and if a is Lipschitz, then k[H, a]∂xj ukL2 ≤ Ck∇x akL∞ kukL2 . For k > 0, writing ∆k u as sum of derivatives, this implies that (3.17). k[H, a]∆k ukL2 ≤ C2−k k∇x akL∞ k∆k ukL2 ,. with C independent of k and H, provided that the symbol h remains in a bounded set of symbols of degree 0. We now proceed to the proof of the proposition. Since Ψ and Q commute with ∆k , one has X (3.18) [Q−s Ψ, Ta ]u = [Q−s Ψ, Sk−3 a]∆k u. k≥3. Moreover, since the spectrum of Sk−3 a∆k u is contained in the annulus 2k−1 ≤ |ξ| ≤ 2k+2 , it follows that (3.19). [Q−s Ψ, Sk−3 a]∆k = 2k(m−s) [Hk , Sk−3 a]∆k. where the symbol of Hk is hk (ξ) = 2k(s−m) q −s (ξ)ψ(ξ)ϕ(2−k ξ) and ϕ supported in a suitable fixed annulus. Note that the family {hk } is bounded in the space of symbols of degree 0, uniformly in k, s ∈ [0, 1] and ψ in a bounded set of symbols of degree m. By (3.17), it follows that k[Hk , Sk−3 a]∆k ukL2 ≤ C2k(m−s−1) k∇Sk−3 akL∞ k∆k ukL2 . 1. Together with (3.9) and Proposition 3.1, this implies that for u ∈ H −s+ 2 log , k[Q−s Ψ, Sk−3 a]∆k uk ≤ C(k + 1)kakLL k∆k ukL2 . Using Proposition 3.2, the estimate (3.16) follows. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(15) 190. F. COLOMBINI AND G. MÉTIVIER. P 3.8. – If a ∈ L∞ ∩ LL is real valued, then Ta − (Ta )∗ ∂xj and ∂xj Ta − 1 1 (Ta )∗ map H s+ 2 log into H s− 2 log and satisfy k Ta − (Ta )∗ ∂xj uk s− 12 log ≤ CkakLL kuk s+ 21 log , H (3.20) H k∂xj Ta − (Ta )∗ uk s− 12 log ≤ CkakLL kuk s+ 21 log . H. H. Proof. – The Sk a are real valued, since a is real, and the ∆k are self adjoint, thus (Ta )∗ u =. ∞ X. ∆k (Sk−3 a) u .. k=3. Therefore, one has X X [Sk−3 a, ∆k ] = [Sk−3 a, ∆k ]Ψk Ta − (Ta )∗ = where Ψk is a Fourier multiplier with symbol ψk = ψ(2−k ξ) and ψ is supported in a suitable annulus. Using again [4] (see (3.17)) yields k[Sk−3 a, ∆k ]∂xj Ψk ukL2 ≤ C(k + 1)kakLL kΨk ukL2 , and a similar estimate when the derivative is on the left of the commutator. Since the spectrum of [Sk−3 a, ∆k ]Ψk u is contained in an annulus of size ≈ 2k , this implies (3.20). 1 P 3.9. – If a and b belong to L∞ ∩ LL, then Ta Tb − Tab ∂xj maps H s+ 2 log 1 into H s− 2 log and (3.21) k Ta Tb − Tab ∂xj uk s− 21 log ≤ C kakLL kbkL∞ + kbkLL kakL∞ kuk s+ 21 log . H. H. Proof. – By Proposition 3.4, it is sufficient to prove the estimate for any paraproduct T N . One has X X TaN TbN ∂xj u = Sk−N a ∆k Sl−N b ∆l ∂xj u . k≥N l≥N. In this sum, terms with |l−k| ≤ 2 vanish, because of the spectral localization of Sl−N b ∆l ∂xj . The commutators [∆k , Sl−N b] contribute to terms which are estimated as in (3.18): k[∆k , Sl−N b]∆l ∂xj ukL2 ≤ C(k + 1)kbkLL k∆l ukL2 . If N is large enough, the spectrum of the corresponding term is contained in an annulus of size ≈ 2k and hence the commutators contribute to an error term in (3.21). Therefore, it is sufficient to estimate X X (3.22) Sk−N aSl−N b − Sk−N (ab) ∆k ∆l ∂xj u. k≥N l≥N. Again, only terms with |l − k| ≤ 2 contribute to the sum. Using (3.5), one has ka − Sk−N akL∞ ≤ C(k + 1)2−k kakLL , kb − Sl−N bkL∞ ≤ C(k + 1)2−k kbkLL , kab − Sk−N (ab)kL∞ ≤ C(k + 1)2−k kabkLL . Thus kSk−N aSl−N b − Sk−N (ab)kL∞ ≤ C(k + 1)2−k kakLL kbkL∞ + kakL∞ kbkLL . 4 e SÉRIE – TOME 41 – 2008 – No 2.

(16) THE CAUCHY PROBLEM FOR WAVE EQUATIONS. 191. Since the terms in the sum (3.22) have their spectrum in annuli of size ≈ 2k , this implies that 1 1 this sum belongs to H 0− 2 log when u ∈ H 0+ 2 log , with an estimate similar to (3.21). 3.3. Positivity estimates The paradifferential calculus sketched above is well adapted to the analysis of high frequencies but does not take into account the low frequencies. For instance, the positivity of the function a does not imply the positivity of the operator Ta in L2 , only the positivity up to a smoothing operator. However, in the derivation of energy estimates, such positivity results are absolutely necessary. To avoid a separate treatment of low frequencies, we introduce modified paraproducts for which positivity results hold (we could also introduce weighted paraproducts as in [10, 11, 12]). Consider a nonnegative integer ν and define the modified paraproducts (3.23). Paν u =. ∞ X. Smax{ν,k−3} a ∆k u = Sν aSν+2 u +. k=0. ∞ X. Sk a ∆k+3 u.. k=ν. Then Paν u − Ta u =. (3.24). ν+2 X. ν X. ∆j a ∆k u. k=0 j=max{0,k−2}. and au − Paν u =. (3.25). ∞ X. ∆j a Sj+2 u.. j=ν+1. The difference (3.24) concerns only low frequencies, and therefore the results of Propositions 3.7, 3.8 and 3.9 are valid if one substitutes Paν in place of Ta , at the cost of additional error terms. In particular, (3.24) and (3.25) immediately imply the following estimates: L 3.10. – (3.26). i) There is a constant C such that for all ν, a ∈ L∞ and all u ∈ L2 ,. k(Paν − Ta )∂xj ukL2 + k∂xj (Paν − Ta )ukL2 ≤ C2ν kakL∞ kukL2 .. ii) There is a constant C0 such that for all ν, for all a ∈ LL and all u ∈ L2 , (3.27). kau − Paν ukL2 ≤ C0 ν2−ν kakLL kukL2 .. We will also use the following extension of Proposition 3.8: P 3.11. – If a ∈ L∞ ∩ LL is real valued, then Paν − (Paν )∗ ∂xj and ∂xj Paν − 1 1 (Paν )∗ map H 0+ 2 log into H 0− 2 log and k Paν − (Paν )∗ ∂xj uk 0− 12 log ≤ CkakLL kuk 0+ 21 log + νkukL2 , H H (3.28) k∂xj Paν − (Paν )∗ uk 0− 12 log ≤ CkakLL kuk 0+ 21 log + νkukL2 . H. H. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(17) 192. F. COLOMBINI AND G. MÉTIVIER. Proof. – One has X Paν − (Paν )∗ ∂xj u = [Sν a, Sν+2 ]∂xj u + [Sk a, ∆k+3 ]∂xj u. k≥ν. The sum over k is treated exactly as in the proof of Proposition 3.8 and contibutes to the same error term. Using again Theorem 35 of [4], the L2 norm of the first term is estimated by Ck∇x Sν akL∞ kukL2 ≤ C(ν + 1)kakLL kukL2 and contibutes to the second error term in (3.28). When the derivative is on the left, the proof is similar. Moreover, a comparison of Paν u with au immediately implies the following positivity estimate. C 3.12. – There is a constant c0 , such that for any positive LL-function a such that δ = min a(x) > 0, all ν such that ν2−ν ≤ c0 δ/kakLL , and u ∈ L2 (Rn ), δ (3.29) Re Paν u, u L2 ≥ kuk2L2 . 2 Here, (·, ·)L2 denotes the scalar product in L2 . This estimate extends to vector valued functions u and matrices a, provided that a is symmetric and positive. 3.4. The time dependent case In the sequel we will consider functions of (t, x) ∈ [0, T ] × Rn , considered as functions of t with values in various spaces of functions of x. In particular we denote by Ta the operator acting for each fixed t as Ta(t) : (3.30). (Ta u)(t) =. ∞ X. Sk−3 (Dx )a(t) ∆k (Dx )u(t).. k=3. The Propositions 3.4, 3.7, 3.8 and 3.9 apply for each fixed t. There are similar definitions for the modified paraproducts Paν ; further, Lemma 3.10 and Corollary 3.12 apply for fixed t. When a is a Lipschitz function of t, the definition (3.30) immediately implies that (3.31). [∂t , Ta ] = T∂t a ,. [∂t , Paν ] = P∂νt a .. When a is only Log Lipschitz this formula does not make sense, since ∂t a is not defined as a function. The idea, already used in [5, 6], is that it is sufficient to commute ∂t with time regularization of a. In our context, this simply means that in (3.30), we will replace the term Sk−3 a, which is a spatial regularization of a, by a space-time regularization, namely Sk−3 ak where ak is a suitable time mollification of a. We now give the details for P ν , as we will need them in the next section. Introduce the mollifiers (3.32). k (t) = 2k (2k t). where  ∈ C0∞ (R) is non negative, with integral over R equal to 1. 4 e SÉRIE – TOME 41 – 2008 – No 2.

(18) THE CAUCHY PROBLEM FOR WAVE EQUATIONS. 193. D 3.13. – Given a ∈ L∞ ∩ LL([0, T0 ] × Rn ), define Z (3.33) ak (t, x) = k ∗t ã = k (t − s)ã(s, x) ds where ã is the LL extension of a given by (3.34). t ≤ 0,. ã(t, x) = a(0, x),. ã(t, x) = a(T0 , x),. t ≥ T0 .. ‹ν is defined by Next, for fixed t, the operator P a(t) ‹ν u = Sν aν Sν+2 u + P a(t). (3.35). ∞ X. Sk ak ∆k+3 u.. k=ν. ‹aν the operator acting on functions of (t, x) by (P ‹aν u)(t) = P ‹ν u(t). We denote by P a(t) P 3.14. – Let a ∈ L∞ ∩ LL([0, T0 ] × Rn ). Then for each t ∈ [0, T0 ], the ν ‹ν )∂x , R2 (t) = ∂x (P ν − P ‹ν ), R3 (t) = (P ‹ν )∗ − operators R1 (t) = (Pa(t) −P j j a(t) a(t) a(t) a(t) ‹ν ∂x , R4 (t) = ∂x (P ‹ν )∗ −P ‹ν ), and R5 (t) = [Dt , P ‹aν ](t) map H 0+ 21 log into H 0− 12 log P a(t). j. j. a(t). a(t). and there is a constant C such that for all t ∈ [0, T0 ] and for k = 1, . . . , 5, kRk uk 0− 21 log ≤ CkakLL kuk 0+ 12 log + νkukL2 . (3.36) H H Proof. – a) First, we recall several estimates from [6]. For a ∈ LL([0, T0 ] × Rn ) the difference a − ak satisfies |a(t, x) − ak (t, x)| ≤ C(k + 1)2−k kakLL ,. (3.37). |∂t ak (t, x)| ≤ C(k + 1)kakLL ,. (3.38). with C independent of t and x. In particular, we note that kSk (a(t) − ak (t))kL∞ ≤ C(k + 1)2−k kakLL .. (3.39) b). In accordance with (3.35), for l = 1, 2, 5, we split Rl into two terms X (3.40) Rl (t)u = Bl u + Hl u, Hl u = wk k≥ν. with Bl u spectrally supported in the ball of radius 2 an annulus |ξ| ≈ 2k . For R1 , B1 u = Sν (a(t) − aν (t)) Sν+2 ∂xj u,. ν+4. and with wk spectrally supported in. wk = Sk (a(t) − ak (t)) ∆k+3 ∂xj u.. With (3.39), this implies that kB1 ukL2 ≤ C(ν + 1)kakLL kukL2 and kwk kL2 ≤ C(k + 1)kakLL k∆k+3 ukL2 , implying that kH1 uk. 1 log. H 0− 2. ≤ CkakLL kuk. 1 log. H 0+ 2. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE. ..

(19) 194. F. COLOMBINI AND G. MÉTIVIER. For R2 , the analysis is similar. One has B2 u = ∂xj Sν (a(t) − aν (t)) Sν+2 u ,. wk = ∂xj Sk (a(t) − ak (t)) ∆k+3 u .. Thanks to the spectral localization, the estimates for B2 u and wk are the same as in the case of R1 , implying that kB2 ukL2 ≤ C(ν + 1)kakLL kukL2. (3.41). kH2 uk. (3.42) c). 1 log. H 0− 2. ≤ CkakLL kuk. 1 log. H 0+ 2. .. For k = 5 we write (3.40) with B5 u = Sν (∂t aν (t)) ∆ν+2 u,. wk = Sk (∂t ak (t)) ∆k+3 u.. Thus the estimates (3.38) imply kB5 ukL2 ≤ C(ν + 1)kakLL kukL2 kH5 uk. 1 log. H 0− 2. d). ≤ CkakLL kuk. 1 log. H 0+ 2. .. One has ν ν R3 (t) = R1 (t) + R2∗ (t) + (Pa(t) )∗ − Pa(t) ∂xj .. The third term is estimated in Proposition 3.11. The operators R1 and R2∗ = B2∗ + H2∗ are estimated in part b), implying that R3 satisfies (3.36) for k = 3. The proof for R4 = R3∗ = ν ν )∗ − Pa(t) is similar. R1∗ + R2 + ∂xj (Pa(t) This finishes the proof of the proposition. L 3.15. – There is a constant C0 such that for any a ∈ LL([0, T0 ]×Rn ), u ∈ L2 (Rn ), ν ≥ 0 and all t ∈ [0, T0 ], one has ‹ν ukL2 ≤ C0 ν2−ν kakLL kukL2 . ka(t)u − P a(t). (3.43) Proof. – We have. ‹aν u = (a − Sν aν )Sν+2 u + au − P. ∞ X. (a − Sk ak ) ∆k+3 u.. k=ν. Combining (3.5) and (3.39), we see that ka(t) − Sk ak (t)kL∞ ≤ Ck2−k kakLL . This implies (3.43). The lemma immediately implies the following positivity estimate. C 3.16. – There is a constant c0 , such that for any positive LL-function a such that δ = min a(t, x) > 0, all ν such that ν2−ν ≤ c0 δ/kakLL , and u ∈ L2 (Rn ), δ ν (3.44) Re Pa(t) u, u L2 (Rn ) ≥ kuk2L2 (Rn ) . 2 The same result holds for vector valued functions u and definite positive square matrices a. Finally, we quote the following commutation result which will be needed in the next section. 4 e SÉRIE – TOME 41 – 2008 – No 2.

(20) THE CAUCHY PROBLEM FOR WAVE EQUATIONS. 195. î ó 1 ‹ν , Λ 12 and P 3.17. – Suppose that a ∈ LL([0, T0 ] × Rn ). Then Λ 2 P a(t) î ó ‹ν , Λ 21 Λ 12 are bounded in L2 and satisfy P a(t) î ó î ó 1 ‹ν , Λ 12 u ‹ν , Λ 12 Λ 12 u + ≤ C ν 2 2−ν kakLL + νkakL∞ kukL2 . Λ2 P P a(t) a(t) 2 2 L. L. ‹aν conProof. – Thanks to the spectral localization, the low frequency part Sν aν Sν+2 in P 2 tributes to terms whose L norm is bounded by CνkukL2 . The commutator with the high frequency part reads X 1 [Λ 2 , Sk ak ]∆k+3 u. k≥ν. We argue as in the proof of Proposition 3.7 and write 1. (3.45). 1. [Λ 2 , Sk ak ]∆k+3 = (k + 1) 2 [Hk , Sk ak ]∆k+3 1. 1. where the symbol of Hk is hk (ξ) = (k + 1)− 2 (Log(2 + |ξ|)) 2 ϕ(2−k ξ) and ϕ is supported in a suitable fixed annulus. Note that the family {hk } is bounded in the space of symbols of degree 0. By (3.17), one has 1. k[Hk , Sk a(t)]Λ 2 ∆k+3 ukL2 ≤ C(k + 1)2−k k∇x Sk ak (t)kL∞ k∆k ukL2 . Since ∇x Sk ak = (∇x Sk a) ∗ k , its L∞ norm is bounded by CkkakLL . Adding up, and using the spectral localization, these terms contribute by a function whose L2 norm is bounded by Cν 2 2−ν kakLL kukL2 . 1 When Λ 2 is on the left of the commutator, the analysis is similar. 4. Proof of the main results 4.1. The main estimate We consider the operator (2.1) with coefficients which satisfy (2.4), (2.5) and (2.6). We fix θ < θ1 in ]1 − α, α[, and with λ to be chosen later, we introduce the notation (4.1). s(t) = θ + tλ.. Recall that. ß ™ θ1 − θ T = min T0 , . λ. (4.2). Note that for t ∈ [0, T ], s(t) remains in [θ, θ1 ] ⊂]1 − α, α[. We will consider solutions of the Cauchy problem (4.3). Lu = f,. u|t=0 = u0 ,. Xu|t=0 = u1. with (4.4) (4.5) (4.6). u ∈ H1−θ+ 12 log,λ (T ), u0 ∈ H 1−θ (Rn ), f = f1 + f2 ,. ∂t u ∈ H−θ+ 12 log,λ (T ), u1 ∈ H −θ (Rn ),. f1 ∈ L−θ,λ (T ),. f2 ∈ H−θ− 21 log,λ (T ),. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(21) 196. F. COLOMBINI AND G. MÉTIVIER. Note that if u and f satisfy (4.4) and (4.6), then (4.7). u ∈ L2 ([0, T ]; H 1−θ1 ),. ∂t u ∈ L2 ([0, T ]; H −θ1 ),. f ∈ L1 ([0, T ]; H −θ2 ). (4.8). for all θ2 ∈ ]θ1 , α[, so that the meaning of the Cauchy condition is clear. The main step in the proof of Theorem 2.6 is the following: T 4.1. – There is a λ0 ≥ 0 of the form (2.30) such that for λ ≥ λ0 there is a constant K such that: for all f , u0 and u1 satisfying (4.5) (4.6), and all u satisfying (4.4) solution of the Cauchy problem (4.3), one has (4.9). u ∈ C1−θ,λ (T ),. ∂t u ∈ C−θ,λ (T ).. Moreover u satisfies the energy estimate (2.29). This theorem contains two pieces of information : first an energy estimate for smooth u, see Propositions 4.3 and 4.4. By a classical argument, smoothing the coefficients and passing to the limit, this estimate allows for the construction of weak solutions, see Section 5.2. The second piece of information contained in the theorem is a “weak=strong” type result showing that for data as in the theorem, any (weak) solution u satisfying (4.4) is the limit of smooth (approximate) solutions, in the norm given by the left hand side of the energy estimate, implying that u satisfies the additional smoothness (4.9) and the energy estimate. This implies uniqueness of weak solutions. The idea is to get an energy estimate by integration by parts, from the analysis of ¨ ∂ (4.10) 2 Re Lu, e−2γt (1 − ∆x )−s(t) Xu where h·, ·i denotes the L2 scalar product in Rn extended to the Hermitian symmetric duality H σ ×H −σ for all σ ∈ R, and ∆x denote the Laplace operator on Rd . This extends the analysis of [6] where X = ∂t . The parameter γ is chosen at the end to absorb classical error terms (present for Lipschitz coefficients) while the parameter λ which enters in the definition of s(t), is chosen to absorb extra error terms coming from the loss of smoothness of the coefficients. To prove Theorem 4.1, the first idea would be to mollify the equation. However, the lack of smoothness of the coefficients does not allow us to use this method directly and we cannot prove that the weak solutions are limits of exact smooth solutions. Instead, the idea is to write the equation as a system (2.16) for (u, v) and mollify this system. This leads to the consideration of the equations: ( ˜ + f, Zv + b̃0 v = L̃2 u − L̃1 u − du (4.11) Y u + c̃0 u = v/a0 + g. In this form, the commutator of spatial mollifiers with ∂t are trivial, and we can prove that weak solutions of (4.11) are limits of smooth solutions, (uε , v ε ) with g ε 6= 0, which thus do not correspond to exact solutions uε of (4.3). 4 e SÉRIE – TOME 41 – 2008 – No 2.

(22) THE CAUCHY PROBLEM FOR WAVE EQUATIONS. 197. Notations. It is important for our purpose to keep track of the dependence of the various constants on the Log-Lipschitz norms. In particular we will use the notations δ0 , δ1 of (2.8) and ALL , AL∞ , B of (2.9) (2.10). To simplify the exposition, we will denote by C, K0 and K constants which may vary from one line to another, C denoting universal constants depending only on the paradifferential calculus; K0 depending also on AL∞ /δ0 ; K, still independent of the parameters (γ, ε), but dependent also on δ0 , δ1 , θ0 , θ1 and the various norms of the coefficients. 4.2. Estimating v First, we give estimates that link v and ∂t u. L 4.2. – Suppose that u satisfies (4.4). Then v = Xu + c0 u belongs to the space H−θ+ 12 log,λ (T ) ⊂ L2 ([0, T ]; H −θ1 ) and for almost all t, kv(t)k −s(t)+ 12 log ≤ CAL∞ ku(t)k 1−s(t)+ 21 log + k∂t u(t)k −s(t)+ 12 log H H H (4.12) + C(ALL + B) ku(t)kH 1−s(t) + k∂t u(t)kH −s(t) , (4.13). k∂t u(t)k. 1 log. H −s(t)+ 2. ≤ K0 ku(t)k. C kv(t)k −s(t)+ 21 log H δ0 + kv(t)kH −s(t) .. 1 log. H 1−s(t)+ 2. + K ku(t)kH 1−s(t). +. There are similar estimates in the spaces H s without the 21 log. If in addition Lu = f with f satisfying (4.6), then ∂t v ∈ L1 ([0, T ]; H −1−θ1 ). Proof. – a) First, we note that the multiplication (a, u) 7→ au is continuous from (L∞ ∩ LL)([0, T ] × Rn ) × H−θ+ 21 log,λ (T ) to H−θ+ 21 log,λ (T ). Indeed, the corresponding norm estimate of the product is clear for smooth u, from (3.15) integrated in time. The claim follows by density. In particular, this shows that a0 ∂t u and the aj ∂xj u belong to H−θ+ 12 log,λ (T ). Similarly, the estimate (4.14). kbu(t)k. 1 log. H −s(t)+ 2. ≤ Ckbu(t)kH 1−s(t) ≤ CkbkC α ku(t)kH 1−s(t). implies that c0 u ∈ H−θ+ 12 log,λ (T ). Therefore v ∈ H−θ+ 12 log,λ (T ) and the estimate (4.12) holds. The proof of (4.13) is similar, noting that ∂t u = b). d X 1 aj c0 v− ∂xj u − u. a0 a a0 j=1 0. As in the proof of Lemma 2.2, we see that the equation implies that ∂t v = f −. d X. ˜ ∂xj (ãj v) − b̃0 v + L̃2 u − L̃1 u − du.. j=1. The conservative form of L̃2 and the multiplicative properties above show that ∂xj (ãj v), L̃2 u ∈ H−θ−1+ 21 log,λ (T ) ⊂ L2 ([0, T ]; H −1−θ1 ). Similarly, L̃1 u and b̃0 v belong to H−θ+ 12 log,λ (T ) and thus to L2 ([0, T ]; H −θ1 ). The last term ũ is in L2 . Therefore, ∂t v−f ∈ L2 ([0, T ]; H −1−θ1 ). Since f ∈ L1 ([0, T ]; H −θ2 ) for θ2 ∈ ]θ1 , α[, the lemma follows. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(23) 198. F. COLOMBINI AND G. MÉTIVIER. Next, we give a-priori estimates in the space H−θ+ 12 log,λ (T ) ∩ C−θ,λ (T ) for smooth solutions of (4.15). (Z + c̃0 )v = ϕ,. v|t=0 = v0 .. We define the operators (Qv)(t) = (1 − ∆x )−s(t)/2 v(t),. (4.16). (Qγ v)(t) = e−γt (Qv)(t).. P 4.3. – Suppose that v ∈ L2 ([0, T ]; H 1 ) and ∂t v ∈ L1 ([0, T ]; L2 ). Then the functions vγ (t) := Qγ v belong to C 0 ([0, T ], L2 ) and satisfy Z t 2 k(γ + λΛ)1/2 vγ (t0 )k2L2 dt0 kvγ (t)kL2 + 2 0 t. Z. h(Z + c̃0 )v(t. ≤2. (4.17). 0. ), Q2γ (t0 )v(t0 )i dt0. +. kvγ (0)k2L2. Z +. 0. t. F (t0 ) dt0. 0. with (4.18). F (t0 ) ≤ K0. Proof. – a). ALL −γt0 1/2 0 2 ke Λ v(t )kH −s(t0 ) + Kkv(t0 )k2H −s(t0 ) . δ0. Since v ∈ L2 ([0, T ]; H 1 ) and ∂t v ∈ L1 ([0, T ]; L2 ), we have ∂t Qγ v = Qγ ∂t v − (γ + λΛ)Qγ v ∈ L1 ([0, T ]; L2 ). (4.19). as immediately seen using the spatial Fourier transform. Moreover, vγ = Qγ v ∈ C 0 ([0, T ]; L2 ) and satisfies the following identity Z t kvγ (t)k2L2 − kvγ (0)k2L2 = 2 Re h∂t Qγ v, Qγ vi dt0 . 0. Thus, t. Z. h∂t v, Q2γ vi dt0 = 2 Re. 2Re 0. (4.20) =. Z. t. hQγ ∂t v, Qγ vi dt0. 0. kvγ (t)k2L2. −. kvγ (0)k2L2. Z +2. t. k(γ + λΛ)1/2 vγ (t0 )k2L2 dt0. 0. b) Next we consider the terms ∂xj (ãj v). We note that they belong to L2 ([0, T ]; H −σ ) for all σ > 0. In particular, since s(t) ≥ θ > 0, we note that the pairing h∂xj (ãj v), Q2γ vi is well defined. We give an estimate for Z t 2 Re h∂xj (ãj v), Q2γ vi dt0 , 0. using the decomposition ãj v = Tãj v + Rãj v. By Proposition 3.4 it follows kRãj v(t)k. 1 log. H 1−s(t)− 2. ≤ Ckãj kLL kv(t)k. 1 log. H −s(t)+ 2. since s(t) ∈ [θ, θ1 ] ⊂]0, 1[. Moreover, kQ2γ v(t)k 4 e SÉRIE – TOME 41 – 2008 – No 2. 1 log. H s(t)+ 2. ≤ Ce−2γt kv(t)k. 1 log. H −s(t)+ 2. ..

(24) 199. THE CAUCHY PROBLEM FOR WAVE EQUATIONS. Thus |h∂xj Rãj v(t), Q2γ,ε v(t)i| ≤ kRãj v(t)k. 1 log. ≤ Ckãj kLL e. kv(t)k2. H 1−s(t)− 2 −2γt. kQ2γ,ε v(t)k. 1 log. H s(t)+ 2. 1 log. H −s(t)+ 2. .. It remains to consider Reh∂xj Tãj v, Q2γ vi = RehQγ ∂xj Tãj v, Qγ vi = Reh∂xj Tãj Qγ v, Qγ vi + Reh∂xj [Qγ , Tãj ]v, Qγ vi. Note that these computations make sense because v(t) ∈ H 1 and all the pairings are well defined. Proposition 3.7 implies that kh∂xj [Qγ , Tãj ]v(t)k0− 21 log ≤ Ce−γt kãj kLL kv(t)k−s(t)+ 21 log and therefore (4.21). |h∂xj [Qγ , Tãj ]v(t), Qγ v(t)i| ≤ Ckãj kLL e−2γt kv(t)k2. 1 log. H −s(t)+ 2. .. Next, for vγ (t) ∈ H 2−θ1 , we have 2 Reh∂xj Tãj vγ , vγ i = Reh(∂xj Tãj − Tã∗j ∂xj )vγ , vγ i = Reh(Tãj − Tã∗j )∂xj vγ , vγ i + Reh[∂xj , Tãj ]∂xj vγ, , vγ i. Using Propositions 3.8 and 3.7, one can bound both terms by the right hand side of (4.21). Adding up, we have proved that Z t Z t 0 h∂xj (ãj v), Q2γ vi dt0 ≤ Ckãj kLL 2 Re ke−γt Λ1/2 v(t0 )k2H −s(t) dt0 . 0. c) that. 0. The zero-th order term is clearly a remainder, and the multiplicative properties imply |hc̃0 v(t), Q2γ v(t)i ≤ Kkv(t)k2H −s(t) .. d). We note that kaj /a0 kLL ≤ kaj kLL kk1/a0 kL∞ + kaj kL∞ kk1/a0 kLL ≤. AL∞ ALL AL∞ ALL ALL ≤2 , + δ0 δ02 δ02. since δ0 ≤ a0 ≤ AL∞ . Using identity (4.20) and the estimates of parts b) and c), implies (4.17) and so the proof of the lemma is complete. 4.3. Estimating ∇x u We now get estimates of ∇x u from the analysis of (4.22). e 2 u, Q2γ Xui = − − 2 RehL. n X. 2 Reh∂xj (ãj,k ∂xk u), Q2γ Xui.. j,k=1. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(25) 200. F. COLOMBINI AND G. MÉTIVIER. P 4.4. – Suppose that u ∈ L2 ([0, T ]; H 2 ) with ∂t u ∈ L2 ([0, T ]; H 1 ). Then uγ := Qγ u ∈ C 0 ([0, T ], H 1 ) and Z t 1 2 δ0 δ1 k∇x uγ (t)kL2 + δ0 δ1 k(γ + λΛ)1/2 ∇x uγ (t0 )k2L2 dt0 2 0 (4.23) Z t Z t e 2 u, Q2γ vi dt0 + CA2L∞ k∇x uγ (0)k2 2 + ≤ −2 Re hL E(t0 ) dt0 , L. 0. 0. where (4.24). 1 kXu(t)k2 −s(t)+ 1 log 2 H δ02 + kXu(t)k2H −s(t) .. |E(t)| ≤ K0 ALL AL∞ e−2γt ku(t)k2. +. 1 log. H 1−s(t)+ 2. + Ke−2γt ku(t)k2H 1−s(t). To simplify the exposition, we note here that all the dualities h·, ·i written below make sense, thanks to the smoothness assumption on u. This will not be repeated at each step. Moreover, in the proof below, we assume that u itself is smooth (in time). Proof. – a) that. We first perform several reductions. Using iii) of Proposition 3.4, one shows h∂xj (ãj,k ∂xk u), Q2γ Xui = h∂xj (Tãj,k ∂xk u), Q2γ Xui + E1. with (4.25). |E1 (t)| ≤ Ckãj,k kLL k∂xk u(t)k. 1 log. H −s(t)+ 2. kQ2γ Xu(t)k. 1 log. H s(t)+ 2. .. Since kãj,k kLL ≤ K0 ALL ≤ K0 ALL AL∞ /δ0 , E1 satisfies (4.24). Similarly, h∂xj (Tãj,k ∂xk u), Q2γ Xui = h∂xj Qγ Tãj,k ∂xk u, Qγ Xui = h∂xj Tãj,k ∂xk Qγ u, Qγ Xui + E2 where E2 also satisfies (4.25), and hence (4.24). b). Next we write X. Xu = Ta0 ∂t u +. Taj ∂xj u + r. and kr(t)k. 1 log. H 1−s(t)− 2. ≤ CALL ku(t)k. 1 log. H 1−s(t)+ 2. + k∂t u(t)k. 1 log. H −s(t)+ 2. . + CBku(t)kH 1−s(t) .. Therefore, r contributes to an error term E3 = h∂xj Tãj,k ∂xk Qγ u, Qγ ri such that |E3 (t)| ≤ e−2γt K0 AL∞ ku(t)k. 1 log. H 1−s(t)+ 2. kr(t)k. 1 log. H 1−s(t)− 2. .. Using (4.13) in the estimate of r, we see that |E3 (t)| ≤ e−2γt K0 AL∞ ALL ku(t)k 1−s(t)+ 12 log H 1 ku(t)k 1−s(t)+ 12 log + δ0 kXu(t)k −s(t)+ 12 log +Kku(t)kH 1−s(t) +KkXu(t)kH −s(t) H. and hence satisfies (4.24). 4 e SÉRIE – TOME 41 – 2008 – No 2. H.

(26) THE CAUCHY PROBLEM FOR WAVE EQUATIONS. c). 201. Consider now the term h∂xj Tãj,k ∂xk Qγ u, Qγ Ta0 ∂t ui = −hTãj,k ∂xk Qγ u, ∂xj Qγ Ta0 ∂t ui = −hTãj,k ∂xk Qγ u, Ta0 ∂xj Qγ ∂t ui + E4 = −h(Ta0 )∗ Tãj,k ∂xk Qγ u, ∂xj Qγ ∂t ui + E4 = −hTa0 Tãj,k ∂xk Qγ u, ∂xj Qγ ∂t ui + E4 + E5 = −hTa0 ãj,k ∂xk Qγ u, ∂xj Qγ ∂t ui + E4 + E5 + E6. where E4 , E5 and E6 are estimated by Proposition 3.7, 3.8 and 3.9 respectively. They all satisfy |Ek (t)| ≤ Ce−2γt Aku(t)k 1−s(t)+ 21 log k∂t u(t)k −s(t)+ 21 log . H. H. with A = kãj,k kLL ka0 kL∞ +kãj,k kL∞ ka0 kLL ≤ K0 AL∞ ALL . Again using (4.13) to replace ∂t u by Xu, one shows that these errors satisfy (4.24). Similarly h∂xj Tãj,k ∂xk Qγ u, Qγ Tal ∂xl ui = −hTal ãj,k ∂xk Qγ u, ∂xl ∂xj Qγ ui + E7 where E7 satisfies |E7 (t)| ≤ Ce−2γt K0 AL∞ ALL ku(t)k2. (4.26). 1 log. H 1−s(t)+ 2. thus (4.24). d). Introduce the notation. (4.27). wj = ∂xj Qγ u.. Because ãj,k = ãk,j , we have RehTal ãj,k wk , ∂xl wj i + RehTal ãk,j wj , ∂xl wk i = Reh((Tal ãj,k )∗ ∂xl − ∂xl Tal ãj,k )wk , wj i := E8 Using Propositions 3.8 and 3.7, one shows that E8 satisfies |E8 (t)| ≤ Ckal ãj,k kLL kwj (t)k. 1 log. H 0+ 2. kwk (t)k. 1 log. H 0+ 2. and therefore E8 also satisfies (4.26) thus (4.24). e) (4.28). It remains to consider the sum S := Re. n X. hTbj,k ∂xk Qγ u, ∂xj Qγ ∂t ui. j,k=1. with bj,k = a0 ãj,k = a0 aj,k + aj ak . By the strict hyperbolicity assumption (2.8), it follows for all ξ ∈ Rn n X bj,k (t, x)ξj ξk ≥ δ0 δ1 |ξ|2 . j,k=1. Therefore, we can use Corollary 3.16. Since kbj,k kLL ≤ 2AL∞ ALL , there exists an integer ν, with 2ν AL∞ ALL (4.29) ≈ , ν δ ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(27) 202. F. COLOMBINI AND G. MÉTIVIER. such that for all t ∈ [0, T0 ] and (w1 , . . . , wn ) in L2 (Rn ), the following estimate is satisfied (4.30). n X. Re. hPbνj,k (t) wk , wj i ≥. j,k=1. δ0 δ1 kwk2L2 . 2. From now on we fix such a ν and use the notation Pb in place of Pbν . Using Lemma 3.10 and Proposition 3.14, we see that ‹b wk k 0− 1 log ≤ Ckbj,k kLL kwk k 0+ 1 log + Kkwk kL2 k∂xj Tbj,k wk − ∂xj P j,k 2 2 H. H. Therefore S = Re. n X. ‹b ∂x Qγ u, ∂x Qγ ∂t ui + E9 hP j j,k k. j,k=1. where. |E9 (t)| ≤ Ce−2γt kbj,k kLL ku(t)k. 1 log. H 1−s(t)+ 2. +e. −2γt. k∂t u(t)k. 1 log. H −s(t)+ 2. νKku(t)kH 1−s(t) k∂t u(t)k. 1 log. H −s(t)+ 2. .. Using (4.13), implies that E9 satisfies (4.24). ‹b + (P ‹b )∗ ) at the cost of ‹b by 1 ∂x (P Next, we use Proposition 3.14 to replace ∂xj P j j,k j,k j,k 2 an error E10 similar to E9 . At this stage, we commute Qγ and ∂t as in (4.19). Using the notation (4.27), yields 2S =. n X. ‹b + (P ‹b )∗ )wk , ∂t wj i Reh(P j,k j,k. j,k=1. +γ. (4.31). n X. ‹b + (P ‹b )∗ )wk , wj i Reh(P j,k j,k. j,k=1. +λ. n X. ‹b + (P ‹b )∗ )wk , Λwj i + 2E9 + 2E10 . Reh(P j,k j,k. j,k=1. We denote by S 1 , S 2 and S 3 the sums on the right hand side. f). The symmetry bj,k = bk,j implies the identity n d X ‹b wk , wj i + E11 RehP j,k dt. S1 =. j,k=1. where. n X. E11 =. ‹b , ∂t ]wk , wj i Reh[P j,k. j,k=1. is estimated using Proposition 3.14: |E11 (t)| ≤ Ckbj,k kLL kw(t)k. 1 log. H 0+ 2. −2γt. ≤ Ce. + νkwkL2 kw(t)k. 1 log. H 0+ 2. kbj,k kLL ku(t)k. ku(t)k. 1 H 1−s(t)+ 2 log. 1 H 1−s(t)+ 2 log. + νku(t)kH 1−s(t). and therefore satisfies (4.24). Moreover, ‹b wk , Λwj i = RehP ‹b Λ 21 wk , Λ 12 wj i + RehΛ 12 [Λ 21 , P ‹b , ]wk , wj i RehP j,k j,k j,k ‹b )∗ wk , Λwj i = RehΛ 2 wk , P ‹b Λ 2 wj i + Rehwk , [P ‹b , Λ 2 ]Λ 2 wj i. Reh(P j,k j,k j,k 1. 4 e SÉRIE – TOME 41 – 2008 – No 2. 1. 1. 1. .

(28) THE CAUCHY PROBLEM FOR WAVE EQUATIONS. 203. We use Proposition 3.17 to estimate the commutators and n X. S3 = 2. ‹b Λ 2 wk , Λ 2 wj i + E12 RehP j,k 1. 1. j,k=1. where |E12 (t)| ≤ Kkw(t)k2L2 ≤ Kku(t)k2H 1−s(t) . Summing up, we have shown that up to an error which satisfies (4.24), the quantity (4.22) under consideration is equal to n n X d X ‹ ‹b wk , wj i RehPbj,k wk , wj i + γ 2 RehP j,k dt. (4.32). j,k=1. j,k=1. +λ. n X. ‹b Λ 2 wk , Λ 2 wj i. 2 RehP j,k 1. 1. j,k=1. By (4.30), the last two sums are larger than or equal to δ0 δ1 kw(t)k2L2 and δ0 δ1 kw(t)k2 0+ 1 log , H 2 respectively. Similarly, integrating the first term between 0 and t and using (4.30) give control of δ02δ1 kw(t)kL2 , finishing the proof of (4.23). 4.4. A-priori estimates for the solutions of (4.11) The proof of Theorem 4.1 is based on a-priori estimates for smooth solutions of the system (4.11). T 4.5. – There are λ0 ≥ 0 of the form (2.30) and γ0 such that for λ ≥ λ0 and γ ≥ γ0 the following is true: for all u ∈ L2 ([0, T ]; H 2 ) and v ∈ L2 ([0, T ]; H 1 ) with ∂t u ∈ L2 ([0, T ]; H 1 ) and ∂t v ∈ L1 ([0, T ]; L2 ) and for all parameters λ, γ and all t ≤ T , the following holds: ã Å 0 1 δ0 δ1 ku(t0 )k2H 1−s(t0 ) + kv(t0 )k2H −s(t0 ) sup e−2γt 2 0≤t0 ≤t Z t 0 + δ0 δ1 e−2γt (λku(t0 )k2 1−s(t0 )+ 1 log + γku(t0 )k2H 1−s(t0 ) ) dt0 H. 0. Z (4.33). +. t. 0. e−2γt (λkv(t0 )k2. H −s(t. 0. 0 )+ 1 log 2. 2. + γkv(t0 )k2H −s(t0 ) ) dt0. ≤ CA2L∞ ku(0)k2H 1−θ + kv(0)k2H −θ + 2 Re. Z. t. hf, Q2γ vi dt0. 0. Z +K 0. t. 0. e−2γt kg(t0 )k1−s(t)− 12 log ku(t0 )k1−s(t)+ 21 log dt0 ,. ˜ ∈ L1 ([0, T ]; H α0 −1 ), g = Y u + c̃0 u − v/a0 ∈ with f = Zv + b̃0 v − L̃2 u + L̃1 u + du 0 L2 ([0, T ]; H α ) for all α0 < α. Proof. – We compute the integral over [0, t] of Rehf, Q2γ vi. Proposition 4.3 takes care of the first term 2 RehZv + b̃0 v, Q2γ vi. We split the second term into three pieces hL̃2 u, Q2γ vi = hL̃2 u, Q2γ Xui − hL̃2 u, Q2γ (a0 g)i + hL̃2 u, Q2γ (c0 u)i ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.

(29) 204. F. COLOMBINI AND G. MÉTIVIER. and use Proposition 4.4 for the first piece. The multiplicative properties imply that |hL̃2 u(t), Q2γ (a0 g)(t)i| ≤ Kkg(t)k1−s(t)− 21 log kL̃2 u(t)k−1−s(t)+ 12 log ≤ Kkg(t)k1−s(t)− 21 log ku(t)k1−s(t)+ 21 log , and |hL̃2 u(t), Q2γ (c0 u)(t)i| ≤ Kku(t)k1−s(t) kL̃2 u(t)k−1−s(t) ≤ Kku(t)k21−s(t) . Next, using the multiplicative properties stated in Corollary 3.6 for the products b̃j ∂xj u and ˜ we see that ∂xj (c̃j u), and the embedding L2 ⊂ H −s for the term du, ˜ e 1 u + du)(t)k k(L H −s(t) ≤ Kku(t)kH 1−s(t) . Thus ˜ |h(L̃1 + d)u(t), Q2γ v(t)i| ≤ Kku(t)k1−s(t) kv(t)k−s(t) ≤ K ku(t)k21−s(t) + kv(t)k2−s(t) . Proposition 4.4 gives an estimate of ∇x u. We also need an estimate for u. The identity (4.20) applied to u yields Z t 0 e−2γt ku(t)k2H −s(t) + e−2γt (λku(t0 )k2 −s(t0 )+ 1 log + γku(t0 )k2H −s(t0 ) ) dt0 H. 0. = kuγ (0)k2H −s(0) + 2 Re. 2. Z. t. h∂t u, Q2γ ui dt0 .. 0. Next, we use the inequality |h∂t u, Q2γ ui| ≤ C ku(t)k2H 1−s(t) + k∂t u(t)k2H −1−s(t) . In addition, we note that the second equation in (4.11) yields k∂t u(t)kH −1−s(t) ≤ K kv(t)k2H −s(t) + ku(t)k2H −s(t) + kg(t)k2H −1−s(t) . We add the various estimates and use Propositions 4.3 and 4.4 to obtain a final estimate. On the left hand side we have 1 0 δ0 δ1 ku(t0 )k2H 1−s(t0 ) + kv(t0 )k2H −s(t0 ) (4.34) sup e−2γt 2 0≤t0 ≤t Z t 0 (4.35) +γ e−2γt δ0 δ1 ku(t0 )k2H 1−s(t0 ) + kv(t0 )k2H −s(t0 ) dt0 0. Z (4.36). +λ. t. 0. e−2γt δ0 δ1 ku(t0 )k2. 1 0 H 1−s(t )+ 2 log. 0. + kv(t0 )k2. On the right hand side, we find the initial data (4.37). CA2L∞ ku(0)k2H 1−s(0) + kv(0)k2H −s(0) ,. the contribution of f Z (4.38). 2 Re 0. 4 e SÉRIE – TOME 41 – 2008 – No 2. t. hf (t0 ), Qγ v(t0 )i dt0 ,. 1 0 H −s(t )+ 2 log. . dt0 ..

(30) THE CAUCHY PROBLEM FOR WAVE EQUATIONS. 205. an estimated contribution of g Z t 0 e−2γt kg(t0 )k1−s(t0 )− 21 log ku(t0 )k1−s(t0 )+ 12 log dt0 , (4.39) K 0. and two types of “remainders”: Z t 0 e−2γt ku(t0 )k2 (4.40) K0 ALL AL∞. H 1−s(t. 0. 0 )+ 1 log 2. +. 1 kvγ (t0 )k2 −s(t0 )+ 1 log dt0 2 H δ02. and Z (4.41). t. K 0. 0 e−2γt kuγ (t0 )k2H 1−s(t0 ) + kv(t0 )k2H −s(t0 ) dt0 .. If (4.42). λ ≥ 2K0. ALL AL∞ δ0 δ1. and λ ≥ 2K0. ALL AL∞ δ02. the term in (4.40) can be absorbed by (4.36). Note that this choice of λ is precisely the choice announced in (2.30), with a new function K0 of AL∞ /δ0 . Finally, if γ is large enough, the term (4.41) is absorbed by (4.35), finishing the proof of the main estimate (4.33). 4.5. Proof of Theorem 4.1 From now on, we assume that λ ≥ λ0 and γ ≥ γ0 are fixed, so that the estimate (4.33) holds. Consider u, f , u0 and u1 satisfying the equation (4.3) and the smoothness assumptions (4.4), (4.5), (4.6). Consider v = Xu + c0 u, which by Lemma 4.2 satisfies v ∈ H−θ+ 21 log , ∂t v ∈ L1 ([0, T ]; H −1−θ1 ), v|t=0 = v0 ∈ H −θ , P with v0 = a0|t=0 u1 + aj |t=0 ∂xj u0 + c0|t=0 u0 . In particular, (u, v, f ) and g = 0 satisfy (4.11). (4.43). We mollify u and v and introduce, for ε > 0, (4.44). uε = Jε u,. vε = Jε v. with Jε = (1 − ε∆x )−1 .. For all ε > 0, (4.4) and (4.43) imply that uε ∈ L2 ([0, T ], H 2 ),. ∂t uε ∈ L2 ([0, T ], H 1 ),. vε ∈ L2 ([0, T ], H 1 ),. ∂t vε ∈ L1 ([0, T ], L2 ),. (see (4.7)). Moreover, using the spatial Fourier transform, one immediately sees that uε converges to u in H1−θ,λ (T ) and vε converges to v in H−θ,λ (T ). Define ˜ ε, fε = Zvε + b̃0 vε − L̃2 uε + L̃1 uε + du gε = Y uε + c̃0 uε − vε /a0 . L 4.6. – Assumptions (4.4) and (4.6) imply that fε = f1,ε + f2,ε with f1,ε → f1 in L−θ,λ (T ) and f2,ε → f2 in H−θ− 12 log,λ (T ). Moreover, gε → 0 in H1−θ− 21 log,λ (T ). ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE.