High frequency oscillations in bounded elastic media

Order no : 2009EVRY0026 THESIS Université d’Evry Val d’Essonne Speciality: Mathematics by Salma Bougacha 14 January 2010 High frequency oscillations in bounded elastic media (Oscillations haute fréquence en milieux élastiques bornés) Commision members Mr. Mr. Mr. Mrs. Mr. Mr. Mr. Mr. Jean-Luc Radjesvarane François Clotilde Kamel Pierre-Gilles James Eric Akian Alexandre Castella Fermanian Kammerer Hamdache Lemarié-Rieusset Ralston Savin (Co-supervisor) (Supervisor) (Referee) (President) (Referee) Thesis prepared in Département de Mathématiques Université d’Evry Val d’Essonne, 91 025 Evry. In cooperation with O.N.E.R.A, Aeroelasticity and Structural Dynamics Department, 92 322 Châtillon and C.N.E.S., 31 000 Toulouse. Oscillations haute fréquence en milieux élastiques bornés Résumé Cette thèse est consacrée à l’étude haute fréquence de problèmes de Dirichlet et Neumann pour le système de l’élasticité. On y étudie le phénomène de réflexion au bord au moyen de deux techniques : la sommation de faisceaux gaussiens et les mesures de Wigner. Dans les chapitres 1 et 2, on commence par étudier le problème plus simple de l’équation des ondes scalaire à une vitesse. Sous certaines hypothèses sur les conditions initiales, on construit des solutions approchées par superposition de faisceaux gaussiens. La justification de l’asymptotique se fonde sur une estimation de normes de certains opérateurs intégraux à phases complexes. Pour des conditions initiales plus générales, on utilise les mesures de Wigner pour calculer la densité d’énergie microlocale. On calcule explicitement les transformées de Wigner d’intégrales de faisceaux gaussiens. Le comportement de la densité d’énergie microlocale de la solution exacte se déduit de celui établi pour la solution approchée. Dans le chapitre 3, on utilise les résultats établis pour les sommes infinies de faisceaux gaussiens pour construire une solution approchée pour les équations de l’élasticité et calculer sa densité d’énergie microlocale. L’existence de deux vitesses différentes dans le système de l’élasticité introduit de nouvelles difficultés qui sont traitées dans ce chapitre. Mots-clefs : élasticité, équation des ondes, conditions de bord, réflexion, faisceaux gaussiens, mesures de Wigner. Abstract This thesis is devoted to the study of the high frequency Dirichlet and Neumann problems for the elasticity system. We study the reflection phenomenon at the boundary by means of two techniques: Gaussian beams summation and Wigner measures. In chapters 1 and 2, we start by studying the simpler problem of the scalar wave equation with one speed. Under some hypotheses on the initial conditions, we build an approximate solution by a Gaussian beams superposition. Justification of the asymptotics is based on norms estimate of some integral operators with complex phases. For more general initial conditions, we use Wigner measures to compute the microlocal energy density. We compute Wigner transforms of Gaussian beams integrals in an explicit way. The behaviour of the microlocal energy density for the exact solution is deduced from the one for the approximate solution. In chapter 3, we use the established results on infinite sums of Gaussian beams to build an approximate solution for the elasticity equations and to compute its microlocal energy density. We treat new difficulties arising from the existence of two different speeds in the elasticity system. Keywords : elasicity, wave equation, boundary conditions, reflection, Gaussian beams, Wigner measures. Contents Introduction 16 I Gaussian beams summation for the wave equation in a convex domain 25 1 Introduction 26 2 Construction of the asymptotic solutions 29 2.1 Gaussian beams for stricly hyperbolic operators . . . . . . . . . . . . . . 29 2.2 Incident and reflected beams for the wave equation . . . . . . . . . . . . 32 2.3 Gaussian beams summation . . . . . . . . . . . . . . . . . . . . . . . . . 38 3 Justification of the asymptotics 43 3.1 Approximation operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2 Error estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 58 II Wigner measures for the wave equation in a convex domain 60 1 Introduction 61 2 Asymptotic solution 64 2.1 First order Gaussian beams . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.2 Gaussian beams summation . . . . . . . . . . . . . . . . . . . . . . . . . 69 3 Wigner transforms and measures 78 3.1 Wigner transform for Gaussian integrals . . . . . . . . . . . . . . . . . . 79 3.2 Wigner measure for superposed Gaussian beams . . . . . . . . . . . . . . 83 3.3 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 91 A Proof of the relation between incident and reflected beams’ phases 94 B Results related to the FBI and the Wigner transforms 96 III Elasticity system : asymptotic solutions and Wigner measures 99 1 Introduction 100 2 Gaussian beams for the elasticity equations 101 2.1 Longitudinal beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 2.2 Transversal beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 2.3 Reflection of a beam L . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 2.4 Reflection of a beam T . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3 Construction of the approximate solution 115 3.1 Gaussian beams summation . . . . . . . . . . . . . . . . . . . . . . . . . 115 3.2 Justification of the asymptotics . . . . . . . . . . . . . . . . . . . . . . . 117 4 Wigner transforms and measures 119 4.1 First order beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.2 Wigner measures for the asymptotic solution . . . . . . . . . . . . . . . . 120 References 128 6 Introduction (french) 7 Position du problème De nombreux phénomènes physiques sont modélisés par des équations d’onde. Dans un milieu élastique, les équations du mouvement linéarisées dans le cas de petites perturbations sont : ρ∂t2 u = divσ(u), (1) où u est le déplacement autour d’une configuration d’équilibre statique considérée comme état de référence, ρ la densité, σ(u) le tenseur des contraintes et divσ(u) le vecteur de composantes (divσ(u))j = 3 q k=1 ∂xk σjk (u) pour 1 ≤ j ≤ 3. La surface de la terre en sismologie, les organes humains en imagerie médicale, ainsi que de nombreuses structures en mécanique industrielle peuvent être considérés comme des milieux élastiques. Si les propriétés du milieu sont les mêmes dans toutes les directions, le milieu est dit isotrope, et le tenseur des contraintes est donné par 1 2 σ(u) = λdivuId + µ ∂x u + ∂x uT , où (∂x u)jk = ∂xk uj et λ(x), µ(x) sont les coefficients de Lamé qui vérifient µ(x) > 0 et λ(x) + 2µ(x) > 0. Les équations (1) s’écrivent alors 1 2 ρ∂t2 u − ∂x (λdivu) − div µ∂x u + µ∂x uT = 0. (2a) On supposera µ Ó= λ + 2µ et la densité et les coefficients du milieu réguliers. Dans le cas d’un milieu homogène, la densité et les coefficients de Lamé ne dépendent pas de la position x. La solution u de (2a) se décompose alors en une somme de deux termes uL et uT de rotationnel et de divergence nuls respectivement. Chacun de ces termes est solution d’une équation d’onde à une vitesse u = uL + uT , ∂t2 uL − c2L △uL = 0, ∂t2 uT − c2T △uT = 0, avec λ + 2µ µ et c2T = . ρ ρ Dans cette thèse, on s’intéresse aux problèmes haute fréquence. Ce type de problème apparaît dans plusieurs applications. L’étude des vibrations de structures industrielles, quand la fréquence d’excitation est importante (chocs) est un exemple d’oscillation à haute fréquence en milieu élastique. On peut également rencontrer ce type de problème dans la propagation d’ondes sismiques quand la longueur d’onde est petite. c2L = On complète ainsi l’équation (2a) par des conditions initiales (uIε , vεI ) u|t=0 = uIε , ∂t u|t=0 = vεI , (2b) qui dépendent d’un paramètre haute fréquence ε ≪ 1. La forme exacte des données initiales n’a pas d’importance dans cette étude. Un exemple typique serait uIε = εaeiφ0 /ε et vεI = beiψ0 /ε . La solution du système de l’élasticité avec ces conditions initiales hautement oscillantes dépend désormais de ε (et sera désignée par uε ). On s’intéresse au comportement de cette solution quand ε est très petit. Pour les structures industrielles (aéronautiques, automobiles, ferroviaires, génie civil, etc) comme en sismologie, les corps élastiques considérés occupent un domaine Ω avec 8 bord et les relations (2a)-(2b) doivent être complétées par des conditions aux limites sur ∂Ω. La condition d’encastrement du bord ou condition de Dirichlet s’écrit uε |∂Ω = 0, (2c) alors qu’un bord libre se traduit par une condition de type Neumann σ(uε )ν|∂Ω = 0, (2c’) où ν est la normale extérieure au bord. On peut bien entendu imposer d’autres conditions aux limites (mixtes, dérivées obliques, etc). Résoudre une équation aux dérivées partielles hyperbolique ou un système d’équations avec des conditions initiales et des conditions aux limites données s’appelle problème mixte hyperbolique. La structure de la solution exacte d’un problème mixte pour les équations d’onde dépend de la géométrie du bord. En effet l’énergie se propage à l’intérieur du domaine le long des bicaractéristiques qui sont des courbes du fibré cotangent T ∗ (R × Ω). Les projections de ces courbes sur R × Ω sont les rayons optiques. Pour l’opérateur d’onde à une vitesse constante c, ces rayons sont des courbes (t, xt ) de Rn+1 qui se déplacent de manière rectiligne à la vitesse c à l’intérieur de Ω. Lors d’un contact transverse avec le bord, les rayons optiques se réfléchissent selon les lois de l’optique géométrique. S’ils rencontrent le bord tangentiellement, ils peuvent donner naissance à des rayons diffractifs qui frôlent le bord sans être déviés. Il peuvent aussi donner lieu à des rayons glissants qui restent dans le bord ∂Ω, et qui sont les limites de rayons se rapprochant du bord et se réfléchissant un grand nombre de fois. L’opérateur de l’élasticité lui possède deux familles de rayons associées à chacune des deux vitesses cL et cT . Au contact avec le bord, les rayons associés à la vitesse cL peuvent donner naissance à des rayons associés à la vitesse cT et inversement. On peut étudier les problèmes haute fréquence en construisant des développements asymptotiques de la solution, valides quand ε est très petit. On approche alors la solution au sens d’une norme bien choisie avec une précision qui augmente avec la fréquence. Il existe également d’autres approches qui s’intéressent uniquement à la limite quand ε → 0 de certaines quantités associées à la solution comme la densité d’énergie locale |uε |2 , il s’agit des approches type mesures de Wigner. Dans cette thèse, nous nous intéressons au comportement des solutions haute fréquence du problème mixte pour l’équation des ondes scalaire ∂t2 uε − ∂x · (c2 (x)∂x uε ) = 0, uε |t=0 = uIε , ∂t uε |t=0 = vεI , uε |∂Ω = 0 ou ∂ν uε |∂Ω = 0, (3) et le système de l’élasticité (2). Le bord est supposé régulier et seul le phénomène de réflexion est étudié. La démarche adoptée est la suivante. On commence par étudier le problème plus simple de l’équation des ondes scalaire à une vitesse, puis les techniques utilisées sont adaptées à l’élasticité. Sous certaines hypothèses sur les conditions initiales, on approche la solution à O(εN ) près pour tout N ∈ N en construisant une famille de solutions asymptotiques. La construction est fondée sur une méthode performante dont l’utilisation est bien maîtrisée dans le cas du phénomène de réflexion au bord : la sommation de faisceaux gaussiens. Pour des conditions initiales plus générales, on utilise les mesures de Wigner pour calculer la densité d’énergie par vecteur d’onde. Le comportement de cette quantité se déduit de celui établi pour la solution approchée par des calculs explicites sur les transformées de Wigner. 9 Sommation de faisceaux gaussiens Il existe plusieurs modèles mathématiques de solutions approchées des équations d’onde quand la fréquence tend vers l’infini. Les solutions exactes pour ces équations sont connues pour certaines configurations appelées problèmes canoniques. Dans un milieu homogène à une vitesse, les solutions canoniques sont les ondes planes. Dans un milieu où la longueur d’onde est petite par rapport à ses hétérogénéités et aux distances de propagation, cette forme des solutions exactes est valide à haute fréquence localement. On peut alors intuiter les formes des solutions. C’est ce qu’on appelle un ansatz. L’étape suivante est alors de trouver les conditions nécessaires pour que l’ansatz trouvé vérifie effectivement l’équation d’onde considérée. Un ansatz très simple est de la forme uε ≃ a0 (t, x)eiψ(t,x)/ε , (4) où a0 est une amplitude scalaire ou vectorielle selon le problème considéré, et ψ une phase scalaire. On appelle cette méthode la méthode de l’optique géométrique, ou encore la méthode WKB ou WKBJ [53], du nom des scientifiques Wentzel, Kramers, Brillouin et Jeffreys qui l’ont indépendemment utilisée dans les années 1920. Pour décrire la réflexion en présence d’un bord, des termes similaires avec des amplitudes et phases réfléchies sont rajoutés dans l’ansatz précédent. Dans le cas de l’équation des ondes à une vitesse c(x), on obtient en appliquant l’opérateur ∂t2 − ∂x · (c2 ∂x ) à cet ansatz les termes suivants organisés selon les puissances de ε è é è é ε−2 c2 |∂x ψ|2 − (∂t ψ)2 a0 + iε−1 2∂t ψ∂t a0 − 2c2 ∂x ψ∂x a0 + (∂t2 ψ − ∂x · (c2 ∂x ψ))a0 + ... En annulant le premier terme, on obtient une équation eikonale sur la phase ψ c2 |∂x ψ|2 − (∂t ψ)2 = 0. Pour les équations de l’élasticité, ce type de calculs mène à la même équation eikonale avec l’une des deux vitesses cL ou cT , couplée avec une information sur la direction de l’amplitude vectorielle a0 . Dans le cas d’une phase ψ réelle, cette équation de type Hamilton-Jacobi possède deux solutions locales qui vérifient c|∂x ψ| ± ∂t ψ = 0, pour une même phase initiale donnée ψ0 . La méthode traditionnelle pour calculer ces solutions est la méthode des caractéristiques. Il s’agit, pour trouver par exemple la solution de c|∂x ψ| + ∂t ψ = 0, de résoudre le système Hamiltonien associé au symbole h+ (x, ξ) = c(x)|ξ| dxt ξ t dξ t = ∂ξ h+ (xt , ξ t ) = c(xt ) t , = −∂x h+ (xt , ξ t ) = −∂x c(xt )|ξ t | dt |ξ | dt avec comme direction initiale ξ 0 = ∂x ψ0 (x0 ), puis d’intégrer l’équation dψ dt = ∂t ψ + ∂x ψ · t dx avec la condition initiale ψ(0, x) = ψ (x), le long des courbes xt . Cependant la phase 0 dt ψ trouvée n’est en général pas globale en temps. En effet l’application x0 Ô→ xt n’est pas toujours bijective, et plusieurs rayons différents peuvent se croiser, formant ainsi ce qu’on appelle une caustique. Il en résulte des fonctions WKB qui ne sont pas valables 10 aux caustiques. D’autre part, la formation de caustiques est une situation récurrente même dans les modèles et structures les plus simples [18]. Si la phase ψ a une partie imaginaire non nulle, la méthode des caractéristiques n’est plus applicable. Cependant les propriétés de Im ψ contrôlent l’enveloppe de la solution asymptotique. En effet, si ψ est réelle sur un rayon (t, xt ) et que la partie imaginaire de sa matrice Hessienne sur ce rayon ∂x2 ψ(t, xt ) est définie positive, alors à tout instant t la principale partie de la densité d’énergie de a0 eiψ/ε est concentrée au voisinage du point x = xt pour ε petit. Il n’est plus nécessaire de vérifier l’équation eikonale de façon exacte mais seulement d’annuler la série de Taylor de c2 |∂x ψ|2 − (∂t ψ)2 jusqu’à un certain ordre R ≥ 2 sur les rayons. On est alors ramené à la résolution de systèmes différentiels qui ont des solutions globales. On obtient ainsi ce qu’on appelle des faisceaux gaussiens, ce nom provenant du fait que leur densité d’énergie à un instant donné est une fonction gaussienne. Ces solutions approchées, qui font partie de l’optique géométrique complexe (voir [57] pour une comparaison entre les différentes méthodes d’optique géométrique complexe), apparaissent aussi sous le nom de "quasiphotons" car à chaque instant t ils sont concentrés au voisinage d’un point qui se déplace selon une certaine géodésique avec une vitesse unitaire et possède plusieurs propriétés des particules (loi de conservation d’énergie, réflexion au bord, etc). Certains auteurs distinguent ces faisceaux gaussiens dépendant du temps et de la variable de l’espace de ceux qui ne dépendent que de la variable de l’espace en les appelant faisceaux gaussiens en temps et en espace, faisceaux gaussiens non-stationnaires ou encore paquets gaussiens et paquets d’onde gaussiens. Historiquement, les faisceaux gaussiens apparaissent dans les travaux de V.M. Babich dans les années 1960 [5] et sont généralisés dans les années 1980 par J. Ralston [85], V.M. Babich et V.V. Ulin [8]. Ces solutions approchées ont été largement utilisées en élasticité [6, 19, 54, 79], et pour les résonateurs optiques [7]. Les faisceaux gaussiens peuvent être adaptés naturellement à d’autres équations, comme les équations de Helmholtz et de Schrödinger. Tout comme les différentes méthodes d’optique géométrique complexe, ils constituent une alternative à l’optique géométrique traditionnelle pour décrire les solutions au delà des caustiques, et ce de manière globale en temps. Ils peuvent aussi être vus comme une base de solutions élémentaires pour la propagation d’ondes et permettre ainsi d’étudier les solutions générales d’équations aux dérivées partielles [81, 85]. La précision de ces solutions peut être améliorée en rajoutant à l’amplitude a0 des termes supplémentaires de puissances de ε supérieures εa1 + ε2 a2 + . . . et en augmentant l’ordre R jusqu’auquel l’équation eikonale est vérifiée sur le rayon. Pour décrire un champ qui n’a pas de profil gaussien, on utilise la méthode de sommation de faisceaux gaussiens [19, 52, 55, 83]. Le champ initial est décomposé en une somme de gaussiennes. Chaque faisceau gaussien individuel est calculé en résolvant les systèmes différentiels associés. Le champ est alors obtenu en un point d’observation en superposant une sélection de faisceaux gaussiens. Les stratégies de sommation sont nombreuses. La somme peut être discrète ou continue, la sélection des faisceaux gaussiens à superposer peut se faire selon plusieurs critères. On peut citer quelques orientations récentes : • la sélection des rayons de direction initiale ∂x ψ0 pour décrire une donnée initiale WKB avec une phase ψ0 [64, 97]. 11 • l’utilisation de la transformée de Fourier [42, 98] • l’utilisation de la transformée FBI (de Fourier-Bros-Iagolnitzer) [89] définie de L2 (Rn ) dans L2 (R2n ) par Tε (f )(x, ξ) = cn ε − 3n 4 Ú Rn f (z)eiξ·(x−z)/ε−(x−z) 2 /(2ε) n 3n dz, cn = 2− 2 π − 4 pour f ∈ L2 (Rn ). Les deuxième et troisième méthodes permettent de se ramener à des données de la forme amplitude multipliée par l’exponentielle d’une phase. Quelle que soit la méthode utilisée, il est important d’évaluer ses performances en estimant l’erreur entre le champ théorique et le champ obtenu par la sommation de faisceaux gaussiens. L’erreur de discrétisation d’une intégrale de faisceaux gaussiens pour l’élasticité a été analysée dans [56]. Récemment, la précision d’une superposition continue de faisceaux gaussiens pour approcher la solution exacte d’une équation d’onde acoustique a été étudiée dans [97, 64]. L’erreur relative à l’utilisation d’une série de Taylor pour les phases et les amplitudes des faisceaux gaussiens a été quantifiée par [77] pour l’équation de Helmholtz. Des études similaires ont été réalisées pour l’équation de Schrödinger dans [59, 65]. Dans le chapitre 1 on utilise la transformée FBI pour contruire une famille de solutions du problème mixte (3) comme une intégrale de faisceaux gaussiens. On prouve l’estimation d’erreur suivante : Théorème 1. [théorème 1.1 du chapitre 1] Supposons vérifiées les hypothèses nécessaires sur le domaine (B1-B3 p.33), notamment la transversalité au bord de tous les rayons provenant de Ω. Supposons que les conditions initiales vérifient les hypothèses suivantes A1. uIε et vεI sont uniformément bornées dans H 1 (Ω) et L2 (Ω) respectivement, A2. uIε et vεI sont nulles en dehors d’un compact fixe de Ω, A3. Tε uIε (x, ξ) et Tε vεI (x, ξ) sont négligeables pour les ξ grands et les ξ proches de zéro (voir p.28). Alors on peut construire pour R ∈ N, R ≥ 2, une solution approchée uR ε du problème de Dirichlet ou de Neumann pour l’équation des ondes scalaire comme une intégrale de faisceaux gaussiens. Cette solution vérifie pour tout T > 0 Sup ëuR ε (t, .) − uε (t, .)ëH 1 (Ω) = O(ε R−1 2 ), t∈[0,T ] et Sup ë∂t uR ε (t, .) − ∂t uε (t, .)ëL2 (Ω) = O(ε R−1 2 ). t∈[0,T ] La démarche est la suivante. On commence par décomposer2les conditions initiales 1 2 . A un coeffien un point z sur la famille des fonctions eiξ·(x−z)/ε−(x−z) /(2ε) (x,ξ)∈R2n cient de normalisation près, ceci est le noyau de l’adjoint de la transformée FBI qui est une isométrie. Les conditions initiales s’écrivent alors comme une intégrale de faisceaux gaussiens pondérés par leurs transformées FBI (à un coefficient près). On construit les faisceaux gaussiens individuels en suivant le formalisme de [85]. La superposition de faisceaux dont les phases vérifient l’équation eikonale à l’ordre R donne 12 une solution approchée uR ε . On estime alors les erreurs dans l’équation à l’intérieur 2 2 R I (∂t − ∂x · (c (x)∂x )) uε , la condition au bord et les conditions initiales uR ε |t=0 − uε et I R ∂t uε |t=0 − vε . Tous ces termes sont les résultats d’une famille d’opérateurs intégraux à phase complexe appliqués aux transformées FBI des données initiales. Les normes de ces opérateurs de L2 (R2n ) dans H s (Rn ) sont calculées en utilisant la régularité des phases et amplitudes des faisceaux gaussiens ainsi que les propriétés des phases. Une fois toutes les erreurs estimées, la différence entre la solution approchée uR ε et la solution exacte est contrôlée par l’estimation d’énergie du problème mixte. On obtient immédiatement R−1 l’ordre ε 2 pour la condition de bord de type Dirichlet. Pour prouver le même ordre pour le problème de Neumann, on a recours à la solution approchée uR+1 qu’on compare ε . à la solution exacte et à uR ε Ces idées s’adaptent naturellement au problème de l’élasticité, en généralisant la notion de transformée FBI aux fonctions vectorielles. On a alors l’estimation suivante en élasticité tridimentionnelle : Théorème 2. [théorème 1.1 du chapitre 3] Supposons vérifiées les hypothèses nécessaires sur les conditions initiales et le domaine (voir p.100-101), excluant notamment les rayons provenant de Ω qui touchent le bord tangentiellement ou à une incidence supérieure ou égale à l’angle critique. On peut construire pour R ∈ N, R ≥ 2, une solution approchée uR ε du problème mixte pour l’élasticité comme une intégrale de faisceaux gaussiens. Cette solution vérifie pour tout T > 0 Sup ëuR ε (t, .) − uε (t, .)ëH 1 (Ω)3 = O(ε R−1 2 ), t∈[0,T ] et Sup ë∂t uR ε (t, .) − ∂t uε (t, .)ëL2 (Ω)3 = O(ε R−1 2 ). t∈[0,T ] Mesures de Wigner Les mesures de Wigner sont des mesures dans l’espace des phases qui permettent de décrire le comportement asymptotique de quantités quadratiques telles que la densité d’énergie locale. La fonction de Wigner a été utilisée en 1932 par E. Wigner [101] en mécanique quantique. Depuis, elle a été appliquée dans divers autres domaines comme l’optique et l’analyse du signal. Dans les années 90, plusieurs mathématiciens s’intéressent aux mesures de Wigner, tels P.-L. Lions, T. Paul [63] et P. Gérard [36] (voir aussi les articles [10, 29, 39] et l’exposé [12]). Les mesures de Wigner sont à rapprocher des H-mesures et mesures de défaut microlocales, introduites par L. Tartar [99] et P. Gérard [37] (voir aussi [34]). A l’O.N.E.R.A.1 des travaux récents ont recours aux mesures de Wigner pour déduire le comportement de l’énergie vibratoire à haute fréquence dans un milieu élastique [94] ou visco-élastique [2, 3]. Ces travaux rejoignent les "approches ingénieur" [41, 100] qui constituent une alternative aux techniques habituellement utilisées pour étudier les vibrations des structures à haute fréquence : l’analyse statistique énergétique (SEA) [66, 67] et les modèles de diffusion d’énergie vibratoire [78, 91]. La SEA constitue une approche globale dans la mesure où elle ne fournit que des estimations des énergies vibratoires moyennes par sous-systèmes mécaniques. La difficulté 1 Office National d’Etudes et de Recherches Aérospatiales 13 principale de la méthode, encore très heuristique, est la détermination des paramètres physiques qui interviennent : facteurs de perte par couplage entre sous-systèmes, densités modales, puissances injectées. Les modèles de diffusion d’énergie vibratoire peuvent être qualifiés de locaux car ils fournissent des estimations des densités d’énergie et d’intensité vibratoire. Néanmoins ils n’ont jusqu’à présent été mis en oeuvre que pour des structures simples (poutres, plaques) car ils reposent sur des hypothèses fortes difficilement vérifiables - voire fausses - pour des structures plus complexes. Ces modèles conduisent à une équation de diffusion pour la densité d’énergie vibratoire. Or l’utilisation des solutions WKB traditionnelles montre cependant que l’équation vérifiée par la densité d’énergie est une équation de transport. Le recours aux mesures de Wigner constitue une alternative rigoureuse pour parer à ces difficultés. De plus cette méthode fournit la direction de propagation de l’énergie. Une mesure de Wigner w[fε ] associée à la suite (fε ) uniformément bornée dans L2 (Rn )p est une limite faible de la suite des transformées de Wigner associées à fε (quitte à extraire une sous-suite) Ú ε ε e−iv·ξ fε (x + v)fε∗ (x − v)dv. wε [fε ](x, ξ) = (2π)−n 2 2 Rn Moyennant certaines hypothèses, la limite (au sens des mesures) quand ε → 0 de la densité d’énergie pour les solutions d’équations d’onde peut s’exprimer en terme de mesures de Wigner. Ainsi, pour l’équation des ondes scalaire, la densité d’énergie à l’instant t converge vers 1Ú 1Ú w[∂t uε (t, .)](x, dξ) + Trw[c∂x uε (t, .)](x, dξ). 2 Rn 2 Rn En élasticité, elle converge vers 3 Ú ρÚ µØ Trw[∂t uε (t, .)](x, dξ) + Trw[∂xj uε (t, .) + ∂x (uε )j (t, .)](x, dξ) 2 Rn 4 j=1 Rn λÚ w[divuε (t, .)](x, dξ). 2 Rn Ces quantités ont été complètement caractérisées pour les équations d’onde dans tout l’espace [39, 80]. En présence d’un bord, l’étude des mesures de Wigner devient techniquement plus difficile. La notion de mesure de Wigner a été utilisée dans le cas de domaines bornés pour l’analyse des propriétés ergodiques des fonctions propres pour les problème de Dirichlet dans [38, 103], de Neumann et de Robin dans [13]. D’autres études se sont intéressées aux mesures de Wigner dans un domaine borné ou avec une interface, comme dans les articles [11, 75, 92] et les thèses [26, 31]. Tous ces travaux se fondent sur l’utilisation du calcul pseudo-différentiel semi-classique. + Dans le chapitre 2, le comportement de la densité d’énergie microlocale pour la solution du problème (3) est décrit en utilisant une autre approche similaire à [16, 89] fondée sur les faisceaux gaussiens. On prouve le théorème suivant : Théorème 3. [théorème 1.1 du chapitre 2] Supposons vérifiées les hypothèses nécessaires sur le domaine (B1-B3 p.61), notamment la transversalité au bord de tous les rayons provenant de Ω. Supposons que les données initiales satisfont A1, A2 et également les conditions suivantes (après extension par 0 en dehors de Ω) 14 C1. Les mesures de Wigner de vεI et ∂xb uIε (b = 1, . . . , n) sont uniques, C2. vεI et ∂xb uIε (b = 1, . . . , n) sont ε-oscillantes (voir les équations (53),Chapitre 2), C3. Les mesures de Wigner de vεI et ∂xb uIε (b = 1, . . . , n) ne chargent pas l’ensemble Rn × {ξ = 0}. Alors la densité d’énergie par vecteur d’onde 12 w[∂t uε (t, .)]+ 12 Trw[c∂x uε (t, .)] s’écrit dans Ω × (Rn \{0}) comme la somme de deux mesures de Wigner initiales transportées le long du flot bicaractéristique brisé obtenu par réflexions successives des rayons au bord. La démonstration se divise en deux étapes : on prouve d’abord le théorème pour des conditions initiales qui vérifient l’hypothèse A3 puis on l’étend à des conditions initiales plus générales. Sous l’hypothèse A3, les mesures de Wigner associées aux dérivées de la solution exacte uε et aux dérivées d’une solution approchée uR ε sont les mêmes. On commence donc par calculer explicitement les transformées de Wigner associées aux dérivées de uR ε dans le cas le plus simple R = 2. Pour cela on suit la démarche de Robinson [89], qui a calculé des quantités similaires pour l’équation de Schrödinger dans tout l’espace. Il a étudié la transformée de Wigner d’une superposition de faisceaux gaussiens pondérés par une transformée FBI et l’a approchée par une intégrale faisant apparaître une quantité proche du carré du module de la transformée FBI transportée. On calcule la limite de cette intégrale en utilisant le théorème de convergence dominée. On prouve ainsi le théorème 3 pour la solution approchée uR ε et par conséquent pour la solution exacte du problème (3) avec des conditions initiales qui vérifient les hypothèses A1-A3 et C1. On veut ensuite s’affranchir de l’hypothèse A3 qui est une hypothèse nécessaire à la sommation des faisceaux gaussiens et non au calcul des transformées de Wigner, et la remplacer par les hypothèses classiques C2, C3 d’ε-oscillation et de non chargement de l’ensemble Rn × {ξ = 0}. Pour cela, on construit une suite de données initiales qui vérifient A3 et telles que les mesures de Wigner associées approchent celles de uIε et vεI . Pour le système de l’élasticité les calculs sont au départ similaires mais il faut tenir compte des changements de modes à la réflexion : les ondes qui se propagent à la vitesse cL donnent naissance à des ondes se propageant à la vitesse cT et inversement. La décomposition de Helmholtz des conditions initiales uIε = fε + Ψε , vεI = gε + Θε avec rotfε = rotgε = 0 et divΨε = divΘε = 0, permet d’identifier les quantités transportées selon les flots associés à chacune des vitesses : les termes de rotationnel nul se propagent à la vitesse cL et les termes de divergence nulle à la vitesse cT . Cependant des termes supplémentaires apparaissent dans la transformée de Wigner. Il s’agit de termes croisés entre des quantités qui se transportent selon les flots réfléchis associés à des vitesses différentes. On a alors besoin d’une hypothèse supplémentaire pour annuler la contribution de ces termes croisés. On prouve le résultat suivant : Théorème 4. [théorème 4.1 du chapitre 3] Supposons vérifiées les hypothèses nécessaires sur les conditions initiales (voir p.100 et p.126) et le domaine (voir p.101), excluant notamment les rayons provenant de Ω qui touchent le bord tangentiellement ou à une incidence supérieure ou égale à l’angle critique. Supposons également que 15 D1. Les mesures de Wigner associées à fε et gε sont nulles, ou D2. Les mesures de Wigner associées à Ψε et Θε sont nulles. Alors on peut calculer la densité d’énergie par vecteur d’onde 3 µ q 4 j=1 ρ Trw[∂t uε (t, .)](x, ξ) 2 + Trw[∂xj uε (t, .) + ∂x (uε )j (t, .)](x, ξ) + λ2 w[divuε (t, .)](x, ξ) pour le problème mixte (2) en fonction des mesures de Wigner des conditions initiales. Ce manuscrit comprend trois chapitres. Les chapitres 1 et 2 sont sous forme d’articles23 . Tous les deux traitent le problème mixte de l’équation des ondes scalaire. Au chapitre 1, une solution approchée est construite par sommation de faisceaux gaussiens. Au chapitre 2, la densité d’énergie microlocale de la solution exacte est calculée au moyen des mesures de Wigner. Le chapitre 3 utilise les mêmes techniques pour l’élasticité. 2 3 Chapitre 1 : à paraître dans Comm. Math. Sci. Chapitre 2 : Prépublication. 16 Introduction 17 Statement of the problem Many physical phenomena are modelled by wave equations. In an elastic medium, the equations of linearized motion in the case of small disturbances are: ρ∂t2 u = divσ(u), (1) where u is the displacement around a static equilibrium configuration considered as a reference state, ρ is the density, σ(u) is the stress tensor and divσ(u) is the vector of components (divσ(u))j = 3 q k=1 ∂xk σjk (u) for 1 ≤ j ≤ 3. Earth surface in seismology, human organs in medical imagery, as well as many structures in industrial mechanics can be considered as elastic media. If the properties of the medium are the same in all the directions, the medium is called isotropic, and the stress tensor is given by 1 2 σ(u) = λdivuId + µ ∂x u + ∂x uT , where (∂x u)jk = ∂xk uj and λ(x), µ(x) are Lamé coefficients satisfying µ(x) > 0 et λ(x) + 2µ(x) > 0. Equations (1) read 1 2 ρ∂t2 u − ∂x (λdivu) − div µ∂x u + µ∂x uT = 0. (2a) We assume µ Ó= λ + 2µ and the density and the Lamé coefficients are smooth. In a homogeneous medium, the density and the Lamé coefficients do not depend on the position x. The solution u of (2a) can be written in this case as the sum of two terms uL and uT , which are curl-free and divergence-free respectively. Each one of these terms is a solution of a wave equation u = uL + uT , ∂t2 uL − c2L △uL = 0, ∂t2 uT − c2T △uT = 0, with c2L = µ λ + 2µ and c2T = . ρ ρ In this thesis, we are interested in high frequency problems which arise in several applications. The study of industrial structures vibrations, when the frequency of excitation is important (shocks) is an example of high frequency oscillations in elastic media. One can also encounter this kind of problem in seismic waves propagation when the wavelength is small. We thus complete equations (2a) with initial conditions (uIε , vεI ) u|t=0 = uIε , ∂t u|t=0 = vεI , (2b) depending on a high frequency parameter ε ≪ 1. The exact form of the initial data is not important here. A typical example is uIε = εaeiφ0 /ε and vεI = beiψ0 /ε , then the solution of the elasticity system with these highly oscillating initial conditions depends on ε (and will be denoted by uε ). We are interested in the behavior of this solution when ε is very small. For industrial structures (aerospace, automotive, railways, civil engineering, etc.) as well as in seismology, the elastic bodies considered occupy a domain Ω with a boundary 18 and the relations (2a)-(2b) must be supplemented by boundary conditions on ∂Ω. The clamped boundary condition or Dirichlet condition reads uε |∂Ω = 0, (2c) whereas a free boundary results in a condition of the Neumann type σ(uε )ν|∂Ω = 0, (2c’) where ν is the normal exterior to the boundary. One can of course impose other boundary conditions (mixed, oblique derivatives, etc). The problem of solving a hyperbolic partial differential equation or system of equations with given initial conditions and boundary conditions is called a hyperbolic mixed problem. The structure of the exact solution of a mixed problem for the wave equation depends on the geometry of the domain. Indeed, in the interior of the domain, the energy is propagated along bicharacteristics which are curves of the cotangent bundle T ∗ (R × Ω). Projections of these curves on R × Ω are the optical rays. For the wave operator with a constant speed c, these rays are curves (t, xt ) of Rn+1 moving in a rectilinear way at the speed c inside Ω. When striking the boundary transversally, the optical rays are reflected according to geometrical optics laws. If they meet the boundary tangentially, they may give rise to diffractive rays which hit the boundary without being deviated. They can also give rise to gliding rays which remain on the boundary ∂Ω, and are limits of rays approaching the boundary and reflected a large number of times. As regards the operator of elasticity, it has two families of rays associated with each one of the speeds cL and cT . When striking the boundary, the rays associated to the speed cL can give rise to rays associated to the speed cT and conversely. One can study high frequency problems by building asymptotic developments of the solution, valid when ε is very small. The solution is thus approximated for a suitable norm with an accuracy increasing with the frequency. There exist also other approaches which are focused only on the limit when ε → 0 of some quantities associated with the solution such as the local energy density |uε |2 , for example the Wigner measures method. In this thesis, we are interested in the high frequency solutions of mixed problems for the scalar wave equation ∂t2 uε − ∂x · (c2 (x)∂x uε ) = 0, uε |t=0 = uIε , ∂t uε |t=0 = vεI , uε |∂Ω = 0 or ∂ν uε |∂Ω = 0, (3) and the system of elasticity (2). The boundary is assumed to be smooth and only the reflection phenomenon is studied. The adopted strategy is the following. We start by studying the simpler problem of the scalar wave equation, then the techniques used are adapted to the elasticity system. Under some hypotheses on the initial conditions, we approach the solution close to O(εN ) for all N ∈ N by building a family of asymptotic solutions. The construction is based on a powerful method well controlled for the phenomenon of reflection at the boundary: the Gaussian beams summation method. For more general initial conditions, we use Wigner measures to compute the microlocal energy density. This quantity is characterized by analyzing the microlocal energy density of the approximate solution by means of explicit computations on the Wigner transforms. 19 Gaussian beams summation There exist several mathematical models of approximate solutions for wave equations when the frequency grows to infinity. The exact solutions for these equations are known for certain configurations called canonical problems. In a homogeneous medium with one wave speed, the canonical solutions are plane waves. In a medium where the wavelength is small compared to its heterogeneities and to the propagation distances, this form of exact solutions is valid for high frequencies locally. One can then guess the shapes of the solutions. This is what is called an ansatz. The following step is then to find some conditions for the ansatz in order to satisfy effectively the considered wave equations. A very simple ansatz is uε ≃ a0 (t, x)eiψ(t,x)/ε , (4) where a0 is a scalar or vector amplitude according to the considered problem, and ψ a scalar phase. This method is called the geometrical optics method, or the WKB or WKBJ method [53], for the scientists Wentzel, Kramers, Brillouin and Jeffreys who independently introduced it in the 1920’s. To describe reflection in the case of a domain with boundary, similar terms with reflected amplitudes and phases are added in the previous ansatz. For the wave equation with speed c(x), one obtains by applying the operator ∂t2 − ∂x · (c2 ∂x ) to this ansatz the following terms organized according to the powers of ε è é è é ε−2 c2 |∂x ψ|2 − (∂t ψ)2 a0 + iε−1 2∂t ψ∂t a0 − 2c2 ∂x ψ∂x a0 + (∂t2 ψ − ∂x · (c2 ∂x ψ))a0 + ... Making the first term vanish, one gets an eikonal equation for the phase ψ c2 |∂x ψ|2 − (∂t ψ)2 = 0. For the elasticity system, similar computations lead to the same eikonal equation with one of the two speeds cL or cT , coupled with an information on the direction of the vectorial amplitude a0 . In the case of a real phase ψ, this Hamilton-Jacobi type equation has two local solutions which satisfy c|∂x ψ| ± ∂t ψ = 0, for one given initial phase ψ0 . The traditional method to compute these solutions is the method of characteristics. In order to find for example the solution of c|∂x ψ| + ∂t ψ = 0, it consists in solving the Hamiltonian system associated with the symbol h+ (x, ξ) = c(x)|ξ| dxt ξ t dξ t = ∂ξ h+ (xt , ξ t ) = c(xt ) t , = −∂x h+ (xt , ξ t ) = −∂x c(xt )|ξ t | dt |ξ | dt with initial direction ξ 0 = ∂x ψ0 (x0 ), and then integrating the equation dψ dt = ∂t ψ + t t ∂x ψ · dx dt with the initial condition ψ(0, x) = ψ0 (x), along the curves x . However the phase ψ is generally not global in time. Indeed the map x0 Ô→ xt is not always one-toone, and several different rays can cross, yielding what is called a caustic. It results in WKB solutions which are not valid at the caustics. On the other hand, the formation of caustics is a recurring situation even in the simplest models and structures [18]. If the phase ψ has a non zero imaginary part, the method of characteristics is no more applicable. However the properties of Im ψ control the envelope of the asymptotic 20 solution. Indeed, if ψ is real on a ray (t, xt ) and if the imaginary part of its Hessian matrix on this ray ∂x2 ψ(t, xt ) is positive definite, then at any instant t the principal part of the energy density of a0 eiψ/ε is concentrated in the vicinity of the point x = xt for small ε. It is not necessary any more to satisfy the eikonal equation exactly but only to make the Taylor series of c2 |∂x ψ|2 − (∂t ψ)2 vanish up to a certain order R ≥ 2 on the ray. One has then to solve differential systems which have global solutions. This is what is called Gaussian beams, which owe their name to the fact that their energy density at a given instant is a Gaussian function. These approximate solutions, which belong to complex geometrical optics (see [57] for a comparison between the various methods of complex geometrical optics), also appear under the name of "quasiphotons" because at every instant t they are concentrated in the vicinity of a point which moves along some geodesic line with a unit speed and has several properties of the particles (energy conservation law, reflection at the boundary, etc.). Some authors distinguish these Gaussian beams depending on the time and the space variable from those which depend only on the space variable by calling them space-time Gaussian beams, nonstationary Gaussian beams, or Gaussian packets and Gaussian wave packets. Historically, Gaussian beams appear in the work of V.M. Babich in the 1960’s [5] and are generalized in the 1980’s by J. Ralston [85], V.M. Babich and V.V. Ulin [8]. These approximate solutions were widely used in elasticity [6, 19, 54, 79], and for optical resonators [7]. Gaussian beams can be adapted naturally to other equations, such as the Helmholtz and Schrödinger equations. Just like the various methods of complex geometrical optics, they constitute an alternative to traditional geometrical optics to describe the solutions beyond the caustics, globally in time. They can also be seen as a basis of elementary solutions for wave propagation, thus allowing to study the general solutions of partial differential equations [81, 85]. The accuracy of these solutions can be improved by adding to the amplitude a0 further terms of higher powers of ε of the form εa1 + ε2 a2 + . . . and by increasing the order R up to which the eikonal equation is satisfied on the ray. To describe a field with non Gaussian profile, one uses the Gaussian beams summation method [19, 52, 55, 83]. The initial field is expanded as a sum of Gaussian beams. Each individual Gaussian beam is computed by solving the associated differential systems. The field is then obtained at an observation point by superposing a selection of Gaussian beams. The summation strategies are numerous. The sum can be discrete or continuous, the selection of the Gaussian beams to be superposed can be done according to several criteria. One can quote some recent orientations: • selection of rays of initial direction ∂x ψ0 to describe WKB initial data with a phase ψ0 [64, 97]; • use of the Fourier transform [42, 98]; • use of the FBI (Fourier-Bros-Iagolnitzer) transform [89] defined from L2 (Rn ) to L2 (R2n ) by 3n Tε (f )(x, ξ) = cn ε− 4 Ú Rn f (z)eiξ·(x−z)/ε−(x−z) 2 /(2ε) n 3n dz, cn = 2− 2 π − 4 for f ∈ L2 (Rn ). The second and third methods allow to get data of the form of an amplitude multiplied by the exponential of a phase. 21 Whatever the method is, it is important to evaluate its performances by estimating the error between the theoretical field and the field obtained by the Gaussian beams summation. The discretization error of an integral of Gaussian beams for elasticity was analyzed in [56]. Recently, the accuracy of a continuous superposition of Gaussian beams to approach the exact solution of the acoustic wave equation was studied in [97, 64]. The error related to the use of Taylor series for the phases and the amplitudes of the Gaussian beams was quantified by [77] for the Helmholtz equation. Similar studies were carried out for the Schrödinger equation in [59, 65]. In chapter 1 we use the FBI transform to construct a family of solutions of the mixed problem (3) as an integral of Gaussian beams. The following error estimate is proved: Theorem 1. [theorem 1.1, chapter 1] Suppose fulfilled the required hypotheses on the domain (B1-B3 p. 33), in particular transversality at the boundary of all rays originating from Ω. Assume that the initial conditions satisfy the following assumptions A1. uIε and vεI are uniformly bounded in H 1 (Ω) and L2 (Ω) respectively, A2. uIε and vεI vanish outside a fixed compact of Ω, A3. Tε uIε (x, ξ) and Tε vεI (x, ξ) are negligible for large ξ and ξ close to zero (see p. 28). Then we can construct for R ∈ N, R ≥ 2, an approximate solution uR ε of the Dirichlet or Neumann problem of the scalar wave equation as an integral of Gaussian beams. This solution satisfies for all T > 0 Sup ëuR ε (t, .) − uε (t, .)ëH 1 (Ω) = O(ε R−1 2 ), t∈[0,T ] and Sup ë∂t uR ε (t, .) − ∂t uε (t, .)ëL2 (Ω) = O(ε R−1 2 ). t∈[0,T ] The strategy is the following. 1One starts by decomposing the initial conditions at point 2 iξ·(x−z)/ε−(x−z)2 /(2ε) . Up to a scaling coefficient, z over the family of functions e (x,ξ)∈R2n this is the kernel of the adjoint of the FBI transform, which is an isometry. The initial conditions are then written as an integral of Gaussian beams weighted by their FBI transforms (modulo a scaling coefficient). One builds the individual Gaussian beams following the formalism of [85]. Superposition of beams of which phases satisfy the eikonal equation up to order R gives an approximate solution uR ε . One then estimates 2 R 2 the errors in the interior equation (∂t − ∂x · (c (x)∂x )) uε , the boundary condition and I R I the initial conditions uR ε |t=0 − uε and ∂t uε |t=0 − vε . All these terms are the results of a family of integral operators with a complex phase applied to the initial data’ FBI transforms. The norms of these operators from L2 (R2n ) to H s (Rn ) are computed by using the smoothness of the Gaussian beams phases and amplitudes and the properties of these phases. Once all the errors are estimated, the difference between the approximate controlled by the mixed problem energy estimate. solution uR ε and the exact solution is R−1 One obtains immediately the order ε 2 for the Dirichlet boundary condition. To prove the same order for the Neumann problem, one resorts to the approximate solution uR+1 ε and compares it to the exact solution and to uR ε. These ideas can be naturally adapted to the elasticity problem, by generalizing the concept of FBI transform to vector functions. We obtain the following estimate for tridimentional elasticity: 22 Theorem 2. [theorem 1.1, chapter 3] Suppose fulfilled the required assumptions on the initial conditions and the boundary (see p.100-101), in particular excluding rays originating from Ω which hit the boundary tangentially or with an incidence equal or larger than the critical angle. One can build for R ∈ N, R ≥ 2, an approximate solution uR ε of the mixed problem for elasticity as an integral of Gaussian beams. This solution satisfies for all T > 0 Sup ëuR ε (t, .) − uε (t, .)ëH 1 (Ω)3 = O(ε R−1 2 ), t∈[0,T ] and Sup ë∂t uR ε (t, .) − ∂t uε (t, .)ëL2 (Ω)3 = O(ε R−1 2 ). t∈[0,T ] Wigner measures Wigner measures are phase space measures which allow to describe the asymptotic behavior of quadratic quantities such as the local energy density. The Wigner function was introduced in 1932 by E. Wigner [101] in quantum mechanics. Since then, it has been applied in various other fields like optics and signal analysis. In the nineties, many mathematicians became interested in Wigner measures, such as P.- L. Lions, T. Paul [63] and P. Gérard [36] (see also the papers [10, 29, 39] and the talk [12]). Wigner measures are related to H-measures and microlocal defect measures, introduced by L. Tartar [99] and P. Gérard [37] (see also [34]). At Onera4 , recent works resort to Wigner measures to deduce the behavior of high frequency vibratory energy in an elastic [94] or viscoelastic [2, 3] medium. These works agree with the engineering approaches [41, 100] which constitute an alternative for the techniques usually used to study the high frequency vibrations of structures: the statistical energy analysis (SEA) [66, 67] and the power flow analysis [78, 91]. The SEA is a global approach insofar as it provides only estimates of average vibratory energies for mechanical subsystems. The main difficulty of the method, which is still very heuristic, is the derivation of the involved physical parameters: subsystems coupling loss factors, modal densities, injected powers. The power flow analysis is a local approach because it provides estimates of the energy densities and the vibratory intensity. Nevertheless, it relies on strong assumptions that can not easily be checked, or are even false for complex structures. That is why it is used only for simple structures (beams, plates). This method leads to a diffusion equation for the vibratory energy density. However the use of traditional WKB solutions shows that the equation satisfied by the energy density is a transport equation. The use of Wigner measures is a rigorous alternative to tackle these difficulties. Moreover this method provides the energy propagation directions and paths. A Wigner measure w[fε ] for a sequence (fε ) uniformly bounded in L2 (Rn )p is a weak limit of the sequence of the Wigner transforms associated with fε (upon extracting a subsequence) wε [fε ](x, ξ) = (2π)−n Ú ε ε e−iv·ξ fε (x + v)fε∗ (x − v)dv. 2 2 Rn Under some assumptions, the limit (in the sense of measures) when ε → 0 of the energy density for wave equation solutions can be expressed in term of Wigner measures. For 4 The French Aerospace Lab 23 the scalar wave equation, the energy density at the instant t converges to 1Ú 1Ú w[∂t uε (t, .)](x, dξ) + Trw[c∂x uε (t, .)](x, dξ). 2 Rn 2 Rn For the elasticity system, it converges to 3 Ú µØ ρÚ Trw[∂t uε (t, .)](x, dξ) + Trw[∂xj uε (t, .) + ∂x (uε )j (t, .)](x, dξ) 2 Rn 4 j=1 Rn λÚ w[divuε (t, .)](x, dξ). + 2 Rn These quantities were fully characterized for wave equations in the whole space domain [39, 80]. In the presence of a boundary, the study of Wigner measures becomes technically more difficult. The concept of Wigner measures was used in bounded domains for the analysis of the ergodic properties of the eigenfunctions for the Dirichlet problem in [38, 103], or the Neumann and Robin problems in [13]. Other studies have been focused on the Wigner measures in a bounded domain or with an interface, such as the papers [11, 75, 92] and the theses [26, 31]. All these works are based on the use of semi-classical pseudo-differential calculus. In chapter 2, the microlocal energy density for the solution of the problem (3) is described by using another approach similar to [16, 89] based on Gaussian beams. The following theorem is proved: Theorem 3. [theorem 1.1, chapter 2] Suppose fulfilled the required hypotheses on the domain (B1-B3 p.61), in particular transversality at the boundary of all rays originating from Ω. Assume that the initial conditions satisfy A1, A2 together with the following assumptions (after extension by zero outside Ω) C1. The Wigner measures of vεI and ∂xb uIε (b = 1, . . . , n) are unique, C2. vεI and ∂xb uIε (b = 1, . . . , n) are ε-oscillatory (see equations (53),chapter 2), C3. The Wigner measures of vεI and ∂xb uIε (b = 1, . . . , n) do not load the set Rn × {ξ = 0}. Then the microlocal energy density 12 w[∂t uε (t, .)] + 12 Trw[c∂x uε (t, .)] is equal in Ω × (Rn \{0}) to the sum of two initial Wigner measures transported along the broken bicharacteristic flow obtained by successively reflecting the rays at the boundary. The proof is divided into two steps: the theorem is firstly proved for initial conditions which satisfy the assumption A3 and then extended to more general initial conditions. Under the assumption A3, Wigner measures associated with the derivatives of the exact solution uε and with the derivative of an approximate solution uR ε are the same. One thus starts by computing explicitly the Wigner transforms associated with the derivatives of uR ε in the simple case R = 2. To do so one follows the ideas of Robinson [89], who computed similar quantities for the Schrödinger equation in the whole space domain. He analyzed the Wigner transform of a superposition of Gaussian beams weighted by a FBI transform and approached it by an integral involving a quantity close to the square modulus of the transported FBI transform. We compute the limit of this integral by 24 using the dominated convergence theorem. Theorem 3 is thus proved for the approximate solution uR ε and consequently for the exact solution of the problem (3) with initial conditions satisfying the assumptions A1-A3 and C1. One wants then to remove the assumption A3 which is needed for the Gaussian beams summation but not for the computation of the Wigner transforms, and to replace it by the traditional assumptions C2, C3 of ε-oscillation and unloading of the set Rn × {ξ = 0}. In order to do that, we build a sequence of initial data which fulfill A3 and such that their associated Wigner measures approach those of uIε and vεI . For the system of elasticity computations are similar at the beginning but one has to take into account the phenomenon of mode conversion at the reflections: the waves propagating at the speed cL give rise to waves propagating at the speed cT and conversely. Helmholtz decomposition of the initial conditions uIε = fε + Ψε , vεI = gε + Θε with rotfε = rotgε = 0 and divΨε = divΘε = 0, allows to identify the quantities transported along the flows associated with each speed: the curl-free terms propagate at the speed cL and the divergence-free terms at the speed cT . However additional cross terms between quantities transported along different flows appear in the Wigner transform. One then needs a further assumption to cancel the contribution of these cross terms. The following result is proved: Theorem 4. [theorem 4.1, chapter 3] Suppose fulfilled the required assumptions on the initial conditions (see p.100 and p.126) and the boundary (see p.101), in particular excluding rays originating from Ω which hit the boundary tangentially or with an incidence equal or larger than the critical angle. Suppose furthermore that D1. The Wigner measures associated with fε and gε are zero, or D2. The Wigner measures associated with Ψε and Θε are zero. Then one can compute the microlocal energy density 3 ρ λ µØ Trw[∂xj uε (t, .) + ∂x (uε )j (t, .)](x, ξ) + w[divuε (t, .)](x, ξ) Trw[∂t uε (t, .)](x, ξ) + 2 4 j=1 2 for the mixed problem (2) by using the Wigner measures of the initial conditions. This thesis contains three chapters. Chapters 1 and 2 are included in paper forms56 . Both of them deal with the mixed problem of the scalar wave equation. In chapter 1, an approximate solution is constructed by Gaussian beams summation. In chapter 2, the microlocal energy density of the exact solution is computed by means of Wigner measures. Chapter 3 uses the same techniques for elasticity. 5 6 Chapter 1: to appear in Comm. Math. Sci. Chapter 2: Preprint 25 Chapter I Gaussian beams summation for the wave equation in a convex domain Contents 1 Introduction 26 2 Construction of the asymptotic solutions 29 2.1 Gaussian beams for stricly hyperbolic operators . . . . . . . . . . . . 29 2.2 Incident and reflected beams for the wave equation . . . . . . . . . . 32 2.2.1 Construction of beams associated to p+ . . . . . . . . . . . . 32 2.2.2 Construction of beams associated to p− . . . . . . . . . . . . 37 2.2.3 Error estimates for individual Gaussian beams . . . . . . . . 38 Gaussian beams summation . . . . . . . . . . . . . . . . . . . . . . . 38 2.3 3 Justification of the asymptotics 43 3.1 Approximation operators . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2 Error estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 P uR ε 3.3 3.2.1 The interior estimate of . . . . . . . . . . . . . . . . . . 48 3.2.2 The boundary estimate of BuR ε . . . . . . . . . . . . . . . . . 50 3.2.3 The initial conditions . . . . . . . . . . . . . . . . . . . . . . 56 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . 58 Introduction 26 1 Introduction In this paper, our aim is to provide asymptotic solutions, in a sense to be precised later, to the following initial-boundary value problem (IBVP) for the wave equation                P uε = ∂t2 uε − ∂x · (c2 (x)∂x uε ) = 0 in [0, T ] × Ω, uε |t=0 = uIε , ∂t uε |t=0 = vεI in Ω, (1) Buε = 0 in [0, T ] × ∂Ω, where B is a Dirichlet or Neumann type boundary operator. Above, T > 0 is fixed, and Ω is a bounded domain of Rn , with n = 2 or n = 3 for important applications to acoustics or elastodynamics problems. We assume the boundary ∂Ω is C ∞ and the domain convex for the bicharacteristic curves of P , see more precisely assumption B1 p.33 below. Furthermore, the coefficient c is assumed to be in C ∞ (Ω̄), though this assumption may be substantially relaxed. Our initial data will depend on a small parameter ε > 0, playing the role of a small wavelength, and our main objective is to study the high frequency limit, corresponding to ε → 0, that is the construction of high frequency solutions. Moreover, we shall assume that uIε , vεI are A1. uniformly bounded respectively in H 1 (Ω) and L2 (Ω), A2. uniformly supported in a fixed compact set K ⊂ Ω. The search for such approximate solutions and related notions of parametrices for the wave equation and similar equations has been an intensive field of activities. A widely used technique to produce such high frequency solutions is given by geometric optics, also called WKB method [73]. This technique is well known in the Physics literature [53]. Then, and in the full space case, approximate solutions are constructed under the form N Ø εj aj eiψ/ε , (2) j=0 with a real phase function ψ and complex amplitudes functions aj . The presence of a boundary may lead to further terms with reflected phases and amplitudes. Typically, initial data should have the same form as in (2), but solutions for more general initial conditions can be obtained by summing an infinite number of WKB solutions. Mathematically, this technique relies on the well known theory of Fourier Integral Operators (FIOs), see for instance [44], see also the earlier works of Maslov and Fedoruk [73] and the recent lecture notes by Rauch and Markus [88]. In general, the global construction of a FIO breaks down at some time, due to generic existence of caustics, see [25]. The caustics problem is also linked to the local solvability of the eikonal equation for the phase, which is derived by substituting the WKB ansatz in the partial differential equation. Indeed, the eikonal equation is solved using the method of characteristics and 27 the phase therefore cannot be defined near every point of the domain, at the exception of some very particular cases. To overcome this difficulty, one either uses a collection of local FIOs or, more generally, constructs a global FIO. This is the way chosen by Chazarain to produce a parametrix for the mixed problem of the wave equation in [21]. Though this method is quite satisfying for the mathematical analysis of propagation of singularities, it does not give approximate solutions directly. A computationally oriented alternative to this mathematical elaborate method is the use of Gaussian beams summation. Gaussian beams are high frequency asymptotic solutions to linear partial differential equations that are concentrated on a single ray. In the mathematical literature, their first use dates back to the 1960s, see [5]. Since then, they have been useful in a variety of problems in mathematical physics such as modelling seismic [43] or electromagnetic [28] wave fields. They also have been used in pure mathematics, such as propagation of singularities [45, 85] and semiclassical measures [81], see [47] and [39] for other methods concerning these problems. One advantage of this method over the WKB precedure is that an individual Gaussian beam has no singularities at caustics. Note that Gaussian beams summation is naturally linked to FIOs with complex phases [44] (see [15, 58, 59, 96] for recent contributions). In a bounded domain of general geometry, both of the WKB and the Gaussian beams ansatzs are inadequate to produce asymptotic solutions. Other models are needed to describe the diffraction phenomena or the gliding of rays along the boundary, such as the Fourier-Airy Integral Operators [74] or the gliding beams [87]. However, in our precise setting of a convex domain with compactly supported initial data, only the reflection effects at the boundary must be considered. Dirichlet or Neumann boundary conditions can be taken into account by combining a finite sum of successively reflected Gaussian beams [51, 68]. Using an infinite sum of Gaussian beams, one can then match quite general initial conditions. This summation can be achieved in different ways, see [19, 52, 55] and the recent [43, 50, 61, 64, 65, 77, 97]. In [64] and [97], superpositions of Gaussian beams are used to solve wave equations with initial data of WKB form. In fact, see Theorem 1.1 below, more general initial conditions are allowed through the use of their FBI transforms, which is also naturally linked with the concept of a Gaussian beam. The FBI or Fourier-Bros-Iagolnitzer transform (see [24, 72, 95]) is, for a given scale ε, the operator Tε : L2 (Rn ) → L2 (R2n ) defined by Tε (a)(y, η) = cn ε − 3n 4 Ú Rn a(w)eiη.(y−w)/ε−(y−w) 2 /(2ε) n 3n dw, cn = 2− 2 π − 4 , a ∈ L2 (Rn ). (3) Its adjoint is the operator 3n Tε∗ (f )(x) = cn ε− 4 Ú R2n f (y, η)eiη.(x−y)/ε−(x−y) 2 /(2ε) dydη, f ∈ L2 (R2n ). (4) As the Fourier Transform, the FBI transform is an isometry, satisfying Tε∗ Tε = Id. Its main property is to decompose an L2x function over the family of functions 2 (eiη.(x−y)/ε−(x−y) /(2ε) )(y,η)∈R2n . For instance, FBI transformation was the method used in [89] to construct an approximate solution for the Schrödinger equation with WKB Introduction 28 initial conditions. The FBI transform is of course again connected with FIOs with complex phases and an interesting result on their global L2 boundedness has been proven recently in [96], regarding the Hermann Kluck propagator. In this paper, our approach to find asymptotic solutions to the problem (1) is to achieve a superposition of incident and reflected Gaussian beams weighted by the FBI transforms of the initial data, satisfying both the condition at the boundary and the initial conditions. Our main result is given by Theorem 1.1. Under assumptions A1 and A2, suppose the FBI transforms of the initial data is infinitely small on the complement of some ring Rη = {η ∈ Rn , r0 ≤ |η| ≤ r∞ }, 0 < r0 < r∞ , in the sense that A3. ëTε uIε ëL2 (Rn ×Rηc ) = O(εs ) and ëTε vεI ëL2 (Rn ×Rηc ) = O(εs ), ∀s ≥ 0. Then for any integer R ≥ 2, there is an asymptotic solution to (1) of the form uR ε (t, x) = qs k R2n akε (t, x, y, η, R)eiψk (t,x,y,η,R)/ε dydη, where akε eiψk /ε are Gaussian beams and the summation over k is finite. uR ε is asymptotic to the exact solution of the IBVP (1) in the following sense Sup ëuR ε (t, .) − uε (t, .)ëH 1 (Ω) = O(ε R−1 2 ), t∈[0,T ] and Sup ë∂t uR ε (t, .) − ∂t uε (t, .)ëL2 (Ω) = O(ε R−1 2 ). t∈[0,T ] Let us note that construction of asymptotic solutions such as a summation of Gaussian beams is certainly not new, but rigorous justification is the main point of our work, together with precise estimates. This paper is organized as follows. In section 2 we recall the construction of Gaussian beams for a strictly hyperbolic differential operator as achieved in [85]. Then, we study the case of the wave equation and construct the incident and reflected beams, and in a final step, we construct approximate solutions for (1) by a Gaussian beams summation. Justification of the asymptotics is given in section 3. Therein, we introduce approximation operators acting from L2 (R2n ) to L2 (Rn ) with a complex phase and compute their norms. We apply these operators on FBI transforms of initial data, and estimate the error of the constructed asymptotic solutions near the boundary, thus taking into account the precise boundary condition, and in the interior set. These estimates are combined with the errors in the initial conditions and yield the justification of the asymptotics by means of energy type estimates. We close this introduction by a short discussion on the notations. Throughout this paper, we will use standard multiindex notations. The inner product of two vectors a, b ∈ Rd will be denoted by a · b. The transpose of a matrix A will be noted AT . If E is a subset of Rd , we denote 1E its characteristic function. For a smooth function 2.1 - Gaussian beams for stricly hyperbolic operators 29 f ∈ C ∞ (Rdx , C), we will use the notation ∂x f to denote its gradient vector (∂xb f )1≤b≤d , ∂x2 f to denote its Hessian matrix (∂xb ∂xc f )1≤b,c≤d and ∂xr f , r > 2 to denote the family (∂xb1 . . . ∂xbr f )1≤b1 ,...,br ≤d . For a vector function F ∈ C ∞ (Rd , Cp ), we denote its Jacobian matrix by DF with (DF )j,k = ∂k Fj and its second derivatives by D2 F with (D2 F )j,k,l = ∂j ∂k Fl . For yε , zε ∈ R+ , we use the notation yε . zε if there exists a constant c > 0 independent of ε such that yε ≤ czε . We write yε . ε∞ or yε = O(ε∞ ) if ∀s ≥ 0 there exists cs > 0 s.t. yε ≤ cs εs for ε small enough . Finally, the word cons denotes a positive constant (different each time it appears). 2 Construction of the asymptotic solutions In this section we first introduce the notion of Gaussian beams for strictly hyperbolic differential operators, following the presentation of [85]. Then the construction of incident and reflected Gaussian beams in the particular case of the wave equation is explained. Finally, the approximate solution for the IBVP (1) is given in the last section as an infinite sum of Gaussian beams. 2.1 Gaussian beams for stricly hyperbolic operators This section follows basically the presentation of [85]. Let P (t, x, ∂t , ∂x ) be a strictly hyperbolic differential operator of order mP and of principal symbol p. That is, we suppose that the roots τ of p(t, x, τ, ξ) = 0 are simple and real for all (t, x) and ξ Ó= 0. The symbol p is assumed real. A Gaussian beam for P is a function of the form wε (t, x) = N Ø j=0 εj aj (t, x)eiψ(t,x)/ε , N ∈ N, (5) satisfying ∃m > 0 s.t. ëP wε ëL2t,x = O(εm ). Note that the above expansion is similar to the usual WKB expansion, but it is required here that: (i) the beam wε is concentrated on some fixed ray (t(s), x(s)) associated to p. Here s is the "time" parameter of this curve. (ii) the phase ψ is a complex-valued function, but real-valued on the ray (t(s), x(s)). The exact definition of a ray (t(s), x(s)) is as follows. First of all, we introduce the so-called null bicharacteristics, which are the curves, solutions of the Hamiltonian equations ṫ(s) = ∂τ p(t(s), x(s), τ (s), ξ(s)), ẋ(s) = ∂ξ p(t(s), x(s), τ (s), ξ(s)), τ̇ (s) = −∂t p(t(s), x(s), τ (s), ξ(s)), ˙ = −∂x p(t(s), x(s), τ (s), ξ(s)), ξ(s) (6) with initial conditions satisfying p(t(0), x(0), τ (0), ξ(0)) = 0. Note that it follows that p(t(s), x(s), τ (s), ξ(s)) = 0, for all s. Then by definition, the projection on Rn+1 t,x of such Construction of the asymptotic solutions 30 a curve (t(s), x(s), τ (s), ξ(s)), that is (t(s), x(s)), is called a ray. We suppose fulfilled the conditions for local existence, uniqueness and smoothness with respect to initial conditions of solutions to the Hamiltonian system (6), see [40]. The construction of a Gaussian beam wε is achieved by making P wε vanish to a certain order on a fixed and given ray (t(s), x(s)). For this purpose, applying P to the form (5) of a Gaussian beam, we obtain a similar form P wε = NØ +mP εj−mP cj eiψ/ε , (7) j=0 where c0 = p(t, x, ∂t ψ, ∂x ψ)a0 , cj = Laj−1 + p(t, x, ∂t ψ, ∂x ψ)aj + gj , j ≥ 1. (8) Above, aj = 0 for j > N , g1 = 0 and gj is a function of ψ, a0 , . . . , aj−2 for j ≥ 2. Furthermore, L is a linear differential operator with coefficients depending on ψ. Using p′ , the symbol of the terms of order mP − 1 of P , L can be written in an explicit way as 1 1 2 2 p(t, x, ∂t ψ, ∂x ψ)∂t,x ψ) + p′ (t, x, ∂t ψ, ∂x ψ). (9) L = ∂τ,ξ p(t, x, ∂t ψ, ∂x ψ) · ∂t,x + T r(∂τ,ξ i 2i For the construction of a Gaussian beam adapted to P , the first step, and by far the most important one, is to build a phase ψ satisfying the eikonal equation p(t, x, ∂t ψ(t, x), ∂x ψ(t, x)) = 0 on (t, x) = (t(s), x(s)) up to order R only, (10) with R ≥ 2, which means α ∂t,x [p(t, x, ∂t ψ(t, x), ∂x ψ(t, x))]|(t(s),x(s)) = 0 for |α| ≤ R. Compare this with the usual eikonal equation p(t, x, ∂t ψ(t, x), ∂x ψ(t, x)) = 0 required by the WKB method in full space. Order 0 of eikonal (10) p (t(s), x(s), ∂t ψ(t(s), x(s)), ∂x ψ(t(s), x(s))) = 0, is fulfilled by setting 1 2 1 2 ∂t ψ, ∂x ψ |(t(s),x(s)) = τ (s), ξ(s) . This constraint insures that d ψ(t(s), x(s)) ds (P.a) is real, which leads by choosing ψ(t(0), x(0)) a real quantity, to the required property ψ(t(s), x(s)) is real. (P.b) Replacing ∂τ,ξ p|(t(s),x(s),τ (s),ξ(s)) by (ṫ(s), ẋ(s)) yields in the differentiation of (10) to the compatibility condition 2 ∂t,x ψ|(t(s),x(s)) A ṫ(s) ẋ(s) B =− A ∂t p ∂x p B |(t(s),x(s),τ (s),ξ(s)) = A τ̇ (s) ˙ ξ(s) B . (11) 2.1 - Gaussian beams for stricly hyperbolic operators 31 It also gives for every function f ∈ C ∞ (Rt × Rnx , C) d f |(t(s),x(s)) . ds α ψ, |α| = 2, we may write order 2 of eikonal (10) as Using this relation on ∂t,x ∂τ,ξ p|(t(s),x(s),τ (s),ξ(s)) · ∂t,x f |(t(s),x(s)) = (12) d 2 2 2 ∂t,x ψ|(t(s),x(s)) + H12 (s)T ∂t,x ψ|(t(s),x(s)) + ∂t,x ψ|(t(s),x(s)) H12 (s) ds 2 2 + ∂t,x ψ|(t(s),x(s)) H22 (s)∂t,x ψ|(t(s),x(s)) + H11 (s) = 0, 2 where H11 (s) = ∂t,x p|(t(s),x(s),τ (s),ξ(s)) , (H12 )bc (s) = (∂τ,ξ )b (∂t,x )c p|(t(s),x(s),τ (s),ξ(s)) and 2 p|(t(s),x(s),τ (s),ξ(s)) . One can substitute for ∂t ∂x ψ|(t(s),x(s)) and ∂t2 ψ|(t(s),x(s)) H22 (s) = ∂τ,ξ from the compatibility condition (11), since ṫ(s) Ó= 0 by the strict hyperbolicity of P . The previous Riccati equation yields then a similar Riccati equation on ∂x2 ψ|(t(s),x(s)) . Although non-linear, this equation has a unique global symmetric solution which satisfies the fundamental property Im ∂x2 ψ|(t(s),x(s)) is positive definite, (P.c) given an initial symmetric matrix ∂x2 ψ|(t(0),x(0)) with a positive definite imaginary part (see the proof of Lemma 2.56 p.101 in [51]). Higher order derivatives of the phase on the ray are determined recursively. For 3 ≤ r ≤ R, order r of the eikonal equation (10) combined with the relation (12) leads to linear inhomogeneous ordinary differential equations (ODEs) on ∂xr ψ|(t(s),x(s)) . They have a unique solution for a fixed initial condition ∂xr ψ|(t(0),x(0)) . The second step of the construction is to make cj , for 1 ≤ j ≤ N + 1, vanish on the ray up to the order R − 2j. The choice of the order R − 2j is related to the quadratic imaginary part in the phase and the study of estimates in Sobolev spaces. This will appear clearly in the justification of the approximation in Lemma 2.2. In any case, the equations on the amplitudes cj = 0 can be solved on the ray at most up to the order 2 R − 2, due to the term ∂t,x ψ in the operator L (9). Taking into account the eikonal equation (10), one gets the following evolution equations on aj , 0 ≤ j ≤ N è1 1 2 2 ψ) p(t,x, ∂t ψ, ∂x ψ)∂t,x ∂τ,ξ p(t, x, ∂t ψ, ∂x ψ) · ∂t,x aj + T r(∂τ,ξ i 2i é + p′ (t, x, ∂t ψ, ∂x ψ) aj + gj+1 = 0 on (t, x) = (t(s), x(s)) up to order R − 2j − 2. (13) This equation uniquely determines the Taylor series of aj on (t(s), x(s)) up to the order R − 2j − 2, given the values of their spatial derivatives at (t(0), x(0)) up to the same order. Remark 2.1. The number N of amplitudes in the ansatz (5) and the order R up to which the eikonal equation (10) is solved are not independent. Indeed, the computations of the amplitudes derivatives require R − 2N − 2 ≥ 0. Another condition ([85] p.219) is assumed to insure that the remainder terms cj , N +2 ≤ j ≤ N +mP , contribute with the right power of ε (see [98] for an alternative justification) R − 2N − 3 ≤ 0. (14) Construction of the asymptotic solutions 32 An essential point for the use of Gaussian beams is the smoothness of the phase and the amplitudes with respect to (w.r.t.) (t(0), x(0)). To this aim, the needed initial values of the derivatives of the phase ∂xr ψ|(t(0),x(0)) , 2 ≤ r ≤ R, and of the amplitudes ∂xr aj |(t(0),x(0)) , 0 ≤ r ≤ R − 2j − 2, are chosen smooth w.r.t. (t(0), x(0)). The phase and the amplitudes are then prescribed to be equal to their Taylor developments (truncated up to fixed orders) on the ray. The final step of the construction is to multiply the amplitudes by a cutoff equal to 1 near the ray. 2.2 Incident and reflected beams for the wave equation The preceeding results will now be applied and detailed for the particular case of the wave equation and the construction of reflected beams. The computations rely on the results of [68] and [85]. We extend c in a smooth way outside Ω̄. Let p(x, τ, ξ) = c2 (x)|ξ|2 − τ 2 be the principal symbol of the wave operator P = ∂t2 − ∂x · (c2 ∂x ). Then τ (s) = τ (0) from the Hamiltonian equations (6). Writing p = −p+ p− with p+ (x, τ, ξ) = c(x)|ξ| + τ and p− (x, τ, ξ) = −c(x)|ξ| + τ, shows that null bicharacteristics s Ô→ (t(s), x(s), τ (0), ξ(s)) for p s.t. τ (0) Ó= 0 are either null bicharacteristics for p+ if τ (0) < 0 or for p− if τ (0) > 0, by using the parametrization s′ = −2τ s. Denote h+ (x, ξ) = c(x)|ξ| and let (xt0 (y, η), ξ0t (y, η)) (or simply (xt0 , ξ0t )) be the Hamiltonian flow for h+ starting from the point (y, η), that is dxt0 ξt dξ0t = ∂ξ h+ (xt0 , ξ0t ) = c(xt0 ) 0t , = −∂x h+ (xt0 , ξ0t ) = −∂x c(xt0 )|ξ0t |, dt |ξ0 | dt xt0 |t=0 = y, (15) ξ0t |t=0 = η, η Ó= 0. Then the null bicharacteristic curve (t(s), x(s), τ (s), ξ(s)) for p starting at s = 0 from ±t (0, y, ∓c(y)|η|, η) is exactly (t, x±t 0 (y, η), ∓c(y)|η|, ξ0 (y, η)) the null bicharacteristic curve for p± . As in [93], one can prove that the Hamiltonian system (15) associated to h+ has a unique solution global in time (by Cauchy-Lipschitz theorem), which depends smoothly on (t, y, η) ∈ R × Rn × Rn \{0}. The remainder of this section is organised as follows. In section 2.2.1, one explains the construction of incident and reflected beams associated to p+ , then section 2.2.2 is a simple repetition for p− and finally in section 2.2.3 we give error estimates for the individual beams gathered in (22). 2.2.1 Construction of beams associated to p+ For the ray (t, xt0 (y, η)) associated with p+ , denote by wε0 (t, x, y, η) a Gaussian beam concentrated on that ray, by ψ0 (t, x, y, η) and a0j (t, x, y, η) its associated phase and am- 2.2 - Incident and reflected beams for the wave equation 33 plitudes. If no confusion is possible, symbols y, η and even t, x, y, η in the notations above will be dropped. The phase ψ0 is determined by solving the eikonal equation (10) on the ray (t, xt0 ) together with the conditions ∂t ψ0 (t, xt0 ) = −h+ (xt0 , ξ0t ), ∂x ψ0 (t, xt0 ) = ξ0t , (P0 .a) and the choice of ψ0 (0, y) a real function , ∂x2 ψ0 (0, y) a symmetric matrix with a positive definite imaginary part, ∂xr ψ0 (0, y), 3 ≤ r ≤ R, permutable families. In particular ψ0 satisfies the important properties ψ0 (t, xt0 ) is real, (P0 .b) Im ∂x2 ψ0 (t, xt0 ) is positive definite. (P0 .c) and The phase ψ0 is assumed to be equal to its Taylor series up to the order R on x = xt0 ψ0 (t, x) = Ø 1 (x − xt0 )α ∂xα ψ0 (t, xt0 ). α! |α|≤R (16) The amplitudes of wε0 (t, x) are also determined by the requirement that the cj , 1 ≤ j ≤ N + 1 in (8) are null up to orders R − 2j on the ray (t, xt0 ), given their initial spatial derivatives on the ray ∂xr a0j (0, y), r = 0, . . . , R − 2j − 2. We choose them as Ø 1 (x − xt0 )α ∂xα a0j (t, xt0 ), j = 0, . . . , N, α! |α|≤R−2j−2 a0j (t, x) = χd (x − xt0 ) (17) where d > 0 and χd is a cut-off of C0∞ (Rn , [0, 1]) satisfying χd (x) = 1 if |x| ≤ d/2 and χd (x) = 0 if |x| ≥ d. Throughout the paper, the parameter d will be adjusted to obtain requested estimates. This construction leads to a beam wε0 (t, x, y, η) called an incident beam for p+ , satisfying sup ëP wε0 (t, .)ëL2 (Ω) = O(εm ) for some m > 0. t∈[0,T ] o Let T ∗ Ω = T ∗ Ω\{η = 0}. To study the reflection on the boundary, we make the following assumptions B1. The domain Ω is convex for the bicharacteristic curves of P , that is for every o (y, η) ∈ T ∗ Ω, xt0 (y, η) cuts the boundary at only two times of opposite signs and transversally, Construction of the asymptotic solutions 34 o B2. For every (y, η) ∈ T ∗ Ω, xt0 (y, η) does not remain in a compact of Rn when t varies in R, B3. The boundary has no dead-end trajectories, that is infinite number of successive reflections cannot occur in a finite time. o For (y, η) ∈ T ∗ Ω, let T1 (y, η) be the instant (that is the exit time) s.t. T (y,η) x0 1 (y, η) ∈ ∂Ω and T1 (y, η) > 0. o o Note that T ∗ Ω is an open set, and thanks to B1, the function (y, η) ∈ T ∗ Ω Ô→ T1 (y, η) is well-defined and C ∞ , as follows from the implicit functions theorem. The reflection involution associated to the considered symbol p is the map o o R : T ∗ Rn |∂Ω → T ∗ Rn |∂Ω (X, Ξ) Ô→ (X, (Id − 2ν(X)ν(X)T )Ξ). Above ν denotes the exterior normal field to ∂Ω. Let ϕt0 = (xt0 , ξ0t ) denote the incident Hamiltonian flow solution of (15). We define the first reflected flow ϕt1 by the condition ϕT1 1 = R oϕT0 1 , that is the Hamiltonian flow for h+ having at t = T1 , position xT0 1 , the direction being given by the reflected vector of ξ0T1 . Then the broken flow is defined recursively after a finite number of successive reflections as follows (see fig.1): for k > 1, Tk and ϕtk = (xtk , ξkt ) are determined by: T (y,η) k Tk (y, η) is the instant s.t. xk−1 k ϕTk k = R oϕTk−1 . (y, η) ∈ ∂Ω and Tk (y, η) > Tk−1 (y, η), The convexity of the boundary B1 implies the non-grazing hypothesis o T (y,η) k ∀(y, η) ∈ T ∗ Ω and k ≥ 1, ẋk−1 where ẋtk−1 denotes d t x . dt k−1 T (y,η) k (y, η) · ν(xk−1 (y, η)) > 0, Assumption B3 leads to o ∀(y, η) ∈ T ∗ Ω, Tk (y, η) → +∞. (18) k→+∞ o It insures that for a fixed point (y, η) in T ∗ Ω, there is a finite number q+ (y, η) of reflections in [0, T ]. Following the method of Ralston in [85] p.220, we shall construct reflected beams which satisfy the boundary estimate wε1 , . . . , wεq+ ′ ∃m′ > 0 and s ≥ 0 s.t. ëB(wε0 + · · · + wεq+ )ëH s ([0,T ]×∂Ω) = O(εm ), together with the interior estimates sup ëP wεk (t, .)ëL2 (Ω) = O(εm ), 1 ≤ k ≤ q+ . t∈[0,T ] 2.2 - Incident and reflected beams for the wave equation 35 Figure 1: successive reflections. For each 1 ≤ k ≤ q+ , the reflected beam wεk will be written as wεk = eiψk /ε (ak0 + · · · + εN akN ). To insure the interior estimates, each phase ψk and amplitudes akj (0 ≤ j ≤ N ) must satisfy equations (10) and (13) on the reflected ray (t, xtk ). As the beams vanish away from their associated rays, the contribution to the boundary norm of wε0 + · · · + wεq+ occurs when t is close to some Tk and then from the beams wεk−1 and wεk . The construction of the reflected beams is completed recursively. Assume that the beam wεk−1 has been constructed and that its associated phase satisfies t t ∂t ψk−1 (t, xtk−1 ) = −h+ (xtk−1 , ξk−1 ), ∂x ψk−1 (t, xtk−1 ) = ξk−1 , t ψk−1 (t, xk−1 ) is real, Im ∂x2 ψk−1 (t, xtk−1 ) is positive definite. (Pk−1 .a) (Pk−1 .b) (Pk−1 .c) One may write on the boundary ∂Ω 1 2 1 2 N k−1 iψk−1 /ε B wεk−1 + wεk = ε−mB dk−1 e −mB + · · · + ε dN 1 2 + ε−mB dk−mB + · · · + εN dkN eiψk /ε , mB being the order of B (mB = 0 for Dirichlet and mB = 1 for Neumann). In order to satisfy the boundary estimate, the first step is to impose on ψk to have k the same time and tangential derivatives as ψk−1 at (Tk , xTk−1 ), up to the order R. Tk More precisely, let us introduce boundary coordinates near xk−1 = xTk k as follows. We partition ∂Ω with a finite number of small open subsets U1 , . . . , UL s.t. there exist C ∞ parametrizations σl : Nl → Rn , l = 1, . . . , L, where Nl are open subsets of Rn−1 , σl (Nl ) = Ul and σl a diffeomorphism from Nl to Ul . k k k Suppose that xTk−1 belongs to Ul0 and denote xTk−1 = σl0 (ẑk ). For x ∈ Rn close to xTk−1 , we may write x = σl0 (v̂) + vn ν(σl0 (v̂)), Construction of the asymptotic solutions 36 with v̂ ∈ Nl0 and vn ∈ R. If we use the notation σ f (t, v̂, vn ) = f (t, x), then we impose α σ α σ ∂t,v̂ ψk (Tk , ẑk , 0) = ∂t,v̂ ψk−1 (Tk , ẑk , 0), |α| ≤ R. (19) k Order 0 of (19) gives a real value for ψk (Tk , xTk−1 ). Order 1 of this same constraint and order 0 of the eikonal equation (10) on ψk are both satisfied by setting ∂t ψk (t, xtk ) = −h+ (xtk , ξkt ), ∂x ψk (t, xtk ) = ξkt . (Pk .a) ψk (t, xtk ) is real. (Pk .b) It follows that Due to the non-grazing hypothesis, (19) and the compatibility condition resulting k from order 1 of the eikonal equation (10) provide with ∂x2 ψk (Tk , xTk−1 ). To solve the t 2 Riccati equation on ∂x ψk (t, xk ) with its given value at t = Tk , we need to study the k imaginary part of ∂x2 ψk (Tk , xTk−1 ). For k ′ = k − 1, k, one has ∂t ∂v̂ σψk′ (t, v̂, 0) = Dσl0 (v̂)T ∂t ∂x ψk′ (t, xtk′ ), and 1 2 ∂v̂2 σψk′ (t, v̂, 0) = D2 σl0 (v̂) ∂x ψk′ (t, xtk′ ) + Dσl0 (v̂)T ∂x2 ψk′ (t, xtk′ )Dσl0 (v̂). Differentiating (Pk−1 .a) and (Pk .a) yields Im ∂t ∂x ψk′ (t, xtk′ ) = −Im ∂x2 ψk′ (t, xtk′ ) ẋtk′ and Im ∂t2 ψk′ (t, xtk′ ) = ẋtk′ · Im ∂x2 ψk′ (t, xtk′ ) ẋtk′ . Denote 2 σ 2 σ Mk = ∂t,v̂ ψk−1 (Tk , ẑk , 0) = ∂t,v̂ ψk (Tk , ẑk , 0). (20) One has therefore Im Mk = 1 −ẋTk′k , Dσl0 (ẑk ) 2T k Im ∂x2 ψk′ (Tk , xTk−1 ) 1 1 −ẋTk′k , Dσl0 (ẑk ) 2 2 . −ẋTk′k , Dσl0 (ẑk ) are non sink gular. Since Im ∂x2 ψk−1 (Tk , xTk−1 ) is positive definite by (Pk−1 .c), it follows that the k same property holds true for Im Mk and consequently for Im ∂x2 ψk (Tk , xTk−1 ). Hence, the t 2 matrix ∂x ψk (t, xk ) solution of a Riccati equation with its given value at t = Tk satisfies The non-grazing hypothesis insures that the matrices Im ∂x2 ψk (t, xtk ) is positive definite. (Pk .c) Higher order derivatives of the reflected phase on the associated ray are determined recursively. For 3 ≤ r ≤ R, ∂xr ψk (t, xtk ) satisfies linear ODEs with a given value at t = Tk . 2.2 - Incident and reflected beams for the wave equation 37 k The second step is to prescribe that dk−1 −mB +j +d−mB +j vanish up to the order R−2j−2 k at (Tk , xTk−1 ). These requirements provide with the derivatives of akj up to the order k R − 2j − 2 at (Tk , xTk−1 ). Hence, for 0 ≤ r ≤ R − 2j − 2, ∂xr akj (t, xtk ) satisfy linear systems of ODEs with initial conditions given at t = Tk . It follows from this construction that the choice of the (truncated up to fixed orders) Taylor series of the phase and the amplitudes of the incident beam on the starting point of the ray determines recursively the (truncated up to fixed orders) Taylor series of successively reflected beams’ phases and amplitudes. Finally, the amplitudes akj are multiplied by a cutoff equal to 1 near xtk . The reflected phases and amplitudes have the same forms as the incident ones ψk (t, x) = and 1 (x − xtk )α ∂xα ψk (t, xtk ), α! |α|≤R Ø 1 (x − xtk )α ∂xα akj (t, xtk ), j = 1, . . . , N. α! |α|≤R−2j−2 akj (t, x) = χd (x − xtk ) 2.2.2 Ø Construction of beams associated to p− For the symbol p− , the same construction applies for the associated incident and reflected beams. An incident beam for p− is a beam concentrated on the ray (t, x−t 0 ), so it is simply wε0 (−t, x). In fact, denoting P wε0 = Nq +2 j=0 εj−2 c0j eiψ0 /ε , one can notice that P [wε0 (−t, x)] = [P wε0 ](−t, x), and the amplitudes c0j (−t, x) vanish on x = x−t 0 up to the required orders. o Reflected beams for p− are obtained by reflecting ϕt0 backwards. For (y, η) ∈ T ∗ Ω, T (y,η) let T−1 (y, η) < 0 be the instant s.t. x0 −1 (y, η) strikes the boundary ∂Ω. Denote by ϕt−1 the Hamiltonian flow for h+ determined by the condition (see fig.1) T T ϕ−1−1 = R oϕ0 −1 . For k > 1, one can define recursively the instants of reflections T−k and the Hamiltonians flows ϕt−k for h+ as follows: T (y,η) −k T−k (y, η) is the instant s.t. x−k+1 (y, η) ∈ ∂Ω and T−k (y, η) < T−k+1 (y, η), T T −k −k = R oϕ−k+1 . ϕ−k Assumption B3 implies that Tk (y, η) → −∞ when k goes to −∞, and thus insures a finite number q− (y, η) of reflections in [−T, 0]. Then we build Gaussian beams wε−k for p− after 1 ≤ k ≤ q− backwards reflections, ′ by imposing ëB(wε0 + · · · + wε−q− )ëH s ([−T,0]×∂Ω) = O(εm ) for some m′ > 0 and s ≥ 0. We write these beams as N −k wε−k = eiψ−k /ε (a−k 0 + · · · + ε aN ). Construction of the asymptotic solutions 38 In particular, for 1 ≤ k ≤ q− , the phase ψ−k satisfies the following properties t t ∂t ψ−k (t, xt−k ) = −h+ (xt−k , ξ−k ), ∂x ψ−k (t, xt−k ) = ξ−k , t ψ−k (t, x−k ) is real, Im ∂x2 ψ−k (t, xt−k ) is positive definite. (P−k .a) (P−k .b) (P−k .c) Noting that (t, xt−k ), k = 1, . . . , q− , are successively reflected rays for p− , the reflected beam of p− after k reflections is simply wε−k (−t, x). 2.2.3 Error estimates for individual Gaussian beams o We fix (y, η) ∈ T ∗ Ω and choose d sufficiently small s.t. for k = 0, . . . , q± , t ∈ [0, T ] and |x − x±t ±k | ≤ d, 2 Im ψ±k (±t, x) ≥ cons(x − x±t (21) ±k ) . One can see that this choice is always possible by the properties (Pk .a)-(Pk .b)-(Pk .c) of each phase ψk , −q− ≤ k ≤ q+ . For t ∈ [0, T ] and x ∈ Rn , let w+ ε (t, x) = q+ q k=0 wεk (t, x) and w− ε (t, x) = q− q k=0 wε−k (−t, x). (22) Then we have the following estimates on these constructed beams n n − 4 +1 Lemma 2.2. 1. ëε− 4 +1 w± ∂t w± ε (t, .)ëH 1 (Ω) . 1 and ëε ε (t, .)ëL2 (Ω) . 1 uniformly w.r.t. t ∈ [0, T ], 1 2 n 2. ëP ε− 4 +1 w± ε (t, .)ëL2 (Ω) . ε 1 R−1 2 2 n uniformly w.r.t. t ∈ [0, T ], −mB −s+ 3. ëB ε− 4 +1 w± ε ëH s ([0,T ]×∂Ω) . ε R+1 2 , s ≥ 0. The proof of this Lemma and other results rely on this standard estimate for p ∈ N |x|p e−x 2 /ε p dx . ε 2 e−x 2 /(2ε) , ∀x ∈ Rn . (23) For more details, we refer the interested reader to [85] or [68]. 2.3 Gaussian beams summation n The constructed functions ε− 4 +1 w± ε are approximate solutions for the IBVP of the wave equation with initial data ε and −n +1 4 1 n w± ε |t=0 2 =ε −n +1 4 N Ø j=0 n −4 ∂t ε− 4 +1 w± ε |t=0 = ±ε εj a0j |t=0 eiψ0 |t=0 /ε N +1 Ø j=0 +ε −n +1 4 q± Ø wε±k |t=0 , q± Ø ∂t wε±k |t=0 , k=1 n εj fj0 eiψ0 |t=0 /ε ± ε− 4 +1 k=1 2.3 - Gaussian beams summation 39 where the fj0 are related to the phase and amplitudes of wε0 . One can show that the / Ω̄ for k Ó= 0. The exponential decrease of the phases assumptions B1-B2 imply that x0k ∈ away from their associated rays leads to ëwεk |t=0 ëH 1 (Ω) . ε∞ and ë∂t wεk |t=0 ëL2 (Ω) . ε∞ , k Ó= 0. n Modulo infinitely small remainders, the initial conditions of ε− 4 +1 w± ε are then  ε +1 −n 4 N Ø j=0 εj a0j |t=0 eiψ0 |t=0 /ε , ±ε −n 4 N +1 Ø j=0  εj fj0 eiψ0 |t=0 /ε  . We wish to consider the IBVP (1) with general initial conditions (uIε , vεI ) in 3n H 1 (Ω) × L2 (Ω). Note that ψ0 |t=0 has properties similar to φ0 , where cn ε− 4 eiφ0 (x,y,η)/ε denotes the kernel of Tε∗ , see formula (4) in the introduction. The first step is to build, o for a fixed point (y, η) ∈ T ∗ Ω, asymptotic solutions with initial conditions close to n n (ε− 4 +1 eiφ0 (.,y,η)/ε , 0) and (0, ε− 4 eiφ0 (.,y,η)/ε ) in H 1 (Ω) × L2 (Ω). Then one expects to fulfill n o more general initial data (uIε , vεI ) by decomposing uIε on the family (ε− 4 +1 eiφ0 /ε ) ∗ (y,η)∈T Ω n and vεI on the family (ε− 4 eiφ0 /ε ) o (y,η)∈T ∗ Ω , indexed by (y, η). Let us recover the notation of the beams referring to the starting points of the incio dent flow. We fix (y, η) ∈ T ∗ Ω and consider the incident beam wε0 (t, x, y, η) associated to the ray (t, xt0 (y, η)) and the reflected beams wε±k (t, x, y, η), k = 1, . . . , q± . Taylor formulae (16) yields at t = 0 ψ0 (0, x, y, η) = Ø 1 (x − y)α ∂xα ψ0 (0, y, y, η). α! |α|≤R If one chooses the following initial spatial derivatives on the ray for the incident beam’s phase ψ0 (0, y, y, η) = 0, ∂x2 ψ0 (0, y, y, η) = iId and ∂xα ψ0 (0, y, y, η) = 0, 3 ≤ |α| ≤ R, then (P0 .a) implies ψ0 (0, x, y, η) = η · (x − y) + i(x − y)2 /2 = φ0 (x, y, η). (24) We assume henceforth that the incident beam’s phase satisfies (24). Consider an approximate solution 1 − n +1 + ε 4 (wε + w− ε ). 2 Its initial data are   n ε− 4 +1 N Ø j=0 εj a0j |t=0 eiφ0 /ε , 0 , n with a redidue of order ε∞ in H 1 (Ω) × L2 (Ω). To get the form (ε− 4 +1 eiφ0 /ε , 0), one has to make a suitable choice for the amplitudes. The expansion (17) at t = 0 yields Ø 1 (x − y)α ∂xα a0j (0, y, y, η), j = 0, . . . , N, α! |α|≤R−2j−2 a0j (0, x, y, η) = χd (x − y) Construction of the asymptotic solutions 40 and one has full choice for the initial spatial derivatives of a0j on the ray up to the order R − 2j − 2. Under the assumptions a00 (0, y, y, η) = 1, ∂xα a00 (0, y, y, η) = 0 for 1 ≤ |α| ≤ R − 2, ∂xα a0j (0, y, y, η) = 0 for |α| ≤ R − 2j − 2, 1 ≤ j ≤ N, one gets N Ø j=0 εj a0j (0, x, y, η) = χd (x − y). (25) Taking advantage of the exponential decrease of eiφ0 (x,y,η)/ε for |x−y| ≥ d/2, one deduces that ëε +1 −n 4 N Ø j=0 n εj a0j (0, ., y, η)eiφ0 (.,y,η)/ε − ε− 4 +1 eiφ0 (.,y,η)/ε ëH 1 (Ω) . ε∞ . We keep the notations a0j and wε0 to denote the amplitudes satisfying (25) and the associated incident beam. For 1 ≤ k ≤ q± , we denote by wε±k the corresponding reflected beams and by w± ε the sum of the incident and reflected beams for p± . Next, we shift to the initial condition on the time derivative, for which we construct ′ ′ a new incident beam wε0 with amplitudes a0j . Indeed, an approximate solution 1 − n +1 + ′ ′ ε 4 (wε − w− ε ), 2 has initial data  −n 4 0, ε N +1 Ø εj j=0 1 ′ i∂t ψ0 a0j ′  2 + ∂t a0j−1 |t=0 eiφ0 /ε  , ′ ′ modulo a remainder of order ε∞ in H 1 (Ω) × L2 (Ω), with a0−1 = a0N +1 = 0. In order n to approach the form (0, ε− 4 eiφ0 /ε ), we derive new initial Taylor series for the incident beam’s amplitudes. As ∂t ψ0 (0, y, y, η) = −c(y)|η|, we impose 1 ′ ′ 2 a00 (0, y, y, η) = i (c(y)|η|)−1 , ∂xα ∂t ψ0 a00 (0, y, y, η) = 0 for 1 ≤ |α| ≤ R − 2, 1 2 ′ ′ ∂xα i∂t ψ0 a0j + ∂t a0j−1 (0, y, y, η) = 0 for |α| ≤ R − 2j − 2, 1 ≤ j ≤ N. One gets N +1 Ø 1 ′ ′ 2 εj i∂t ψ0 a0j + ∂t a0j−1 (0, x, y, η) = 1 + j=0 N Ø j=0 +ε N +1 εj Ø |α|=R−2j−1 (x − y)α zα (x, y, η) ′ ∂t a0N (0, x, y, η), (26) where zα are smooth remainders that vanish for |x − y| ≥ d. Making use of (14) and (23), one can show that n ëε− 4 N +1 Ø j=0 ′ 1 ′ ′ 2 n εj i∂t ψ0 a0j + ∂t a0j−1 (0, ., y, η)eiφ0 (.,y,η)/ε − ε− 4 eiφ0 (.,y,η)/ε ëL2 (Ω) . ε ′ R−1 2 . ′ Let wε±k , 1 ≤ k ≤ q± , be the reflected beams associated to wε0 and denote by w± ε the sum of the so obtained incident and reflected beams for p± . Hence, the approximate solutions 1 − n +1 + ′ 1 − n +1 + ′ ε 4 (wε + w− ε 4 (wε − w− ε )(t, x, y, η) and ε )(t, x, y, η), 2 2 2.3 - Gaussian beams summation 41 have the required initial data 1 2 n 1 n 2 ε− 4 +1 eiφ0 (x,y,η)/ε , 0 and 0, ε− 4 eiφ0 (x,y,η)/ε , modulo remainders of respective orders ε∞ and ε R−1 2 in H 1 (Ω) × L2 (Ω). To fulfill general initial conditions (uIε , vεI ), the previous computations together with the identity Tε∗ Tε = Id, suggest that we look for an approximate solution such as 1 2 cn − 3n Ú I + − ε 4 o Tε uε (y, η) w ε (t, x, y, η) + w ε (t, x, y, η) dydη 2 T ∗Ω 1 2 cn 3n Ú I +′ −′ + ε− 4 (t, x, y, η) − w (t, x, y, η) dydη. εT v w o ε ε ε ε 2 T ∗Ω Let us notice that it is not clear that the previous integral is well defined. (′ ) (t, x, y, η) breaks down when y approaches the Firstly, the construction of w± ε boundary ∂Ω because the numbers of reflections in [0, ±T ] become infinitely large. Next we need to tackle the problem of integration for large η. One way to overcome these two problems is to require that the initial FBI transforms are compactly supported modulo small remainders. This requirement is in the spirit of considering only compactly supported symbols in the study of the FIOs of [59]. Nevertheless, this restriction was removed recently by Rousse and Swart in [90]. In particular, a general boundedness result of FIOs with complex phases for subquadratic Hamiltonians was established therein. The proof is rather subtle and relies in particular on Cotlar-Stein estimate. The same arguments can be used for the constant coefficient wave equation but seem not to work for the general wave equation. In fact, in this case, the second derivatives of the Hamiltonian are not bounded and thus the proof of [90] needs to be adapted. A last problem related to the wave equation is the integration for small η. In view of all these difficulties, this explains why we have made in the introduction the assumptions A2 and A3 on the initial data, which we recall uIε and vεI are supported in a fixed compact K ⊂ Ω, ëTε uIε ëL2 (Rn ×Rηc ) = O(ε∞ ) and ëTε vεI ëL2 (Rn ×Rηc ) = O(ε∞ ), where Rη = {η ∈ Rn , r0 ≤ |η| ≤ r∞ }, 0 < r0 < r∞ . These assumptions are satisfied for instance by a large class of WKB functions aeiΦ/ε , a ∈ C0∞ (Ω). Indeed the nonstationary phase lemma implies that the FBI transform of such a function is of order O(ε∞ ) outside the compact set A × B = {y ∈ Rn , dist(y, suppa) ≤ c} × {η ∈ Rn , dist(η, ∂x Φ(A)) ≤ c}, c > 0, see Lemmas 4.2 and 4.3 of [89]. Thus aeiΦ/ε satisfies assumption A3, provided that ∂x Φ does not vanish on suppa. Remark 2.3. Another strategy can be used to match initial conditions of WKB form in a Gaussian beams summation [64, 97]. It consists of integrating the beams associated to rays that start from y ∈ suppa with the direction η = ∂x Φ(y). The accuracy of such obtained solutions faces a damage caused by caustics, namely an extra factor Construction of the asymptotic solutions 42 1−n ε 4 appears in the error estimate. This loss originates from the restriction to rays x±t ±k (y, ∂x Φ(y)) (k = 0, . . . , N± ), which technically leads to considering the deformation matrices ∂y [x±t ±k (y, ∂x Φ(y))] singular at caustics (see Lemma 5.1 of [64]). The summation over rays starting with general directions η independent of y uses the symplectic maps ϕ±t ±k and thus provides a phase space description of the solution that ŤunfoldsŤ the caustics. Let ρ be a cut-off of C0∞ (Rn , [0, 1]) supported in a compact Ky ⊂ Ω and satisfying ρ(y) = 1 if dist(y, K) < ∆ for a small ∆ > 0, (27) and φ a cut-off of C0∞ (Rn , [0, 1]) supported in a compact Kη ⊂ Rn \{0} s.t. φ = 1 on Rη . One can establish that the assumptions A2 and A3 imply ë (1 − ρ(y)φ(η)) Tε uIε ëL2y,η . ε∞ and ë (1 − ρ(y)φ(η)) Tε vεI ëL2y,η . ε∞ . In fact, viewing the FBI transform as the Fourier Transform of some auxiliary function yields by Parseval equality the following result Lemma 2.4. Let a be a positive real and G a measurable subset of Rn s.t. dist(G, K) ≥ a. If u ∈ L2 (Rnw ) is supported in K then n ë1G (y)Tε uëL2y,η = cn ε− 4 ë1G (y)u(w)e−(w−y) 2 /(2ε) ëL2w,y . e−a 2 /(4ε) ëuëL2w . On the other hand, if (y, η) varies in Ky × Kη , then q+ (y, η) is uniformly bounded. In fact, for j ≥ 1, the Tj are positive, depend continuously on (y, η) and the property (18) insures that Tj ր +∞ when j → +∞. Thus they uniformly go to +∞ on the compact Ky × Kη , by Dini’s theorem on the sequence (1/Tj )j≥1 . As Tq+ ≤ T , it follows that sup q+ < +∞. The same result holds true for q− . Denote N± = sup q± . The Ky ×Kη Ky ×Kη construction of the reflected beams in section 2.2 may be continued up to N± reflections. The final result of the discussion above is an approximate solution proposed as uR ε (t, x) 5 N+ N− Ø Ø 1 3n Ú ′ ′ ρ(y)φ(η) εTε vεI (y, η)( wεk (t, x, y, η) − wε−k (−t, x, y, η)) = ε− 4 c n 2 R2n k=0 k=0 + Tε uIε (y, η)( N+ Ø k=0 wεk (t, x, y, η) + N− Ø k=0 6 wε−k (−t, x, y, η)) dydη. (28) This approximate solution is indexed by R, the order of vanishing of the eikonal equation (10) on the ray. The incident beams’ phase fulfills the initial conditions (24) and their ′ amplitudes satisfy respectively (25) for wε0 and (26) for wε0 for every (y, η) ∈ suppρ ⊗ φ,. The size d ∈]0, 1] of the support of the cut-offs multiplying the amplitudes no longer depends on (y, η) and would be chosen sufficiently small to satisfy various constraints we set out in the following section. In the sequel, we prove that this family of functions (uR ε ) indeed allows to approach the exact solution of the IBVP problem (1) to any arbitrary power of ε by choosing the order R. The difference between the asymptotic solutions and the exact one is investigated in C([0, T ], H 1 (Ω)) × C 1 ([0, T ], L2 (Ω)) by means of error estimates in the interior equation, the boundary condition and the initial conditions. The only assumptions needed on the initial data are A1, A2 and A3. 3.1 - Approximation operators 3 43 Justification of the asymptotics R We aim to estimate ëuR ε (t, .) − uε (t, .)ëH 1 (Ω) and ë∂t uε (t, .) − ∂t uε (t, .)ëL2 (Ω) for t ∈ [0, T ]. It follows from standard results [22] that the IBVP for the wave equation is wellposed, and furthermore one has the energy estimate (as a consequence of [60] p.185 for the Dirichlet problem and of [9] p.224 for the Neumann problem) R Sup ëuR ε (t, .) − uε (t, .)ëH 1 (Ω) + Sup ë∂t uε (t, .) − ∂t uε (t, .)ëL2 (Ω) . t∈[0,T ] t∈[0,T ] R Sup ëP uR ε ëL2 (Ω) + ëBuε ëH s ([0,T ]×∂Ω) (29) t∈[0,T ] I R I + ëuR ε (0, .) − uε ëH 1 (Ω) + ë∂t uε (0, .) − vε ëL2 (Ω) , where s = 1 for Dirichlet and s = 1 2 for Neumann. The asymptotics will be proven by estimating each term of the r.h.s. of this energy estimate. Since the error estimates in the interior and near the boundary use similar computations, a unified framework will be used by considering the more general problem of estimates linked with a suitable family of approximation operators Oα in section 3.1. Then in section 3.2 we use these estimates for the interior term ëP uR ε ëL2 (Ω) in 3.2.1, the R boundary term ëBuε ëH s ([0,T ]×∂Ω) in 3.2.2 and the initial conditions errors in 3.2.3. All these estimates are gathered in section 3.3 to prove Theorem 1.1. 3.1 Approximation operators Let Kz,θ be a compact of R2n and Er = {(x, z, θ) ∈ Rn × Kz,θ , |x − z| ≤ r}, r > 0. Consider a complex phase function Φ smooth on an open set containing Er0 for some r0 ∈]0, 1]. We assume, for (z, θ) ∈ Kz,θ ,that ∂x Φ(z, z, θ) = θ, Φ(z, z, θ) is real, ∂x2 Φ(z, z, θ) has a positive definite imaginary part. (Q1) Taylor expansion of Φ together with assumptions (Q1) imply the existence of some constant r[Φ] ∈]0, r0 ] s.t. for (x, z, θ) ∈ Er[φ] Im Φ(x, z, θ) ≥ cons(x − z)2 . Consider a sequence lε ∈ C ∞ (Rnx × R2n z,θ , C). We assume that lε (x, z, θ) = 0 if (x, z, θ) ∈ / Er[Φ] , lε is uniformly bounded in L∞ (R3n ). (Q2) For a given multi-index α, let the operators O0 (lε , Φ/ε) and Oα (lε , Φ/ε) be given by è é O0 (lε , Φ/ε) h (x) = Ú R2n h(z, θ)lε (x, z, θ)eiΦ(x,z,θ)/ε dzdθ, h ∈ L2 (R2n ), Justification of the asymptotics 44 and [Oα (lε , Φ/ε) h] (x) = Ú R2n h(z, θ)lε (x, z, θ)(x − z)α eiΦ(x,z,θ)/ε dzdθ, h ∈ L2 (R2n ), with x ∈ Rn . Let us show that these are operators from L2 (R2n ) to L2 (Rn ). For x ∈ Rn we have Ú |lε eiΦ/ε |dzdθ . and thus Ú Similarly, for (z, θ) ∈ Kz,θ Ú (z,θ)∈Kz,θ e−cons(x−z) 2 /ε dzdθ, n |lε eiΦ/ε |dzdθ . ε 2 . Ú n |lε eiΦ/ε |dx . ε 2 . It is then immediate by Schur’s lemma, that n ëO0 (lε , Φ/ε) ëL2 (R2n )→L2 (Rn ) . ε 2 . Similar arguments lead to the estimate n ëOα (lε , Φ/ε) ëL2 (R2n )→L2 (Rn ) . ε 2 + |α| 2 . However, the use of the module inside the previous integrals makes one lose the highly oscillatory character of eiΦ/ε , that is the contribution of eiθ·(x−z)/ε . In fact, a better estimate on the norms of these operators is available if a precise control on lε is assumed. This is stated in the following lemma k Lemma 3.1. Assume that ε 2 ∂xkb lε (b = 1, . . . , n) is uniformly bounded in L∞ (R3n ), at any order k ∈ N. Then, one has 3n 1. ëO0 (lε , Φ/ε) ëL2 (R2n )→L2 (Rn ) . ε 4 , 3n 2. ëOα (lε , Φ/ε) ëL2 (R2n )→L2 (Rn ) . ε 4 + |α| 2 . Proof. 1. Let h ∈ L2 (R2n ). We shall use the notations f (x) for f (x, z, θ) and f ′ (x) for f (x, z ′ , θ′ ). First of all, we explicit the L2 norm of O0 (lε , Φ/ε) h as ëO 0 (lε , Φ/ε)hë2L2 èÚ Rn = Ú ′ R4n ′ ′ ′ ′ hh eiΦ(z)/ε−iΦ (z )/ε ei(θ .z −θ.z)/ε ′ ′ ′ (30) é ′ lε (x)lε (x)ei(θ−θ ).x/ε eiΘ(x,z,θ,z ,θ )/ε dx dzdz ′ dθdθ′ , where Θ(x, z, θ, z ′ , θ′ ) = q |α|=2 (x − z)α − s1 2 (1 − s)∂ α Φ(z q 0 α! |α|=2 x + s(x − z), z, θ)ds s (x − z ′ )α 1 2 (1 − s)∂ α Φ(z ′ 0 α! x + s(x − z ′ ), z ′ , θ′ )ds. 3.1 - Approximation operators 45 Let Iε denote the integral inside the brackets, that we begin to estimate. For 1 ≤ b ≤ n and K ∈ N, successive integrations by parts give Iε (z, z ′ , θ, θ′ )iK ε−K (θb −θb′ )K = Ø K (−1) N +N ′ =K A A K N BÚ Rn ′ ′ ′ ei(θ−θ ).x/ε ∂xNb [eiΘ/ε ]∂xNb [lε lε ]dx, B K where denotes the standard binomial coefficient. To estimate ∂xNb [eiΘ/ε ], N ∈ N, N we use the following result, of which proof is postponed to the end of this section Lemma 3.2. Let p ∈ N∗ and consider a complex phase function Fp of the form Ø Fp (x, z) = (x − z)α fα (x, z), |α|=p with fα smooth on some open set of R2n containing a subset S and ∂xk fα bounded on S for any k ≥ 0. Then for (x, z) ∈ S, |x − z| ≤ 1, small ε, N ∈ N and b = 1, . . . , n, one has |∂xNb [eiFp /ε ]| ≤ max |α|=p 0≤s≤N 1≤k≤N 1 2k 3 Ø sup|∂xsb fα | S N p We write Θ = F2 − F̄2′ with F2 (x, z, θ) = Ø |α|=2 α (x − z) ≤k≤N Ú 1 0 ε−k |x − z|kp−N + Ø 1≤k< N p 4 ε−N/p |eiFp /ε |. 2 (1 − s)∂xα Φ(z + s(x − z), z, θ)ds, α! for (x, z, θ) ∈ Er[Φ] . By Leibnitz formula, ∂xNb [eiΘ/ε ] is a sum of terms of the form ′ ∂xNb1 [eiF2 /ε ]∂xNb2 [e−iF̄2 /ε ], 0 ≤ N1 , N2 ≤ N, N1 + N2 = N. Note that Im F2 = Im Φ. Lemma 3.2 yields for N1 ∈ N and (x, z, θ) ∈ Er[Φ]   |∂xNb1 [eiF2 /ε ]| .  Ø N1 ≤k≤N1 2  −cons(x−z) ε−k |x − z|2k−N1 + ε−N1 /2  e Hence |∂xNb1 [eiF2 /ε ]| . ε− N1 2 e−cons(x−z) 2 /ε 2 /ε . . ′ A similar estimate may be obtained for |∂xNb2 [eiF̄2 /ε ]| when (x, z ′ , θ′ ) ∈ Er[Φ] . It follows, for (x, z, θ), (x, z ′ , θ′ ) ∈ Er[Φ] , that ′ |∂xNb1 [eiF2 /ε ]∂xNb2 [e−iF̄2 /ε ]| . ε− N1 +N2 2 e−cons(x−z) 2 /ε ′ 2 /ε e−cons(x−z ) and thus N ′ 2 /ε |∂xNb [eiΘ/ε ]| . ε− 2 e−cons(2x−z−z ) ′ 2 /ε e−cons(z−z ) , N ∈ N. , Justification of the asymptotics 46 Since ε N′ 2 ′ ∂xNb [lε ¯lε′ ], N ′ ∈ N, is uniformly bounded ′ ′ |∂xNb [eiΘ/ε ]∂xNb [lε lε ]| . ε− N +N ′ 2 ′ 2 /ε e−cons(2x−z−z ) ′ 2 /ε e−cons(z−z ) , and we deduce that BK A θb − θb′ √ |Iε (z, z , θ, θ ) ε ′ ′ n ′ 2 /ε | . ε 2 e−cons(z−z ) , for b = 1, . . . , n and K ∈ N. Choosing K > n and coming back to (30) gives n ëO0 (lε , Φ/ε) hë2L2 . ε 2 Ú ′ 2 /ε R4n |h||h′ |e−cons(z−z ) K dzdz ′ (1 + (θ − θ′ )2 /ε)− 2 dθdθ′ . Upon using the change of variables: √ √ √ √ (z, z ′ ) = (u + εv, u − εv) and (θ, θ′ ) = (σ + εδ, σ − εδ), we have ëO 0 (lε , Φ/ε) hë2L2 . ε 3n 2 Ú Ú R2n −consv 2 e R2n |h(u + √ εv, σ + √ εδ)||h(u − √ εv, σ − √ εδ)|dudσ K (1 + 4δ 2 )− 2 dvdδ, from which, using Cauchy-Schwartz inequality for the first integral, we get 3n ëO0 (lε , Φ/ε) hë2L2 . ε 2 ëhë2L2 . 2. Arguments are similar to the previous case. For a multi-index α, we have ëOα (lε , Φ/ε) hë2L2 = Ú R4n ′ ′ ′ − θb′ )K ′ Iεα (z, z ′ , θ, θ′ )dzdz ′ dθdθ′ , where, for b = 1, . . . , n and K ∈ N Iεα (z, z ′ , θ, θ′ )iK ε−K (θb ′ hh eiΦ(z)/ε−iΦ (z )/ε ei(θ .z −θ.z)/ε Ø K = (−1) N +N ′ =K A K N BÚ ′ Rn ei(θ−θ ).x/ε ′ ′ ∂xNb [(x − z)α (x − z ′ )α eiΘ/ε ]∂xNb [lε lε ]dx. We note that ∂xNb [(x − z)α (x − z ′ )α eiΘ/ε ] is a finite sum of terms of the form b b (x − z)α−ke (x − z ′ )α−le ∂xmb [eiΘ/ε ], where k, l ≤ αb , k + l + m = N and eb denotes the vector of Rn s.t. eba = δab . For (x, z, θ), (x, z ′ , θ′ ) ∈ Er[Φ] , it follows that N ′ 2 /ε |∂xNb [(x − z)α (x − z ′ )α eiΘ/ε ]| . ε|α|− 2 e−cons(2x−z−z ) Since ε N′ 2 ′ 2 /ε e−cons(z−z ) . ′ ∂xNb [lε lε′ ] is uniformly bounded ′ ′ |∂xNb [(x − z)α (x − z ′ )α eiΘ/ε ]∂xNb [lε lε ]| . ε|α|− N +N ′ 2 ′ 2 /ε e−cons(2x−z−z ) ′ 2 /ε e−cons(z−z ) , 3.1 - Approximation operators 47 and thus |Iεα (z, z ′ , θ, θ′ ) A θb − θb′ √ ε BK n ′ 2 /ε | . ε 2 +|α| e−cons(z−z ) and finally , 3n ëOα (lε , Φ/ε) hë2L2 . ε 2 +|α| ëhë2L2 . Similar computations can be carried out for a phase Φ and a sequence of amplitudes lε that depend on a parameter m ∈ [0, M ]. In this case, we consider for m ∈ [0, M ] a compact Kz,θ (m) ⊂ R2n and denote for r > 0 Er = {(m, x, z, θ) ∈ [0, M ] × R3n , (z, θ) ∈ Kz,θ (m), |x − z| ≤ r}. We are interested in a phase function Φ smooth on an open set containing Er0 for some r0 ∈]0, 1]. We make the further assumption Er0 is compact, which is obviously fulfilled when no parameter m interferes. Assuming, for m ∈ [0, M ] and (z, θ) ∈ Kz,θ (m), that ∂x Φ(m, z, z, θ) = θ, Φ(m, z, z, θ) is real, ∂x2 Φ(m, z, z, θ) has a positive definite imaginary part, (Q1’) one can find r[Φ] ∈]0, r0 ] s.t. for (m, x, z, θ) ∈ Er[φ] Im Φ(m, x, z, θ) ≥ cons(x − z)2 . Similarly, the sequence lε will be assumed to belong to C ∞ ([0, M ] × Rnx × R2n z,θ , C) and to satisfy for m ∈ [0, M ], lε (m, x, z, θ) = 0 if (m, x, z, θ) ∈ / Er[Φ] , (Q2’) ∞ 3n lε is uniformly bounded in L ([0, M ] × R ). One can then define, for every given m ∈ [0, M ] and α multiindex (|α| ≥ 0), the operators Oα (lε (m, .), Φ(m, .)/ε), for which the following estimate may be established k Lemma 3.3. Assume that ε 2 ∂xkb lε (b = 1, . . . , n) is uniformly bounded in L∞ ([0, M ] × R3n ), at any order k ∈ N. Then, one has 3n ëOα (lε (m, .), Φ(m, .)/ε) ëL2 (R2n )→L2 (Rn ) . ε 4 + |α| 2 , uniformly w.r.t. m ∈ [0, M ]. In fact, all the estimates used in the proof of Lemma 3.1 hold true with a parameter m ∈ [0, M ], since Er[φ] is still compact, owing to the compactness of Er0 . We now give the proof of Lemma 3.2. Using the formula of composite functions’ high derivatives (see, e.g., [33] p.161), the N th partial derivative of eiFp /ε is ∂xNb [eiFp /ε ] = N 3 4k Ø i k=1 ε Ù j 1 +···+j k =N j 1 ,...,j k ≥1 N! k!j 1 ! . . . j k ! 1 k ∂xj b Fp . . . ∂xj b Fp eiFp /ε , N ∈ N∗ . Justification of the asymptotics 48 Each derivative ∂xj b Fp is a linear combination of b (x − z)α+(s−j)e ∂xsb fα , |α| = p, 0 ≤ s ≤ j and αb ≥ j − s. 1 k The product ∂xj b Fp . . . ∂xj b Fp is then a linear combination of (x − z)α 1 +(s1 −j 1 )eb +···+αk +(sk −j k )eb k 1 ∂xsb fα1 . . . ∂xsb fαk , where for i = 1, . . . , k, |αi | = p, 0 ≤ si ≤ j i and αbi ≥ j i − si . As j 1 + · · · + j k = N , then for N/p ≤ k ≤ N and |x − z| ≤ 1 one has |(x − z)α 1 +(s1 −j 1 )eb +···+αk +(sk −j k )eb | ≤ |x − z|kp−N . Thus for N ∈ N∗ , (x, z) ∈ S, |x − z| ≤ 1 and small ε |∂xNb [eiFp /ε ]| ≤ max |α|=p 0≤s≤N 1≤k≤N 1 2k sup|∂xsb fα | S 3 Ø N p −k ≤k≤N kp−N ε |x − z| + Ø 1≤k< N p ε −N/p 4 |eiFp /ε |, which of course is also valid for N = 0. 3.2 Error estimates The different terms of the energy estimate (29) will be estimated separately. Our main interest is to prove that the interior and boundary errors given for individual beams in Lemma 2.2 hold true for an infinite sum of beams, when the starting points of the incident flow vary in the compact Ky × Kη . The control we have is that we can make the Gaussian beams vanish outside the very neighbourhood of their associated rays, by making the parameter d as small as needed. 3.2.1 The interior estimate of P uR ε In this section, we will prove that Sup ëP uR ε (t, .)ëL2 (Ω) . ε R−1 2 . t∈[0,T ] For 0 ≤ k ≤ N+ , one has by construction P wεk = N +2 Ø εj−2 ckj eiψk /ε , j=0 where ckj is null on (t, xtk ), up to the order R − 2j, for j = 0, . . . , N + 1. One may write P wεk (t, x) = N +1 Ø j=0 ε j−2 3 Ø |α|=R−2j+1 (x − xtk )α rαk (t, x)eiψk (t,x)/ε 4 + εN ckN +2 (t, x)eiψk (t,x)/ε , 3.2 - Error estimates 49 where rαk denotes the remainder in the Taylor formulae of ckj near xtk . Applying P to (28) gives then terms of the form pj,k ε (t, x) =ε Ø − 3n −1+j 4 |α|=R−2j+1 Ú R2n ρ(y)φ(η)hε (y, η) (x − xtk )α rαk (t, x, y, η)eiψk (t,x,y,η)/ε dydη, with j = 0, . . . , N + 1, and 3n pεN +2,k (t, x) = ε− 4 +N +1 Ú R2n ρ(y)φ(η)hε (y, η)ckN +2 (t, x, y, η)eiψk (t,x,y,η)/ε dydη, where hε is either ε−1 Tε uIε or Tε vεI and 0 ≤ k ≤ N+ . Other terms of the same form come ′ ′ from P wεk , 0 ≤ k ≤ N+ , and P [wε−k( ) (−t, .)], 0 ≤ k ≤ N− . Let f˜(t, x, z, θ) = f (t, x, {ϕtk }−1 (z, θ)). Using the volume preserving change of variables (z, θ) = ϕtk (y, η) in the definition of pj,k ε (t, x), 0 ≤ j ≤ N + 1, writes it as a sum of terms of the form ε − 3n −1+j 4 Ú ρ] ⊗ φ(t, z, θ)h̃ε (t, z, θ)(x − z)α r̃αk (t, x, z, θ)eiψ̃k (t,x,z,θ)/ε dzdθ, R2n with |α| = R − 2j + 1. Similarly, pεN +2,k (t, x) is a sum of terms of the form 3n ε− 4 +N +1 Ú R2n ρ] ⊗ φ(t, z, θ)h̃ε (t, z, θ)c̃kN +2 (t, x, z, θ)eiψ̃k (t,x,z,θ)/ε dzdθ. We want to estimate these integrals with the help of the operators Oα applied to 1supp] h̃ . Clearly 1supp] Tç v I (t, .) is uniformly bounded (w.r.t. ε and t) in ρ⊗φ(t,.) ε ρ⊗φ(t,.) ε ε L2 (R2n ). But more work is needed for estimating ε−1 1supp] Tç uI (t, .), which is ρ⊗φ(t,.) ε ε given in the following result Lemma 3.4. ëε−1 Tε uIε ëL2 (R2n ) . 1. Proof. Differentiating (3) w.r.t. yb , 1 ≤ b ≤ n, yields 1 1 3n ε 2 ∂yb (Tε uIε ) = iηb ε− 2 Tε uIε − cn ε− 4 Ú 1 Rn uIε (w)ε− 2 (yb − wb ) eiη.(y−w)/ε−(y−w) 2 /(2ε) dw. The l.h.s. is bounded in L2y,η because ∂yb (Tε uIε ) = Tε (∂wb uIε ). The second term of the r.h.s. is the Fourier transform of a bounded function in L2w , thus it can be estimated using Parseval equality. One gets ëε − 3n 4 Ú 1 Rn uIε (w)ε− 2 (yb − wb ) eiη.(y−w)/ε−(y−w) 1 2 /(2ε) dwëL2y,η . ëuIε ëL2w . 1 Thus ëε− 2 ηb Tε uIε ëL2y,η . 1 and consequently ëε− 2 φ(η)Tε uIε ëL2y,η . 1. Assumption A3 yields 1 ëε− 2 Tε uIε ëL2y,η . 1. √ Hence ëuIε ëL2 . ε. Reproducing the same arguments on the following equality ∂yb (Tε uIε ) = iηb ε −1 leads to ëuIε ëL2 . ε. Tε uIε − cn ε − 3n 4 Ú Rn 1 1 2 1 i 1 2 ε− 2 uIε (w)ε− 2 (yb − wb ) e ε η.(y−w)− 2ε (y−w) dw, Justification of the asymptotics 50 Let us now check if a family of operators Oα may be used. First, each phase ψ˜k is smooth on an open set containing E1 = {(t, x, z, θ) ∈ [0, T ] × R3n , (z, θ) ∈ ϕtk (Ky × Kη ), |x − z| ≤ 1}. E1 is compact, since the map (t, y, η) Ô→ (t, ϕtk (y, η)) is continuous. For t ∈ [0, T ] and (z, θ) ∈ ϕtk (Ky × Kη ), one has by (Pk .a), (Pk .b) and (Pk .c) ∂x ψ˜k (t, z, z, θ) = ξ˜kt (z, θ) = θ, ψ˜k (t, z, z, θ) is real, ∂x2 ψ˜k (t, z, z, θ) has a positive definite imaginary part. Hence ψ̃k satisfies properties (Q1’). We fix some r[ψ̃k ] ∈]0, 1] so that Im ψ˜k (t, x, z, θ) ≥ cons(x − z)2 for every (t, x, z, θ) ∈ Er[ψ̃k ] . (31) Next, for R − 2N − 1 ≤ |α| ≤ R + 1, let lα,k (t, x, z, θ) = ρ] ⊗ φ(t, z, θ)r̃αk (t, x, z, θ), t ∈ [0, T ], and l0,k (t, x, z, θ) = ρ] ⊗ φ(t, z, θ)c̃kN +2 (t, x, z, θ), t ∈ [0, T ]. Then the lα,k , |α| = R − 2N − 1, . . . , R + 1, and l0,k are smooth w.r.t. all their variables. Assume that d ≤ r[ψ̃k ], k = 0, . . . , N+ . (32) Because of the cut-offs χd in the beams’ amplitudes, it follows that ⊗ φ(t, z, θ) = 0 for c̃kN +2 (t, x, z, θ) = r̃αk (t, x, z, θ) = 0 if |x − z| ≥ r[ψ̃k ]. Furthermore, ρ] α,k t (z, θ) ∈ / ϕk (Ky × Kη ). The l satisfy therefore assumptions (Q2’). It follows that the operators Oα can be used to obtain for t ∈ [0, T ] and x ∈ Rn 3n − 4 −1+j pj,k ε (t, x) = ε Ø |α|=R−2j+1 è 1 2 é Oα lα,k (t, .), ψ˜k (t, .)/ε 1supp] h̃ (t, .) (x), ρ⊗φ(t,.) ε with j = 0, . . . N + 1, and è 1 2 é 3n pεN +2,k (t, x) = ε− 4 +N +1 O0 l0,k (t, .), ψ˜k (t, .)/ε 1supp] h̃ (t, .) (x). ρ⊗φ(t,.) ε Applying Lemma 3.3 and making use of (14) yields ëpj,k ε (t, .)ëL2 (Ω) . ε 3.2.2 R−1 2 , uniformly w.r.t. t ∈ [0, T ], for j = 0, . . . , N + 2. The boundary estimate of BuR ε We will now estimate BuR ε |∂Ω , B = D or N standing for Dirichlet and Neumann operators respectively. We shall prove that ëDuR ε ëH 1 ([0,T ]×∂Ω) . ε R−1 2 and ëNuR ε ëH 1/2 ([0,T ]×∂Ω) . ε R−2 2 . (33) 3.2 - Error estimates 51 To this end, we note that the boundary operator B applied to (28) is a sum of terms arising from Bwεk , 0 ≤ k ≤ N+ such as 3n bjε (t, x) = ε− 4 +1−mB +j Ú R2n ρ(y)φ(η)hε (y, η) N+ Ø k=0 ′ dk−mB +j (t, x′ , y, η)eiψk (t,x ,y,η)/ε dydη, (34) ′ with j = 0, . . . , N + mB , and others with the same form arising from Bwεk , 0 ≤ k ≤ N+ , ′ and B[wε−k( ) (−t, .)], 0 ≤ k ≤ N− . Above and as in the previous section, hε is either ε−1 Tε uIε or Tε vεI and thus is uniformly bounded in L2 . We first study the support of the amplitudes. Next we use local boundary coordinates to expand the boundary phases and introduce a change of variables on (y, η) that makes the obtained phases satisfy properties (Q1). The previous results on the approximation operators Oα are then used to estimate the boundary norms. Support of the amplitudes Due to assumptions B1-B2-B3, the rays stay away from the boundary except for times near the instants of reflections. For (y, η) ∈ Ky × Kη and t ∈ [0, T ] near some Tk (y, η), 0 ≤ k ≤ N+ , only xtk (y, η) and, if k Ó= 0, xtk−1 (y, η) approach the boundary. This suggests that the meaningful contributions to the boundary iψk−1 (.,y,η)/ε + dk−mB +j (., y, η)eiψk (.,y,η)/ε near norm of bjε are the quantities dk−1 −mB +j (., y, η)e Tk (y, η), k = 1, . . . , N+ . Furthermore, for t in the neighbourhood of Tk (y, η) and x′ ∈ ∂Ω, Tk (y,η) k−1 one expects d−m (t, x′ , y, η) and dk−mB +j (t, x′ , y, η) to vanish away from xk−1 (y, η), B +j because of the cut-offs in the amplitudes. In the remainder, we show that these two intuitive points are true. The key argument is that (t, y, η) vary in a compact set. The first point is rather easy to see. For (y, η) ∈ Ky × Kη , let us consider a period smaller than any lapse of time between two successive reflections, say β(y, η) = min (Tk (y, η) − Tk−1 (y, η)) /3, (T0 = 0), and define the intervals 0≤k≤N+ I0 (y, η) = ∅, Ik (y, η) = [Tk (y, η) − β(y, η), Tk (y, η) + β(y, η)] for k = 1, . . . , N+ , and IN+ +1 (y, η) = ∅. For each k = 0, . . . , N+ , let o o Ak = {(t, y, η) ∈ [0, T ] × Ky × Kη , t ∈ / I k (y, η) ∪ I k+1 (y, η)}. For (t, y, η) ∈ Ak , dist(xtk (y, η), ∂Ω) > 0 and has then a positive lower bound by continuity on the compact Ak . One has by (31) and (32) ψk (t, x, y, η) ≥ cons(x − xtk (y, η))2 , for (t, x, y, η) ∈ [0, T ] × Rn × Ky × Kη s.t. |x − xtk (y, η)| ≤ d. Thus ′ |dk−mB +j (t, x′ , y, η)eiψk (t,x ,y,η)/ε | ≤ e−cons/ε for (t, y, η) ∈ Ak and x′ ∈ ∂Ω. All we have to care about is then the contribution to the norm at the boundary of iψk−1 /ε dk−1 and dk−mB +j eiψk /ε at times in the interval Ik , k = 1, . . . , N+ . Let −mB +j e 3n qεj,k = ε− 4 +1−mB +j Ú R2n iψk−1 /ε ρ ⊗ φhε 1Ik (t)(dk−1 −mB +j e + dk−mB +j eiψk /ε )dydη. Justification of the asymptotics 52 Summing over k = 1, . . . , N+ yields ëbjε ëL2 ([0,T ]×∂Ω) . N q+ k=1 ëqεj,k ëL2 ([0,T ]×∂Ω) + ε∞ . (35) For the second point, we partition the set of starting points (y, η) according to the part of the boundary the rays xtk−1 (y, η) reach at t = Tk (y, η). Let (ul ) be a C ∞ partition of unity associated to the covering (Ul ) introduced in subsection 2.2.1 and k πl (y, η) = ρ(y)φ(η)ul (xTk−1 (y, η)). Then ëqεj,k ëL2 ([0,T ]×∂Ω) . L Ø l=1 ëmj,k,l ε ëL2 ([0,T ]×∂Ω) , where mj,k,l ε =ε − 3n +1−mB +j 4 Ú R2n k−1 hε πl 1Ik (t)(d−m eiψk−1 /ε b +j + dk−mb +j eiψk /ε )dydη. (36) We fix 1 ≤ l ≤ L and 1 ≤ k ≤ N+ . For 0 < δ < min β, let Ky ×Kη Bδ = {(t, y, η) ∈ [0, T ] × suppπl , t ∈ Ik (y, η)\]Tk (y, η) − δ, Tk (y, η) + δ[}. If (t, y, η) is in the compact set Bδ , then dist(xtk (y, η), ∂Ω) > 0. Let d(δ) ∈]0, δ] s.t. d(δ) < min dist(xtk (y, η), ∂Ω) and consider the set (t,y,η)∈Bδ Sδ = {(t, x′ , y, η) ∈ [0, T ] × ∂Ω × suppπl , t ∈ Ik (y, η) and |x′ − xtk (y, η)| ≤ d(δ)}. If (t, x′ , y, η) ∈ Sδ then t ∈]Tk (y, η) − δ, Tk (y, η) + δ[ and consequently T (y,η) k |x′ − xk−1 (y, η)| ≤ |x′ − xtk−1 (y, η)| + |t − Tk (y, η)| sup s∈[t,Tk (y,η)] |ẋsk−1 (y, η)| ≤ (1 + ëcë∞ )δ, T (y,η) k which implies that x′ ∈ Ul for sufficiently small δ,1since xk−1 (y, η) varies in a compact 2 set of Ul . Assume that d ≤ d(δ). Thus, supp πl (y, η)1Ik (y,η) (t)dk−mB +j (t, x′ , y, η) is included in Sδ . On the other hand, as σl is a diffeomorphism between Nl and Ul , one has |σl (v̂) − σl (v̂ ′ )| ≥ cons|v̂ − v̂ ′ | for every v̂, v̂ ′ ∈ Nl . Therefore, there exists κ > 0 s.t. πl (y, η)1Ik (y,η) (t)dk−mB +j (t, σl (v̂), y, η) = 0 if |t − Tk (y, η)| ≥ δ or |v̂ − ẑk (y, η)| ≥ κδ, T (y,η) k where σl (ẑk (y, η)) = xk−1 (y, η). k−1 The same result holds true for πl (y, η)1Ik (y,η) (t)d−m (t, σl (v̂), y, η), assuming that d ≤ B +j ′ ′ ′ t d (δ) with d (δ) ∈]0, δ] and d (δ) < min dist(xk−1 (y, η), ∂Ω). Furthermore (t,y,η)∈Bδ ′ ′ / Ul . mj,k,l ε (t, x ) = 0 if x ∈ 3.2 - Error estimates 53 Expansion of the boundary phases For simplicity of notation, we shall drop the exponents and indexes l. We expand the phase σψk−1 on [0, T ] × N × {0} near (Tk , ẑk ) σ ψk−1 (t, v̂, 0) = ψk−1 (t, σ(v̂) + vn ν(σ(v̂)))|vn =0 = σψk−1 (Tk , ẑk , 0) + (t − Tk , v̂ − ẑk ) · (τ, θ̂k ) 1 + (t − Tk , v̂ − ẑk ) · Mk (t − Tk , v̂ − ẑk ) 2 + Ø |α|=3 (t − Tk , v̂ − ẑk )α Ú 1 0 3 α σ (1 − s)2 ∂t,v̂ ψk−1 (Tk + s(t − Tk ), ẑk + s(v̂ − ẑk ), 0)ds, α! Tk where θ̂k = Dσ(ẑk )T ξk−1 and the matrix Mk defined in (20) has a positive definite imaginary part. Remember that all the quantities of the previous formulae depend on k (y, η) ∈ (xTk−1 )<−1> (U). For the purpose of obtaining a phase satistfying (Q1), the form of σψk−1 |vn =0 suggests the change of variables (C) : (z, θ) = ϑ(y, η), with k ϑ : (y, η) ∈ (xTk−1 )<−1> (U) Ô→ (Tk , ẑk , τ, θ̂k ). k Because tangential rays are avoided, the function Tk ∈ C ∞ ((xTk−1 )<−1> (U)) so ϑ is C ∞ . 1 Tk Tk Note that ξk−1 = Σ(ẑk )θ̂k + (ν(σ(ẑk )) · ξk−1 )ν(σ(ẑk )) with Σ = Dσ Dσ T Dσ ϑ is bijective and its inverse is given by k ϑ−1 : (Tk , ẑk , τ, θ̂k ) ∈ ϑ((xTk−1 )<−1> (U)) 2−1 . Hence 1 k Ô→ {ϕTk−1 }−1 (σ(ẑk ), Σ(ẑk )θ̂k + (τ 2 /c2 (σ(ẑk )) − |Σ(ẑk )θ̂k |2 ) 2 ν(σ(ẑk ))). 1 2 k )<−1> (U) because the square root in the previous expression ϑ−1 is C ∞ on ϑ (xTk−1 never vanish. Consequently, ϑ is a C ∞ diffeomorphism. Let v = (t, v̂), z = (Tk , ẑk ) and θ = (τ, θ̂k ) and denote f˜(v, z, θ) = f (v, ϑ−1 (z, θ)). We may write σψ̃k−1 |vn =0 as ψ̃k−1 (z, 0, z, θ) + θ · (v − z) + 12 (v − z)M̃k (z, θ)(v − z) q 1 α σ (v − z)α ∂t,v̂ ψ̃k−1 (z, 0, z, θ) + r̃k−1 (v, z, θ) + α! σ σ ψ̃k−1 (v, 0, z, θ) = 3≤|α|≤R := λ̃(v, z, θ) + r̃k−1 (v, z, θ). σ σ Since ψk and ψk−1 have by construction the same derivatives w.r.t. v up to the order R at (z, 0), the expansion of σψ̃k |vn =0 involves the same derivatives up to the order R and a remainder r̃k σ ψ̃k (v, 0, z, θ) = λ̃(v, z, θ) + r̃k (v, z, θ). With the change of variables (C), σmj,k ε may be written on [0, T ] × N × {0} as 3n − 4 +1−mB +j mj,k ε = ε σ Ú R2n k−1 h̃ε π̃1I˜k (t)(σd˜−m ei(λ̃+r̃k−1 )/ε B +j + σd˜k−mB +j ei(λ̃+r̃k )/ε )| det ϑ|dzdθ, where I˜k denotes [T̃k − β̃, T̃k + β̃]. We split the previous integral into two integrals which can be estimated using the operators Oα 3n ε− 4 +1−mB +j 3n Ú R2n Ú ε− 4 +1−mB +j σ ˜k i(λ̃+r̃k−1 )/ε h̃ε π̃1I˜k (t)(σd˜k−1 | det ϑ|dzdθ := ①, −mB +j + d−mB +j )e R2n h̃ε π̃1I˜k (t)σd˜k−mB +j eiλ̃/ε (eir̃k /ε − eir̃k−1 /ε )| det ϑ|dzdθ := ②. Justification of the asymptotics 54 Estimate of ①: The phase λ̃ + r̃k−1 is smooth on an open set containing Er0 = {(v, z, θ) ∈ Rn × suppπ̃, |v − z| ≤ r0 } for some r0 ∈]0, 1]. Furthermore, λ̃ + r̃k−1 satisfies the required properties (Q1). We fix r[λ̃ + r̃k−1 ] ∈]0, r0 ]. σ k Since σdk−1 −mB +j + d−mB +j is zero at v = z up to the order R − 2j − 2 by construction, one has 1 2 Ø σ ˜k−1 d−mB +j + σd˜k−mB +j (v, z, θ) = (v − z)α s̃kα (v, z, θ), |α|=R−2j−1 where s̃kα are smooth remainders. Let aα,k (v, z, θ) = π̃(z, θ)1I˜k (t)s̃kα (v, z, θ)| det ϑ(z, θ)|. The aα,k are smooth and aα,k (t, v̂, Tk , ẑk , θ) = 0 if |t − Tk | ≥ δ or |v̂ − ẑk | ≥ κδ or (z, θ) ∈ / supp(π̃). Then the aα,k satisfy the properties (Q2), assuming δ small enough to insure |(δ, κδ)| ≤ r[λ̃ + r̃k−1 ]. Therefore 3n ① = ε− 4 +1−mB +j Ø 1 2 Oα aα,k , (λ̃ + r̃k−1 )/ε 1suppπ̃ h̃ε . |α|=R−2j−1 One deduces ë①ëL2 ([0,T ]×N ) . ε R+1 −mB 2 ëhε ëL2 . (37) Estimate of ②: This is the term for which Lemma 3.1 is fully used. We write λ̃ as λ̃ = β + 2γ where 1 1 γ = (v − z)M̃k (z, θ)(v − z) and β = λ̃ − (v − z)M̃k (z, θ)(v − z). 4 2 The part β + γ will play the role of the phase for the operators Oα , while eiγ/ε will be enclosed in the amplitude to give it a good behavior. The phase β + γ is smooth on an open set containing Er0 and satisfies the properties (Q1). We associate to this phase some constant r[β + γ] and impose on δ to satisfy |(δ, κδ)| ≤ r[β + γ]. Let − cj,k ε = ε R−1 2 π̃1I˜k (t) σd˜k−mB +j eiγ/ε (eir̃k /ε − eir̃k−1 /ε )| det ϑ|. One has − |cj,k ε | . ε R−1 2 e−cons(v−z) 2 /ε |eir̃k /ε − eir̃k−1 /ε |. If δ is small enough, e−cons(v−z) 2 /ε |eir̃k /ε − eir̃k−1 /ε | . ε−1 |v − z|R+1 e−cons(v−z) 2 /(2ε) so that |cj,k ε | . 1. Hence cj,k ε is smooth and satisfies the properties (Q2): / supp(π̃), cj,k ε (v, z, θ) = 0 if |v − z| ≥ r[β + γ] or (z, θ) ∈ ∞ 3n j,k cε is uniformly bounded in L (R ). , 3.2 - Error estimates 55 N To make use of the estimates of Lemma 3.1, we aim to show that for N ∈ N, ε 2 ∂vNb cj,k ε (b = 1, . . . , n) is uniformly bounded in L∞ (R3n ). For this purpose, we write ∂vNb [eiγ/ε (eir̃k /ε − eir̃k−1 /ε )] as a sum of terms of the form ∂vNb1 [eiγ/ε ]∂vNb2 [eir̃k /ε − eir̃k−1 /ε ], 0 ≤ N1 , N2 ≤ N, N1 + N2 = N. As the remainders r̃k and r̃k−1 are of order R + 1, Lemma 3.2 yields for N1 , N2 ∈ N, (z, θ) ∈ suppπ̃, |v − z| ≤ |(δ, κδ)| and δ sufficiently small |∂vNb1 [eiγ/ε ]| . ε− |∂vNb2 [eir̃k /ε N1 2 e−cons(v−z) ir̃k−1 /ε −e ]| . 3 2 /ε , Ø −k N2 ≤k≤N2 R+1 k(R+1)−N2 ε |v − z| Ø + ε N 2 − R+1 N 1 2 1≤k< R+1 4 2 |eir̃k /ε | + |eir̃k−1 /ε | . The second sum in the last inequality is zero when N2 /(R + 1) ≤ 1. Remember that R ≥ 2. If N2 /(R + 1) > 1 then N2 (R − 1)/(2(R + 1)) > (R − 1)/2 and consequently −N2 /(R + 1) > −N2 /2 + (R − 1)/2. Thus |∂vNb2 [eir̃k /ε ir̃k−1 /ε −e ]| . 3 Ø N2 ≤k≤N2 R+1 −k k(R+1)−N2 ε |v − z| 1 Hence, for (z, θ) ∈ suppπ̃ and |v − z| ≤ |(δ, κδ)| |∂vNb1 [eiγ/ε ]∂vNb2 [eir̃k /ε − eir̃k−1 /ε ]| . ε− It follows that +ε − N2 + R−1 2 2 4 2 |eir̃k /ε | + |eir̃k−1 /ε | . N1 N − 22 + R−1 2 2 . N −2 . |∂vNb cj,k ε | . ε One can use the operator O0 to write 3n ② = ε− 4 +1−mB +j ε and thus R−1 2 1 2 O0 cj,k ε , (β + γ)/ε 1suppπ̃ h̃ε , ë②ëL2 ([0,T ]×N ) . ε R+1 −mB +j 2 Using (37) and (38) yields ëmj,k,l ε ëL2 ([0,T ]×∂Ω) . ε ëhε ëL2 . R+1 −mB 2 (38) ëhε ëL2 . One has a similar bound for qεj,k by summing over l = 1, . . . , L, ëqεj,k ëL2 ([0,T ]×∂Ω) . ε R+1 −mB 2 ëhε ëL2 . Plugging this into (35) gives ëbjε ëL2 ([0,T ]×∂Ω) . ε R+1 −mB 2 . All in all, we have shown that ëBuR ε ëL2 ([0,T ]×∂Ω) . ε R+1 −mB 2 . This result can be adapted to the integer Sobolev spaces as follows ëBuR ε ëH s ([0,T ]×∂Ω) . ε R+1 −mB −s 2 , s ∈ N. An interpolation argument ([62], p.49) enables the same estimate for non integer Sobolev spaces H s ([0, T ] × ∂Ω), s > 0. This proves (33). Justification of the asymptotics 56 3.2.3 The initial conditions I I R In this section we estimate the difference between (uR ε |t=0 , ∂t uε |t=0 ) and (uε , vε ) in H 1 (Ω) × L2 (Ω). By construction, 6 5Ø N+ N− Ø 1 − 3n Ú ′ ′ wε−k (0, x, y, η) wεk (0, x, y, η) − ρ(y)φ(η)εTε vεI (y, η) ε 4 cn 2 R2n k=0 k=0 uR ε (0, x) = 5Ø N+ wεk (0, x, y, η) +ρ(y)φ(η)Tε uIε (y, η) + N− Ø 6 wε−k (0, x, y, η) k=0 k=0 dydη. (′ ) As dist(x0±k (y, η), Ω̄) > 0 for (y, η) ∈ Ky × Kη , k = 1, . . . , N± , wε±k (0, x, y, η) are uniformly exponentially decreasing for x ∈ Ω and (y, η) ∈ Ky × Kη . Thus, only the incident beams contribute to uR ε (0, x) in Ω and Ú 3n − 4 uR cn ε (0, x) = ε R2n ρ(y)φ(η)Tε uIε (y, η)wε0 (0, x, y, η)dydη + O(ε∞ ), uniformly w.r.t. x ∈ Ω. The initial values for the phase and the amplitudes of wε0 have been fixed in (24) and (25). Hence 3n − 4 uR cn ε (0, x) = ε Ú R2n ρ(y)φ(η)Tε uIε (y, η)χd (x − y)eiφ0 (x,y,η)/ε dydη + O(ε∞ ), uniformly w.r.t. x ∈ Ω. It follows, uniformly for x ∈ Ω, that uR ε (0, x) = Tε∗ ρ ⊗ φTε uIε (x) +ε − 3n 4 cn Ú R2n ρ(y)φ(η)Tε uIε (y, η)(χd (x − y) − 1) eiφ0 (x,y,η)/ε dydη + O(ε∞ ). One wants to get rid of the second integral by making use of the exponential decrease of eiφ0 (x,y,η)/ε for |x − y| ≥ d/2. The following estimate is immediate by Cauchy-Schwartz inequality: Lemma 3.5. Let a be a positive real and h ∈ L2 (R2n y,η ). Then ë Ú |x−y|≥a h1Ky ×Kη (y, η)e−(x−y) 2 /(2ε) dydηëL2x . ëhëL2y,η e−a 2 /(4ε) . The previous Lemma leads to ∗ I ∞ ëuR ε |t=0 − Tε ρ ⊗ φTε uε ëL2 (Ω) . ε , by using the boundedness of Tε∗ from L2 (R2n ) to L2 (Rn ) (this result follows, e.g., from [72] p.97). On the other hand, ρ ⊗ φTε uIε approaches Tε uIε up to a small remainder. In fact, as ρ(y) = 1 if dist(y, K) < ∆, one has by Lemma 2.4 and assumption A3 ëTε uIε − ρ ⊗ φTε uIε ëL2y,η . ε∞ , 3.2 - Error estimates 57 and consequently I ∞ ëuR ε |t=0 − uε ëL2 (Ω) . ε . Moving to the spatial derivatives of uR ε , one has ∂xb uR ε (0, x) =ε − 3n 4 cn Ú R2n N Ø I ρ(y)φ(η)Tε uε (y, η) εj ∂xb j=0 è é a0j (0, x, y, η)eiφ0 (x,y,η)/ε dydη +O(ε∞ ), uniformly w.r.t. x ∈ Ω. Plugging the initial condition (25) for the incident amplitudes into the previous equation yields a simpler expression ∂xb uR ε (0, x) =ε − 3n 4 1 cn Ú R2n 1 2 ρ(y)φ(η)Tε uIε (y, η)∂xb χd (x − y)eiφ0 (x,y,η)/ε dydη + O(ε∞ ), 2 1 uniformly w.r.t. x ∈ Ω. 2 Since ∂xb χ(x − y)eiφ0 /ε = −∂yb χ(x − y)eiφ0 /ε , integration by parts leads to ∂xb uR ε (0, x) =ε − 3n 4 cn Ú R2n 1 2 ∂yb ρTε uIε φχd (x − y)eiφ0 (x,y,η)/ε dydη + O(ε∞ ), uniformly w.r.t. x ∈ Ω. Application of Lemma 3.5 and then Lemma 2.4 shows that the term involving ∂yb ρ has an exponentially decreasing contribution in L2 (Ω). On the other hand, the y derivative of the FBI transform is the FBI transform of the derivative. Thus 3n − 4 cn ë∂xb uR ε |t=0 − ε Ú R2n 2 1 ρ ⊗ φTε ∂xb uIε χd (x − y)eiφ0 (x,y,η)/ε dydηëL2 (Ω) . ε∞ . Again, Lemmas 3.5-2.4 and assumption A3 imply ∞ I ë∂xb uR ε |t=0 − ∂xb uε ëL2 (Ω) . ε . Time differentiation of uR ε is somewhat different. The contribution of reflected beams is still uniformly exponentially decreasing for x ∈ Ω 3n − 4 cn ∂t uR ε |t=0 (x) = ε Ú R2n with ′ ε∂t wε0 = ′ ρ(y)φ(η)Tε vεI (y, η)ε∂t wε0 (0, x, y, η)dydη + O(ε∞ ), N +1 Ø 1 ′ ′ 2 εj i∂t ψ0 a0j + ∂t a0j−1 eiψ0 /ε . j=0 ′ The initial values (24) and (26) for the phase and amplitudes of wε0 yield ′ ε∂t wε0 (0, x, y, η) = eiφ0 (x,y,η)/ε + N Ø j=0 εj Ø |α|=R−2j−1 (x − y)α zα (x, y, η)eiφ0 (x,y,η)/ε ′ + εN +1 ∂t a0N (0, x, y, η)eiφ0 (x,y,η)/ε , where zα are smooth remainders that vanish for |x − y| ≥ d. We can use the operators Oα to estimate the contribution of the terms (x − y)α zα to the norm of uR ε |t=0 ëε − 3n 4 Ú R2n 3n ρ ⊗ φTε vεI εj (x − y)α zα eiφ0 /ε dydηëL2x = ε− 4 +j ëOα (ρ ⊗ φzα , φ0 /ε) Tε vεI ëL2x . ε R−1 2 , for j = 0, . . . , N. Justification of the asymptotics 58 We also have ëε − 3n 4 Ú R2n ′ ρ ⊗ φTε vεI εN +1 ∂t a0N |t=0 eiφ0 /ε dydηëL2x 3n 1 2 ′ = ε− 4 +N +1 ëO0 ρ ⊗ φ∂t a0N |t=0 , φ0 /ε Tε vεI ëL2x . εN +1 . It follows, with the help of (14), that ∗ I ë∂t uR ε |t=0 − Tε ρ ⊗ φTε vε ëL2 (Ω) . ε R−1 2 , and finally, from Lemma 2.4 and assumption A3, I ë∂t uR ε |t=0 − vε ëL2 (Ω) . ε R−1 2 . Hence I R I ë∂t uR ε |t=0 − vε ëL2 (Ω) + ëuε |t=0 − uε ëH 1 (Ω) . ε 3.3 R−1 2 . Proof of the main theorem Now we may collect the previous estimates in order to bound the difference between uε the exact solution for (1) and uR ε the approximate solution of order R. For the Dirichlet case, the errors in the interior, at the boundary and in the initial conditions exhibit the same scale of ε, and the energy estimate leads to Sup ëuε (t, .) − uR ε (t, .)ëH 1 (Ω) . ε R−1 2 , Sup ë∂t uε (t, .) − ∂t uR ε (t, .)ëL2 (Ω) . ε R−1 2 . (39) t∈[0,T ] t∈[0,T ] For the Neumann case, one looses an order the energy estimate yields Sup ëuε (t, .) − uR ε (t, .)ëH 1 (Ω) . ε R−2 2 t∈[0,T ] √ ε in the boundary estimate, and thus , Sup ë∂t uε (t, .) − ∂t uR ε (t, .)ëL2 (Ω) . ε R−2 2 . t∈[0,T ] However, when comparing the ansatz at order R and R + 1 in the difference between t α and uR uR+1 ε , we can make use of further powers of ((x − xk ) )|α|=R+1 between the phases ε and ((x − xtk )α )|α|=R−2j−1 in the amplitudes. Using the approximation operators yields uniformly in time ëuR+1 (t, .) − uR ε ε (t, .)ëH 1 (Ω) . ε R−1 2 , ë∂t uR+1 (t, .) − ∂t uR ε ε (t, .)ëL2 (Ω) . ε R−1 2 . Hence one may improve the estimate for the Neumann case by using the approximate solution at the next order R + 1 R+1 ëuε (t, .) − uR (t, .)ëH 1 (Ω) + ëuR+1 (t, .) − uR ε (t, .)ëH 1 (Ω) . ëuε (t, .) − uε ε ε (t, .)ëH 1 (Ω) . ε R−1 2 . This leads to the same estimate (39) for the Neumann case. 3.3 - Proof of the main theorem 59 Remark 3.6. The FBI transforms of uIε and vεI are uniformly locally infinitely small outside the frequency sets F s(uIε ) and F s(vεI ) respectively, as ε tends to 0 (see [72] p.98). These sets may be phase space submanifolds of lower dimensions. For instance, for WKB initial data, F s(aeiΦ/ε ) = {(y, ∂x Φ(y)), y ∈ suppa}. For numerical computations, one has therefore to discretize neighbourhoods of (Ky ×Kη )∩F s(uIε ) and (Ky ×Kη )∩F s(vεI ). Studying numerically the behaviour of FBI transforms in the associated computational domains could lead to interesting results on the optimal mesh size. Details on numerical FBI transforms are given in [61]. 60 Chapter II Wigner measures for the wave equation in a convex domain Contents 1 Introduction 61 2 Asymptotic solution 64 2.1 First order Gaussian beams . . . . . . . . . . . . . . . . . . . . . . . 64 2.1.1 Beams in the whole space . . . . . . . . . . . . . . . . . . . . 64 2.1.2 Incident and reflected beams in a convex domain . . . . . . . 66 2.1.3 General relation between incident and reflected beams’ phases 67 Gaussian beams summation . . . . . . . . . . . . . . . . . . . . . . . 69 2.2.1 Construction of the approximate solution . . . . . . . . . . . 69 2.2.2 Expression of the phases . . . . . . . . . . . . . . . . . . . . . 72 2.2.3 Expression of the amplitudes . . . . . . . . . . . . . . . . . . 73 2.2.4 Gaussian integrals . . . . . . . . . . . . . . . . . . . . . . . . 75 2.2 3 Wigner transforms and measures 78 3.1 Wigner transform for Gaussian integrals . . . . . . . . . . . . . . . . 79 3.2 Wigner measure for superposed Gaussian beams . . . . . . . . . . . 83 3.3 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . 91 A Proof of the relation between incident and reflected beams’ phases 94 B Results related to the FBI and the Wigner transforms 96 61 1 Introduction In this article, we are interested in the high frequency limit of the initial-boundary value problem (IBVP) for the wave equation P uε = ∂t2 uε − ∂x .(c2 (x)∂x uε ) = 0 in [0, T ] × Ω, Buε = 0 in [0, T ] × ∂Ω, uε |t=0 = uIε , ∂t uε |t=0 = vεI in Ω, (1a) (1b) (1c) where B stands for a Dirichlet or Neumann type boundary operator. Above, T > 0 is fixed, and Ω is a bounded domain of Rn with a C ∞ boundary. The coefficient c is assumed to be in C ∞ (Ω̄), though this assumption may be relaxed. Herein, the initial data depend on a small parameter ε > 0, playing the role of a small wavelength, the high frequency limit corresponding to ε → 0. In any case, we shall assume that uIε , vεI are A1. uniformly bounded respectively in H 1 (Ω) and L2 (Ω), A2. uniformly supported in a fixed compact set of Ω. We shall assume that the following hypotheses holds on the domain Ω: B1. Ω is convex with respect to the bicharacteristics of the wave operator, that is every ray originating from Ω hits the boundary twice and transversally, B2. No ray remains in a compact of Rn for increasing times, B3. The boundary has no dead-end trajectories, that is infinite number of successive reflections cannot occur in a finite time. These geometric hypotheses insure that the rays starting from the compact support of the initial data do not face diffraction on the boundary, neither do they glide along ∂Ω. The only phenomena occuring at the boundary is reflection according to geometrical optics laws. We investigate the high frequency limit in terms of Wigner measures. The Wigner function is a phase space distribution introduced by E. Wigner [101] in 1932 to study quantum corrections to classical statistical mechanics. In the 90’s, mathematicians became increasingly interested by the Wigner transforms and related measures. In [63, 69, 70, 71], those transforms are applied to the semiclassical limit of Schrödinger equations. A general theory for their use in the homogenization of energy densities of dispersive equations was laid out by Gérard et al. in [39], see also [35, 36]. Wigner measures are related to the H-measures and microlocal defect measures introduced in [99] and [37], see also [4, 12]. Whereas there is no notion of scale for the latter measures, Wigner transforms are associated to a parameter ε → 0. In quantum mechanics, this parameter is the rescaled Planck constant, while it will be the distance between two points of the medium’s periodic structure for homogenisation problems. Introduction 62 The Wigner transform, at the scale ε, is defined for a given sequence (aε , bε ) in S (Rn )p × S ′ (Rn )p by the duality weak formula ′ −n wε (aε , bε )(x, ξ) = (2π) Ú Rn ε ε e−iv·ξ aε (x + v)b∗ε (x − v)dv. 2 2 If (aε ) is uniformly bounded w.r.t. ε in L2 (Rn )p , then wε [aε ] = wε (aε , aε ) converges as ε goes to 0 to a positive hermitian matrix measure in S ′ (Rnx × Rnξ ) (modulo the extraction of a subsequence). This measure is called a Wigner measure associated to (aε ) and denoted w[aε ]. The Wigner measures associated to the solution of the wave equation (and hyperbolic problems in general, see e.g. [39],[80]) are related to the energy density in the high frequency limit. More precisely, under suitable hypotheses, the density of energy converges in the sense of measures to (proposition 1.7 in [39]) Ú Rn E t (x, dξ), where 1 1 E t = w[∂t uε (t, .)] + Trw[c∂x uε (t, .)]. 2 2 Above, the involved Wigner measures are obtained after extending ∂t uε and c∂xb uε , b = 1, . . . , n, to functions of L2 (Rn ) by setting ∂t uε = 1Ω ∂t uε , ∂x uε = 1Ω ∂x uε and extending c outside Ω̄ in a smooth way. Wigner measures for the wave equation have been studied by Miller [75] who proved refraction results for sharp interfaces and Burq [11] who described their support for a Dirichlet boundary condition. Similar results have been established for other problems [27, 32], in particular eigenfunctions for the Dirichlet problem [38, 103] and for the Neumann and Robin problems [13]. All these works are based on pseudo-differential calculus. In this paper, we shall investigate the Wigner measure by means of direct computations on an approximate solution of the IBVP for the wave equation. The approximate or asymptotic solutions used here are obtained by superposition (or mixing) of Gaussian beams, and more precisely by a weighted integral of Gaussian beams suitably designed to fit initial data as in chapter 1. Gaussian beams are waves with a Gaussian shape at any instant, localized near a single ray [5, 85]. The summation of different beams allows to approach non localized wave fields, see e.g. [19, 52, 55] and the recent [43, 65, 77, 97]. Gaussian beams (or the related coherent states) can be treated as a basis of fundamental solutions of wave motion and used to study general solutions of partial differential equations. They hence allowed amongst others to describe propagation of singularities [85], to prove lack of observability [68] and to study semiclassical measures [81] and trace formulaes [23, 102]. This feature seems to be very well suited for the study of Wigner measures. Indeed, the Wigner transform of two different beams vanishes when ε goes to zero. Even better, the Wigner measure of one individual Gaussian beam associated to the wave equation is a Dirac mass localized on the corresponding bicharacteristic. Thus Gaussian beams form a sort of an orthogonal family for the Wigner measure. The appealing to these elementary solutions for studying Wigner measures is not new; they have been used in the whole space domain by Robinson [89] for the Schrödinger equation and more recently by Castella [16] who used a coherent states approach for the Helmholtz equation. 63 In view of known results, one expects that the Wigner measure of a summation of Gaussian beams would give easily that the associated weights are transported along the broken bicharacteristic flow (see p.66 for the construction of reflected flows and p.90 for the definition of the broken flow). Unfortunately this result is not immediate as even different beams become infinitely close to each other. We show however by elementary computations that this intuition is indeed true and that the Wigner measure of the considered approximate solution is transported along the broken bicharacterisitic flow. Since the asymptotic solution is close to the exact one, we may deduce the same outcome for the Wigner measure E t . In particular, we shall prove the following theorem Theorem 1.1. Set v Iε = 1Ω vεI and uIε = 1Ω uIε . Assume the conditions A1 and A2 fulfilled, and furthermore that: C1. The Wigner measures of v Iε and ∂xb uIε , b = 1, . . . , n, are unique, C2. v Iε and ∂xb uIε , b = 1, . . . , n are ε-oscillatory (see equation (53)), C3. The Wigner measures of v Iε and ∂xb uIε , b = 1, . . . , n do not charge the set Rn ×{ξ = 0}. Let E ± = 21 w[v Iε ± ic|D|uIε ], and denote by ϕtb the broken bicharacteristic flow associated to −i∂t − c|D| obtained after successives reflections on the boundary ∂Ω. Then Et = 2 11 + −1 E o(ϕ−t + E − o(ϕtb )−1 in Ω × (Rn \{0}). b ) 2 For general properties of Wigner measures and transforms, we will refer to the usual framework [39]. The rest of the paper is organized as follows. In the first section, we recall the construction of first order Gaussian beams and the structure of the asymptotic solutions obtained as an infinite sum of such beams. The derivatives of the asymptotic solutions are then expressed using what we call Gaussian integrals. We simplify the expression of the Wigner transform of such integrals in section 3, following initial computations of [89] in the Schrödinger case. We then compute the scalar Wigner measure for the asymptotic solution by exploiting the expressions of the beams’ phases and amplitudes and using the dominated convergence theorem. Finally, we prove the propagation of the Wigner measure along the broken flow for the exact solution of the IBVP (1) with the help of assumptions C2 and C3 on the initial data. A few useful notations will be used hereafter. The inner product of two vectors a, b ∈ Rd will be denoted by a · b. The transpose of a matrix A will be noted AT . If E is a subset of Rd , we denote E c its complementary and 1E its characteristic function. For a function f ∈ L2 (Ω), we denote f = 1Ω f . For r > 0, χr denotes a cut-off of C0∞ (Rn , [0, 1]) satisfying χr (x) = 1 if |x| ≤ r/2 and χr (x) = 0 if |x| ≥ r. We use the following definition of the Fourier transform Fx u(ξ) = Ú Rd u(x)e−ix·ξ dx for u ∈ L2 (Rd ). Asymptotic solution 64 If no confusion is possible, we shall omit the reference to the lower index x. For a smooth function f ∈ C ∞ (Rdx , C), we will use the notation ∂x f to denote its gradient vector (∂xb f )1≤b≤d and ∂x2 f to denote its Hessian matrix (∂xb ∂xc f )1≤b,c≤d . For a function F ∈ C ∞ (Rd , Cp ), the notation DF is used for its Jacobian matrix. We use the letter C to denote a positive constant (different each time it appears). We specify the parameters some constants depend on by denoting them C(V ), where V may be a variable or a set of Rd . For yε and zε sequences of R+ with ε ∈]0, ε0 ], we use the notation yε . zε if there exists a constant C > 0 independent of ε such that yε ≤ Czε for ε small enough. We write yε . ε∞ or yε = O(ε∞ ) if for any s ≥ 0 there exists cs > 0 s.t. for ε small enough y ε ≤ c s εs . Finally, if E is in an open subset of R2n and νε , νε′ are two distributions s.t. lim(νε − νε′ ) = 0 in E, ε→0 we shall write νε ≈ νε′ in E. 2 Asymptotic solution In this section, we explain the notion of Gaussian beam for the wave equation focusing on first order beams. We then construct the asymptotic solution as a superposition of these beams and express its time and spatial derivatives with the help of Gaussian integrals. 2.1 First order Gaussian beams We recall the construction of individual first order Gaussian beams in section 2.1.1, and apply it to describe the incident beam and the reflected beams in section 2.1.2. A useful general relation linking reflected beams’ phases to the phase of an incident beam is given for first order and higher order beams. 2.1.1 Beams in the whole space Denote h+ (x, ξ) = c(x)|ξ| and let (xt , ξ t ) be a Hamiltonian flow for h+ , that is a solution of the system t dξ t dxt t t t ξ = ∂ξ h+ (x , ξ ) = c(x ) t , = −∂x h+ (xt , ξ t ) = −∂x c(xt )|ξ t |. dt |ξ | dt The curves (t, x±t ) of Rn+1 are called the rays of P . An individual first order (Gaussian) beam for the wave equation associated to a ray (t, xt ) has the following form ωε (t, x) = a0 (t, x)eiψ(t,x)/ε , (2) 2.1 - First order Gaussian beams 65 with a complex phase function ψ real-valued on (t, xt ) and an amplitude function a0 null outside a neighbourhood of (t, xt ). It satisfies sup ëP ωε (t, .)ëL2 (Ω) = O(εm ), t∈[0,T ] for some m > 0. The construction is achieved by making the amplitudes of P ωε vanish on the ray up to fixed suitable orders [51, 68, 85] 3 −2 −1 2 4 P ωε = ε p(x, ∂t ψ, ∂x ψ)a0 + ε i[2∂t ψ∂t a0 − 2c ∂x ψ∂x a0 + P ψa0 ] + h.o.t. eiψ/ε , (3) where p(x, τ, ξ) = c2 (x)|ξ|2 − τ 2 is the principal symbol of P . The first equation is then the eikonal equation p(x, ∂t ψ(t, x), ∂x ψ(t, x)) = 0, (4) on x = xt up to order 2 (see Remark 2.1 in chapter 1 for an explanation of the choice of this specific order), which means ∂xα [p(x, ∂t ψ(t, x), ∂x ψ(t, x))]|x=xt = 0 for |α| ≤ 2. Orders 0 and 1 of the previous equation are fulfilled on the ray by setting ∂t ψ(t, xt ) = −h+ (xt , ξ t ) and ∂x ψ(t, xt ) = ξ t . (P.a) It follows, by choosing ψ(0, x0 ) a real quantity, that ψ(t, xt ) is real. (P.b) Order 2 of eikonal (4) on the ray may be written as a Riccati equation d 2 ∂ ψ(t, xt ) + H21 (xt , ξ t )∂x2 ψ(t, xt ) + ∂x2 ψ(t, xt )H12 (xt , ξ t ) dt x + ∂x2 ψ(t, xt )H22 (xt , ξ t )∂x2 ψ(t, xt ) + H11 (xt , ξ t ) = 0, A (5) B H11 H12 is the Hessian matrix of h+ . Although non-linear, this H21 H22 Riccati equation has a unique global symmetric solution which satisfies the fundamental property 1 2 Im ∂x2 ψ t, xt is positive definite, (P.c) where H = given an initial symmetric matrix ∂x2 ψ (0, x0 ) with a positive definite imaginary part (see the proof of Lemma 2.56 p.101 in [51]). The phase is defined beyond the ray as a polynomial of order 2 with respect to (w.r.t.) (x − xt ) [98] 1 ψ(t, x) = ψ(t, xt ) + ξ t · (x − xt ) + (x − xt ) · ∂x2 ψ(t, xt )(x − xt ). 2 (6) Next, we make the term associated to the power ε−1 in the expansion (3) vanish on (t, xt ) 2∂t ψ∂t a0 − 2c2 ∂x ψ∂x a0 + P ψa0 = 0 on (t, xt ), (7) Asymptotic solution 66 which leads to a linear ordinary differential equation (ODE) on a0 (t, xt ). The amplitude is then chosen under the form a0 (t, x) = χd (x − xt )a0 (t, xt ), where d > 0 will be fixed later. The constructed beams are thus defined for all (t, x) ∈ Rn+1 and they satisfy the estimate √ n ëε− 4 +1 P ωε (t, .)ëL2 (Ω) = O( ε) uniformly w.r.t. t ∈ [0, T ]. Gaussian beams for P associated to the ray (t, x−t ) are ωε (−t, x). 2.1.2 Incident and reflected beams in a convex domain Construction of flows and beams We suppose c(x) constant for dist(x, Ω̄) larger o than some D > 0. Given a point (y, η) in the phase space T ∗ Rn where o T ∗ U denotes U × (Rn \{0}) if U is an open set of Rn , an incident beam is a beam associated to the ray (t, xt0 (y, η)) satisfying: dxt0 ξt dξ0t = c(xt0 ) 0t , = −∂x c(xt0 )|ξ0t |, dt |ξ0 | dt xt0 |t=0 = y, ξ0t |t=0 = η, η Ó= 0. The Hamiltonian flow ϕt0 = (xt0 , ξ0t ) for h+ is called an incident flow. The associated beam is denoted ωε0 and called an incident beam. Since we have dependence w.r.t. the initial conditions (y, η), we shall write the incident beam as ωε0 (t, x, y, η) = a0 (t, x, y, η)eiψ0 (t,x,y,η)/ε . Let R be the reflection involution o o R : T ∗ Rn |∂Ω → T ∗ Rn |∂Ω (X, Ξ) Ô→ (X, (Id − 2ν(X)ν(X)T )Ξ). Above ν denotes the exterior normal field to ∂Ω. We restrain the study to starting o points (y, η) ∈ B = ∪t∈R ϕt0 (T ∗ Ω). Each associated flow ϕt0 (y, η) strikes the boundary twice. Reflection of ϕt0 (y, η) at the exit time t = T1 (y, η) s.t. T (y,η) x0 1 T (y,η) (y, η) ∈ ∂Ω and ẋ0 1 T (y,η) (y, η) · ν(x0 1 (y, η)) > 0, gives birth to the reflected flow ϕt1 (y, η) = (xt1 (y, η), ξ1t (y, η)) defined by the condition T (y,η) ϕ1 1 T (y,η) (y, η) = R oϕ0 1 (y, η). Similarly, we also define the reflection time T−1 (y, η) and the flow ϕt−1 (y, η) by reflecting ϕt0 (y, η) as follows x0 −1 T (y,η) T (y,η) ϕ−1−1 T (y, η) ∈ ∂Ω and ẋ0 −1 T (y, η) = R oϕ0 −1 (y,η) (y,η) T (y, η) · ν(x0 −1 (y, η). (y,η) (y, η)) < 0, 2.1 - First order Gaussian beams 67 We denote, for k = ±1, the reflected beams by ωεk (t, x, y, η) = ak0 (t, x, y, η)eiψk (t,x,y,η)/ε . These beams are associated to the reflected bicharacteristics ϕtk . Let us introduce, for k = 0, ±1, the boundary amplitudes dk−mB +j s.t. Bωεk = mB Ø j=0 ε−mB +j dk−mB +j eiψk /ε . Above, mB denotes the order of B (mB = 0 for Dirichlet and mB = 1 for Neumann). The construction of the reflected phases and amplitudes is achieved by imposing that the time and tangential derivatives of ψk equal at (Tk , xT0 k ) those of ψ0 up to order 2, d0−mB + dk−mB vanish at (8) (Tk , xT0 k ), (9) for k = ±1. These constraints uniquely determine the reflected phases and amplitudes, once the incident ones are fixed [85]. Moreover, if T is sufficiently small, the reflected rays is in the interior of the domain at the instant T (x±T ±1 (y, η) ∈ Ω), and the following boundary estimates are satisfied [85] n n 3 ëB(ε− 4 +1 ωε0 (., y, η) + ε− 4 +1 ωε1 (., y, η))ëH s ([0,T ]×∂Ω) = O(ε−mB −s+ 2 ), n n 3 and ëB(ε− 4 +1 ωε0 (., y, η) + ε− 4 +1 ωε−1 (., y, η))ëH s ([−T,0]×∂Ω) = O(ε−mB −s+ 2 ) for s ≥ 0. Let us point out that the construction of the reflected beams is also valid for other boundary conditions if the IBVP is well-posed ([85] p.221). Remark 2.1. Higher order beams, possibly with more than one amplitude, can be constructed to satisfy better interior and boundary estimates. In this case, the eikonal equation (4) must be satisfied up to order larger that 2 on the rays. If r ≥ 3, the equations ∂xα [p(x, ∂t ψ, ∂x ψ)] (t, xt ) = 0 for |α| = r give systems of linear ODEs on (∂xα ψ(t, xt ))|α|=r with a second member involving lower order derivatives of the phase. The key observation to prove this statement is to replace each term ∂τ p(ϕt )∂xα ∂t ψ(t, xt ) + ∂ξ p(ϕt ) · ∂xα ∂x ψ(t, xt ), |α| = r, by 2c(xt )|ξ t | dtd (∂xα ψ(t, xt )). We refer to [85] for further details. 2.1.3 General relation between incident and reflected beams’ phases In this paragraph, we give a useful relation between an incident phase ψinc and a reflected phase ψref for beams of any order. This relation provides with the derivatives of the reflected phase up to order R, which may be useful in other applications of Gaussian beams. Here we will apply the obtained results for first order beams to compute the Hessian matrices of ψ±1 on the rays. The matrices ψ±1 (t, x±t ±1 ) can also be computed by solving the Riccati equations with the proper values at the instants of reflections t = T±1 (see eg. [79]). Asymptotic solution 68 Consider the following auxiliary function linking ϕt1 to ϕt0 for any fixed time t s1 : B → B −T (y,η) T (y,η) (y, η) Ô→ ϕ0 1 o R o ϕ0 1 (y, η). For a given point (y, η) ∈ B, s1 (y, η) is its "image by the mirror" ∂Ω. For instance, Chazarain used this type of auxiliary functions in [21] to show propagation of regularity for wave type equations in a convex domain. By the Implicit functions theorem, T1 is C ∞ on the open set B and so is s1 . T (y,η) Since ϕt0 o s1 satisfies the same Hamiltonian equations as ϕt1 and ϕ1 1 (y, η) = T1 (y,η) ϕ0 o s1 (y, η) for (y, η) ∈ B, one has ϕt1 = ϕt0 o s1 . Besides, noting that T1 (ϕt0 ) = T1 − t, one also has ϕt1 = s1 oϕt0 . (10) ϕt0 and ϕt1 are symplectic C ∞ diffeomorphisms from B to B [49], and so is s1 . One can define a similar auxiliary function s−1 : B → B s.t. ϕt−1 = ϕt0 o s−1 and ϕt−1 = s−1 oϕt0 for t ∈ R. Let us introduce the components of s1 as s1 = (r, λ). 1 2 For every functions f, g ∈ C ∞ Rnu × (Rnξ \{0}), Cp and phase function V ∈ C ∞ (Rnu , Cnξ ) s.t. V (u0 ) ∈ Rnξ \{0}, we introduce the notation m f (u, V (u)) u=u ≍ g(u, V (u)), 0 to denote that the formal derivatives of f (u, V (u)) and g(u, V (u)) up to order m coincide on u0 . The derivation here is viewed formally, since V may be complex valued out of u0 , which makes f (u, V (u)) and g(u, V (u)) not defined for u Ó= u0 . However, on the exact point u0 , one can always use the formulae of composite functions’ derivatives to get a formal expression of the derivatives. We will use the same notation m f (t, x, V (t, x)) ≍ t g(t, x, V (t, x)), x=x 1 2 for functions f, g ∈ C ∞ Rt × Rnx × (Rnξ \{0}), Cp and phase function V ∈ C ∞ (Rt × Rnx , Cnξ ) s.t. for t ∈ R, V (t, xt ) ∈ Rnξ \{0} to denote that the formal derivatives of f (t, x, V (t, x)) and g(t, x, V (t, x)) w.r.t. x up to order m coincide on (t, xt ) for t ∈ R. In the following, we will be sloppy with respect to the notation of the dependence of ∂t ψ0 and ∂x ψ0 on their variables (t, x). Since the reflection R conserves |Ξ|, one has for every (x, ξ) ∈ B and τ ∈ R∗ p(r(x, ξ), τ, λ(x, ξ)) = p(x, τ, ξ). Thus ∞ p(r(x, ∂x ψ0 ), ∂t ψ0 , λ(x, ∂x ψ0 )) ≍ t p(x, ∂t ψ0 , ∂x ψ0 ), x=x0 (11) 2.2 - Gaussian beams summation 69 which implies, by construction of ψ0 2 p(r(x, ∂x ψ0 ), ∂t ψ0 , λ(x, ∂x ψ0 )) ≍ t 0. x=x0 Compare this with the equation 2 p(r(x, ∂x ψ0 ), ∂t ψ1 (t, r(x, ∂x ψ0 )), ∂x ψ1 (t, r(x, ∂x ψ0 ))) ≍ t 0, x=x0 resulting from the construction of ψ1 and (10). This suggests the following Lemma Lemma 2.2. Let R be an integer larger than 1 and ψinc and ψref an incident and a reflected phase of C ∞ (Rt × Rnx , C) satisfying ∂t ψinc (t, xt0 ) = −c(xt0 )|ξ0t | and ∂t ψref (t, xt1 ) = −c(xt1 )|ξ1t |, ∂x ψinc (t, xt0 ) = ξ0t and ∂x ψref (t, xt1 ) = ξ1t , R R x=x0 x=x1 p(x, ∂t ψinc , ∂x ψinc ) ≍ t 0 and p(x, ∂t ψref , ∂x ψref ) ≍ t 0, and having the same time and tangential derivatives at the instant and the point of R−1 reflection (T1 , xT0 1 ) up to the order R. Then ∂t ψref (t, r(x, ∂x ψinc )) ≍ t ∂t ψinc and x=x0 R−1 ∂x ψref (t, r(x, ∂x ψinc )) ≍ t λ(x, ∂x ψinc ). x=x0 The proof is postponed to Appendix A. A similar result linking the reflected phase associated to the ray (t, x−t −1 ) to the incident phase can be established. 2.2 Gaussian beams summation We begin this section by a reminder of the construction of asymptotic solutions to the IBVP (1a), (1b) with some initial conditions (1c’) (see below). These solutions are obtained by a Gaussian beams summation as achieved in chapter 1. We focus on a superposition of first order beams, for which exact expressions of the phases and amplitudes are displayed in 2.2.2. √ These beams lead to a first order approximate solution, close to the exact one up to ε. We end the section approaching the derivatives of the first order solution by some Gaussian type integrals we introduce. 2.2.1 Construction of the approximate solution In chapter 1, we have constructed a family of asymptotic solutions to the IBVP for the wave equation for initial data satisfying A1, A2 and an additionnal hypothesis A3 concerning their FBI transforms. Let us recall here that the FBI transform (see [72]) is, for a given scale ε, the operator Tε : L2 (Rn ) → L2 (R2n ) defined by 3n Tε (a)(y, η) = cn ε− 4 Ú Rn i k 2 n 3n a(x)e ε η·(y−x)− 2 (y−x) dx, cn = 2− 2 π − 4 , a ∈ L2 (Rn ). (12) Asymptotic solution 70 Its adjoint is the operator 3n Tε∗ (f )(x) = cn ε− 4 Ú i R2n k 2 f (y, η)e ε η·(x−y)− 2 (x−y) dydη, f ∈ L2 (R2n ). (13) As the Fourier Transform, the FBI transform is an isometry, satisfying Tε∗ Tε = Id. The extra assumption needed in chapter 1 is A3. ëTε uIε ëL2 (Rn ×Rc ) = O(ε∞ ) and ëTε v Iε ëL2 (Rn ×Rc ) = O(ε∞ ), where Rc denotes the complementary in Rn of some ring R = {η ∈ Rn , r0 ≤ |η| ≤ r∞ }, 0 < r0 < r∞ . In general, this assumption may be not satisfied. We thus construct a family of initial I ) close to (uIε , vεI ) that satisfy assumptions A1 and A2 and furthermore data (uIε,γ , vε,γ have FBI transforms small in L2 (Rn × Rc ). Letting r0 go to 0 and r∞ go to +∞ will make these data approach (uIε , vεI ) in a sense that will be specified in section 3.3. In any case, the needed convergence is weaker than a L2 convergence since we are interested in the study of Wigner measures. Let us first truncate Tε uIε and Tε v Iε outside R by multiplying them by a cut-off γ ∈ C0∞ (Rn , [0, 1]) supported in the interior of R γ = χr∞ /2 (1 − χ4r0 ). (14) Lemma B.4 yields ëTε Tε∗ γ(η)Tε uIε ëL2 (Rn ×Rc ) = O(ε∞ ) and ëTε Tε∗ γ(η)Tε v Iε ëL2 (Rn ×Rc ) = O(ε∞ ). (15) In order to satisfy A2, we multiply (Tε∗ γ(η)Tε uIε , Tε∗ γ(η)Tε v Iε ) by a cut-off ρ ∈ C0∞ (Rn , [0, 1]) supported in Ω, and consider I = ρTε∗ γ(η)Tε v Iε . uIε,γ = ρTε∗ γ(η)Tε uIε , vε,γ (1c’) We index this initial data by γ because the parameters r0 and r∞ will vary in section 3.3. We suppose that ρ(v) = 1 if dist(v, suppuIε ∪ suppvεI ) ≤ D for some D > 0. The required estimate I A3’. ëTε uIε,γ ëL2 (Rn ×Rc ) = O(ε∞ ) and ëTε vε,γ ëL2 (Rn ×Rc ) = O(ε∞ ), is fulfilled since Lemma B.3 implies that ë(1 − ρ)Tε∗ γ(η)Tε uIε ëL2x . e−C/ε and ë(1 − ρ)Tε∗ γ(η)Tε v Iε ëL2x . e−C/ε . (16) Using the boundedness of the operator Tε∗ γTε from L2 (Rn ) to L2 (Rn ) and the relations ∂yb Tε = Tε ∂xb , ∂xb Tε∗ = Tε∗ ∂yb , (17) obtained by integrations by parts in the expressions of Tε and Tε∗ , one can show that the assumption A1 is also fulfilled by these new initial data. Let ρ′ be a cut-off of C0∞ (Rn , [0, 1]) supported in a compact Ky ⊂ Ω and satisfying ρ′ (y) = 1 if dist(y, suppρ) < ∆ for some ∆ > 0, 2.2 - Gaussian beams summation 71 and γ ′ a cut-off of C0∞ (Rn , [0, 1]) supported in Kη ⊂ Rn \{0} s.t. γ ′ ≡ 1 on R. Without loss of generality, we assume that the reflected rays at the instant T (x±T ±1 (y, η)) remain in the interior of the domain when y varies in Ky and η in Rn \{0}. This is always possible upon reducing T because the number of reflections for initial position and speed varying in Ky × (Rn \{0}) is uniformly bounded (see section 2.3 of chapter 1 for similar arguments). Then, the IBVP (1a), (1b) with initial conditions (1c’) has a 0 1 1 2 family of approximate solutions uappr ε,γ in C ([0, T ], H (Ω)) ∩ C ([0, T ], L (Ω)) obtained as a summation of first order beams. A general result using a superposition of beams of any order was proven in chapter 1, and it reads for first order beams as follows Proposition 2.3. [Theorem 1.1, chapter 1] Denote for t ∈ [0, T ] and x ∈ Rn the following superposition of Gaussian beams uappr ε,γ (t, x) 6 5Ø Ø 1 − 3n +1 Ú k′ ′ ′ I k′ 4 ωε (−t, x, y, η) ρ (y)γ (η)Tε vε,γ (y, η) cn ωε (t, x, y, η) − = ε 2 R2n k=0,−1 k=0,1 ′ ′ + ρ (y)γ (η)ε −1 5Ø Tε uIε,γ (y, η) ωεk (t, x, y, η) + Ø k=0,−1 k=0,1 6 ωεk (−t, x, y, η) dydη. ′ Above, ωε0 , ωε0 are incident Gaussian beams with the same phase ψ0 satisfying at t = 0 i ψ0 (0, x, y, η) = η · (x − y) + (x − y)2 , 2 (18) ′ and different amplitudes a00 , a00 satisfying ′ a00 (0, x, y, η) = χd (x − y), [i∂t ψ0 a00 ](0, x, y, η) = χd (x − y) + O(|x − y|). (19) ′ ωε±1 and ωε±1 denote the associated reflected beams. Then uappr ε,γ is asymptotic to uε,γ the exact solution of (1a)-(1b) with initial conditions (1c’) in the sense that √ √ appr sup ëuε,γ −uappr ε,γ ëH 1 (Ω) ≤ C(γ, Ω, T ) ε and sup ë∂t uε,γ −∂t uε,γ ëL2 (Ω) ≤ C(γ, Ω, T ) ε. t∈[0,T ] t∈[0,T ] Remark 2.4. The final error is obtained by summing the errors in the interior equation, the boundary√condition and the initial conditions. Note that the asymptotics is of the same order ε for both Dirichlet and Neumann boundary conditions. More generally, the construction may be achieved for any boundary condition if the IBVP is well-defined in C 0 ([0, T ], H 1 (Ω)) ∩ C 1 ([0, T ], L2 (Ω)) for second members, initial data and boundary data in some Sobolev spaces C 0 ([0, T ], H s1 (Ω)) × H s2 ([0, T ] × ∂Ω) × (H s3 (Ω) × H s4 (Ω)) (see e.g. [76] for boundary conditions with energy estimates in this type of spaces). It may look surprising, but the asymptotics does not depend on the orders si of these Sobolev spaces. Indeed, using higher order beams increases the accuracy √ of the Gaussian beams integral and for sufficiently high order, the error falls to O( ε). On the other hand, √ summation of higher order beams and summation of first order ones are close to order ε. So the error √ in the approximate solution obtained by superposing first order beams is at any case O( ε). The proof relies on the use of a family of approximate operators acting from L2 (R2n ) to L2 (Rn ) (chapter 1). We recall a simple version of the estimate of the norm of these operators established therein. Asymptotic solution 72 For t ∈ [0, T ], let Kz,θ (t) be a compact of R2n and consider the set E1 = {(t, x, z, θ) ∈ [0, T ] × R3n , (z, θ) ∈ Kz,θ (t), |x − z| ≤ 1}, which we assume compact. Let Φ be a phase function smooth on an open set containing E1 and satisfying, for t ∈ [0, T ] and (z, θ) ∈ Kz,θ (t) ∂x Φ(t, z, z, θ) = θ, Φ(t, z, z, θ) is real, ∂x2 Φ(t, z, z, θ) has a positive definite imaginary part. (20) Then there exists r[Φ] ∈]0, 1] s.t. Im Φ(t, x, z, θ) ≥ C(x − z)2 for t ∈ [0, T ], (z, θ) ∈ Kz,θ (t) and |x − z| ≤ r[Φ]. Let lε ∈ C ∞ ([0, T ] × R3n , C) satisfying For t ∈ [0, T ], lε (t, x, z, θ) = 0 if (z, θ) ∈ / Kz,θ (t) or |x − z| > r[Φ], k k ∞ ε 2 ∂xb lε is uniformly bounded in L ([0, T ] × R3n ) for every 1 ≤ b ≤ n and k ∈ N. If Oα (lε (t, .), Φ(t, .)/ε) denotes, for a given multiindex α and t ∈ [0, T ], the operator [Oα (lε (t, .), Φ(t, .)/ε) h] (x) = Ú R2n h(z, θ)lε (t, x, z, θ)(x − z)α eiΦ(t,x,z,θ)/ε dzdθ, h ∈ L2 (R2n ), then, under the previous hypotheses on Φ and lε , we have the following estimate Proposition 2.5. [Lemma 3.3, chapter 1] |α| 3n ëOα (lε (t, .), Φ(t, .)/ε) ëL2 (R2n )→L2 (Rn ) . ε 4 + 2 uniformly w.r.t. t ∈ [0, T ]. 2.2.2 Expression of the phases Incident beams’ phases By the requirement (P.a) for the incident phase, one has d ψ (t, xt0 ) = ∂t ψ0 (t, xt0 ) + ∂x ψ0 (t, xt0 ) · x˙0 t = 0. Taking into account the initial null value dt 0 ψ0 (0, y) = 0 chosen in (18), one gets a null phase on the ray ψ0 (t, xt0 ) = 0. With the aim of computing ∂x2 ψ0 (t, xt0 ), let us examine the Jacobian matrix of the bicharacteristic F0t = Dϕt0 . The matrix F0t satisfies the linear ODE I d t F = JH(xt0 , ξ0t )F0t , dt 0 0 F0 = Id, Writing F0t as F0t = A ∂y xt0 ∂η xt0 ∂y ξ0t ∂η ξ0t B . , leads to the following system of ODEs on (U0t , V0t ) = (∂y xt0 + i∂η xt0 , ∂y ξ0t + i∂η ξ0t ) d t U = H21 (xt0 , ξ0t )U0t + H22 (xt0 , ξ0t )V0t , dt 0 d t V = −H11 (xt0 , ξ0t )U0t − H12 (xt0 , ξ0t )V0t . dt 0 (21) (22) 2.2 - Gaussian beams summation 73 Moreover, F0t is a symplectic matrix in that (F0t )T JF0t = J. Due to the symetry of the following matrices (∂y xt0 )T ∂y ξ0t , (∂η xt0 )T ∂η ξ0t , ∂y xt0 (∂η xt0 )T , and ∂y ξ0t (∂η ξ0t )T , and the relations (∂y xt0 )T ∂η ξ0t − (∂y ξ0t )T ∂η xt0 = Id and ∂y xt0 (∂η ξ0t )T − ∂η xt0 (∂y ξ0t )T = Id, one has (U0t )T V0t = (V0t )T U0t , (V0t )T Ū0t − (U0t )T V̄0t = 2iId, U0t is invertible. (23) Putting together (21), (22) and (23) shows that V0t (U0t )−1 is a symmetric matrix with a positive definite imaginary part and fulfills the Riccati equation (5) with initial value iId. Since this is the initial condition for ∂x2 ψ0 (t, xt0 ) given by (18), it follows that ∂x2 ψ0 (t, xt0 ) = V0t (U0t )−1 . Reflected beams’ phases The expression of the reflected phases ψk , k = ±1, is similar. In fact, since dtd ψk (t, xtk ) = 0 and ψk (Tk , xT0 k ) = ψ0 (Tk , xT0 k ) by (8), we get ψk (t, xtk ) = 0. The relation connecting the incident and the reflected phases stated in Lemma 2.2 gives at order one 1 2 ∂x2 ψ1 (t, xt1 ) ∂x r(xt0 , ξ0t ) + ∂ξ r(xt0 , ξ0t )∂x2 ψ0 (t, xt0 ) = ∂x λ(xt0 , ξ0t ) + ∂ξ λ(xt0 , ξ0t )∂x2 ψ0 (t, xt0 ), and one has a similar relation for ∂x2 ψ−1 (t, xt−1 ). One obtains by plugging the expression of ∂x2 ψ0 (t, xt0 ) and using (10) ∂x2 ψk (t, xtk ) = Vkt (Ukt )−1 where Ukt = ∂y xtk + i∂η xtk and Vkt = ∂y ξkt + i∂η ξkt . As ϕtk is symplectic, it follows that (Ukt , Vkt ) share the same properties (23) as (U0t , V0t ) (Ukt )T Vkt = (Vkt )T Ukt , (Vkt )T Ūkt − (Ukt )T V̄kt = 2iId, Ukt is invertible. 2.2.3 (24) Expression of the amplitudes Incident beams’ amplitudes Using (P.a) and the Hamiltonian system satisfied by (xt0 , ξ0t ), the equation (7) at order zero implies the following transport equation for the value of the amplitude on the ray [51] 1 d 0 (′ ) (′ ) a0 (t, xt0 ) + T r(H21 (xt0 , ξ0t ) + H22 (xt0 , ξ0t )∂x2 ψ0 (t, xt0 ))a00 (t, xt0 ) = 0, dt 2 (25) Asymptotic solution 74 which may be written, using the matrices U0t and V0t , as 51 6 ′ 2 d 0 (′ ) 1 t t t t t t t t −1 0 ( ) a (t, x0 ) + T r H21 (x0 , ξ0 )U0 + H22 (x0 , ξ0 )V0 (U0 ) a0 (t, xt0 ) = 0. dt 0 2 The time evolution for U0t , see (21), combined with the choice of the initial values ′ a00 (0, y) = 1 and a00 (0, y) = (−ic(y)|η|)−1 from (19), yields ′ 1 1 a00 (t, xt0 ) = [det U0t ]− 2 and a00 (t, xt0 ) = i(c(y)|η|)−1 [det U0t ]− 2 . Above the square root is defined by continuity in t from 1 at t = 0. reflected beams’ amplitudes The first reflected amplitudes evaluated on the ray satisfy a transport equation similar to (25), which may be written as 51 6 ′ 2 d k (′ ) 1 () a0 (t, xtk ) + T r H21 (xtk , ξkt )Ukt + H22 (xtk , ξkt )Vkt (Ukt )−1 ak0 (t, xtk ) = 0. dt 2 One can obtain a similar equation to (21) on Ukt involving H21 (xtk , ξkt ) and H22 (xtk , ξkt ), by using the relation ϕtk = ϕt0 o sk . On the whole (′ ) ak0 (t, xtk ) = (′ ) ak0 (Tk , xT0 k ) C det Ukt det UkTk D− 1 2 , k = ±1, where the square root is obtained by continuity from 1 at t = Tk . On the other hand, for k = ±1 d0−mB + dk−mB = b(x, ∂x ψ0 )a00 + b(x, ∂x ψk )ak0 , where b denotes the principal symbol of B. Thus, the condition (9) required for the (′ ) (′ ) construction of the reflected amplitudes implies that ak0 (Tk , xT0 k ) = sa00 (Tk , xT0 k ), with s = −1 for Dirichlet conditions and s = 1 for Neumann conditions. In order to find the relationship between UkTk and U0Tk for k = ±1, we differentiate the equality xTk k = xT0 k ∂y xTk k + ẋTk k (∂y Tk )T = ∂y xT0 k + ẋT0 k (∂y Tk )T , ∂η xTk k + ẋTk k (∂η Tk )T = ∂η xT0 k + ẋT0 k (∂η Tk )T , and compute the derivatives of Tk from the condition xT0 k ∈ ∂Ω ∂y,η Tk = − (x˙0 Tk 1 (∂y,η xT0 k )T ν(xT0 k ), Tk · ν(x0 )) to get after elementary computations UkTk = (Id − 2ν(xT0 k )ν(xT0 k )T )U0Tk . (26) Hence 1 ′ 1 ak0 (t, xtk ) = −si[det Ukt ]− 2 and ak0 (t, xtk ) = s(c|η|)−1 [det Ukt ]− 2 for k = ±1, 1 where the square root is defined by continuity from i[det U0Tk ]− 2 at t = Tk . The previous form of the beams is summarized in the following result 2.2 - Gaussian beams summation 75 Lemma 2.6. For k = 0, ±1, the incident and reflected beams ωεk have the form (′ ) (′ ) ωεk (t, x) = βk χd (x − xtk )ak (t)eiψk /ε , with β0 = 1, β1 = β−1 = −si, 1 1 ak (t) = [det Ukt ]− 2 , a′k (t) = i(c(y)|η|)−1 [det Ukt ]− 2 , i ψk = ξkt · (x − xtk ) + (x − xtk ) · Λk (t)(x − xtk ), Λk (t) = −iVkt (Ukt )−1 . 2 2.2.4 Gaussian integrals It follows that the approximate solution uappr ε,γ has the form (recall the dependence of Gaussian beams w.r.t. variables (y, η)) Ø 1 − 3n +1 Ú 1 ′ ε 4 cn χd (x − xtk )βk pε,k (t, y, η)eiψk (t,x,y,η)/ε ρ (y)γ ′ (η) uappr (t, x) = ε,γ 2n 2 R k=0,1 + ρ′ (y)γ ′ (η) Ø 2 iψk (−t,x,y,η)/ε χd (x − x−t dydη, k )βk qε,k (−t, y, η)e k=0,−1 with I pε,k (t, y, η) = ak (t, y, η)ε−1 Tε uIε,γ (y, η) + a′k (t, y, η)Tε vε,γ (y, η) I and qε,k (t, y, η) = ak (t, y, η)ε−1 Tε uIε,γ (y, η) − a′k (t, y, η)Tε vε,γ (y, η). (27) In the remainder of this section, we write the derivatives of the approximate solution using Gaussian type integrals Iε (h, Φ) that we define by 3n Iε (h, Φ)(t, x) = ε− 4 cn Ú R2n h(t, z, θ)eiΦ(t,x,z,θ)/ε dzdθ, for phase functions Φ ∈ C ∞ (Rn+1 t,x × B, C) satisfying properties (20) and polynomial )) supported for of order 2 in x − z and amplitude functions h ∈ C 0 ([0, T ], L2 (R2n z,θ s every fixed t ∈ [0, T ] in a compact of B. By Proposition 2.5, ë R2n h(t, z, θ)χ(x − z)eiΦ(t,x,z,θ)/ε dzdθëL2x . ëh(t, .)ëL2z,θ . Noting that the phase Φ provides with an exponentially decreasing function for |x − z| ≥ 1, one can use the following crude estimate ë Ú |x−z|≥a h(t, z, θ)eiΦ(t,x,z,θ)/ε dzdθëL2x . e−C/ε ëh(t, .)ëL2zθ for a > 0, (28) to deduce that Iε (h, Φ)(t, .) is uniformly bounded in L2x . The same notation Iε will be also used for vector valued functions h. For a function f depending on (t, x, z, θ) ∈ Rn+1 × B and k = 0, ±1, denote fåk (t, x, z, θ) = f (t, x; {ϕtk }−1 (z, θ)). k (t) = ϕtk (Ky × Kη ). Let Πk (t) be a cutoff of Cc∞ (R2n , [0, 1]) supported in B and Set Kz,θ k satisfying Πk (t) ≡ 1 on Kz,θ (t). We prove the following Lemma Asymptotic solution 76 2 n Lemma 2.7. ∂t uappr ε,γ (t, .) is uniformly bounded in L (R ) and satisfies √ 1 + 2 n − ∂t uappr ε,γ (t, x) = (vt (t, x) − vt (−t, x)) + O( ε) in L (R ) uniformly w.r.t. t ∈ [0, T ], 2 where vt+ and vt− are uniformly bounded sequences of L2 (Rn ) given by vt+ = Ø k=0,1 vt− = Ø k=0,−1 k k k k k æ ′ ⊗ γ′ p ç βk Iε (−ic(z)|θ|Πk ρ^ ε,k , ψk ), k æ ′ ⊗ γ′ q ç βk Iε (−ic(z)|θ|Πk ρ^ ε,k , ψk ). 2 n n Likewise, ∂x uappr ε,γ (t, .) is uniformly bounded in L (R ) and satisfies √ 1 + − 2 n n ∂x uappr ε,γ (t, x) = (vx (t, x) + vx (−t, x)) + O( ε) in L (R ) uniformly w.r.t. t ∈ [0, T ], 2 where vx+ and vx− are uniformly bounded sequences of L2 (Rn )n given by Ø vx+ = k=0,1 vx− = Ø k k k æ ′ ⊗ γ′ p ç βk Iε (iθΠk ρ^ ε,k , ψk ), k=0,−1 k k k æ ′ ⊗ γ′ q ç βk Iε (iθΠk ρ^ ε,k , ψk ). Proof. Because of (6), time derivative of uappr ε,γ may be written as a sum of integrals of the form 3n ε− 4 +j Ú R2n k ρ′ (y)γ ′ (η) fε (y, η)rj,α (t, x, y, η)(x − xtk )α eiψk (t,x,y,η)/ε dydη, with j, k = 0, 1 and |α| ≤ 2, arising from ∂t ωε0 (t, .) and ∂t ωε1 (t, .). Other terms of the same form come from the derivatives of ωε0 (−t, .) and ωε−1 (−t, .) w.r.t. t. fε stands for I k ε−1 Tε uIε,γ or Tε vε,γ and the rj,α are smooth functions vanishing for |x − xtk | ≥ d. The volume preserving change of variables (Ck ) : (z, θ) = ϕtk (y, η), writes the previous integral as 3n ε− 4 +j Ú R2n k 2 k k k1 æk (t,x)/ε α iψ k æ ^ ′ ⊗ γ′ f Πk (t)ρ^ dzdθ. ε rj,α (x − z) e (29) k æ is smooth and satisfies by (P.a)-(P.b)-(P.c) the following Then, the transported phase ψ k k (t) properties, for t ∈ [0, T ] and (z, θ) ∈ Kz,θ k æ (t, z, z, θ) = θ, ∂x ψ k k æ (t, z, z, θ) is real, ψ k k æ (t, z, z, θ) has a positive definite imaginary part. ∂x2 ψ k k k æ (t, x, z, θ) ≥ C(x − z)2 for t ∈ [0, T ], (z, θ) ∈ æ ] ∈]0, 1] so that Im ψ We fix some r[ψ k k k k æ Kz,θ (t) and |x − z| ≤ r[ψk ]. 2.2 - Gaussian beams summation 77 k 1 2 k k ′ ⊗ γ ′ (t, z, θ) ^ For t ∈ [0, T ], Πk ρ^ (t, x, z, θ) depends smoothly on its variables and rj,α k k æ ], (t). Hence, upon choosing d ≤ min r[ψ vanishes for |x − z| > d or (z, θ) ∈ / Kz,θ k k=0,±1 k 1 2 k k æ k and amplitude Π ρ^ ′ ⊗ γ′ ^ every phase ψ rj,α satisfy the properties formulated in k k k I Proposition 2.5. Let us check if 1B fæε is uniformly bounded in L2 (R2n ). Clearly Tε vε,γ I by Lemma B.5. One can then use the is, and the property holds true for ε−1 Tε vε,γ α approximation operators O to write the integral (29) as ε +j − 3n 4 Ú k R2n k æk α iψ ] k æk (r ′ ⊗ γ′ f Πk ρ^ ε j,α ) (x − z) e k /ε dzdθ k k 3n ] k æk æk ′ ⊗ γ ′ (r = ε− 4 +j Oα [Πk ρ^ j,α ) (t, .), ψk (t, .)/ε] 1B fε , The estimate established in Proposition 2.5 yields 3n ëε− 4 +j s k R2n k æk α iψ ] k æk (r ′ ⊗ γ′ f Πk ρ^ ε j,α ) (x − z) e k /ε dzdθëL2x . εj+ |α| 2 . k √ ] k Hence, only (r ) contributes to ∂t uappr ε. Since 0,0 ε,γ , the residue being of order i (′ ) k r0,0 (t, x, y, η) = cn βk ∂t ψk (t, xtk )χd (x − xtk )ak (t, y, η) 2 i (′ ) = − cn βk c(xtk )|ξkt |χd (x − xtk )ak (t, y, η), 2 by (P.a), it follows that ∂t uappr ε,γ (t, x) Ø k 1 − 3n Ú ′ ⊗ γ ′ (t, z, θ) (−i)βk c(z)|θ|χd (x − z)Πk (t, z, θ)ρ^ = ε 4 cn 2 R2n k=0,1 æ k k iψk (t,x,z,θ)/ε pç dzdθ ε,k (t, z, θ)e Ú Ø k 1 3n ′ ⊗ γ ′ (−t, z, θ) + ε− 4 c n iβk c(z)|θ|χd (x − z)Πk (−t, z, θ)ρ^ 2 R2n k=0,−1 k æ (−t,x,z,θ)/ε )dzdθ k iψk qç ε,k (−t, z, θ)e √ + O( ε), in L2 (Rn ), uniformly for t ∈ [0, T ]. Similar arguments apply to the spatial derivatives of uappr ε,γ and lead to ∂xb uappr ε,γ (t, x) Ø k 1 − 3n Ú ′ ⊗ γ ′ (t, z, θ) = ε 4 cn iβk (θk )b χd (x − z)Πk (t, z, θ)ρ^ 2 R2n k=0,1 æ k k iψk (t,x,z,θ)/ε pç dzdθ ε,k (t, z, θ)e Ú Ø k 1 3n ′ ⊗ γ ′ (−t, z, θ) + ε− 4 c n iβk (θk )b Πk (−t, z, θ)ρ^ 2 R2n k=0,−1 k æ (−t,x,z,θ)/ε )dzdθ k iψk qç ε,k (−t, z, θ)e √ + O( ε) in L2 (Rn ), uniformly w.r.t. t ∈ [0, T ]. appr One can get rid of the cutoff χd (x − z) appearing in ∂t uappr ε,γ (t, x) and ∂xb uε,γ (t, x) by using the estimate (28). Wigner transforms and measures 78 3 Wigner transforms and measures In this section, we compute the scalar measures associated to ∂t uappr ε,γ (t, .) and ± (t, .). As |β | = 1, the Wigner transforms associated to v c∂x uappr (±t, .) are finite k t ε,γ sums of terms of the form wε (Iε (ftk , Φk )(±t, .), Iε (gtl , Φl )(±t, .)), k l k l æ k and ′ ⊗ γ′ p ′ ′ çl , Φ = ψ ^ ç where k, l = 0, ±1, ftk = c|θ|Πk ρ^ ε,k , gt = c|θ|Πl ρ ⊗ γ q ε,l k k l æ . As regards the Wigner transforms associated to cv ± (±t, .), since c is uniformly Φl = ψ l x continuous on Rn , one has by a classical result ([39] p.8) wε (cvx± (±t, .), cvx± (±t, .)) ≈ c2 wε (vx± (±t, .), vx± (±t, .)) in R2n , so that the involved quantities have the form c2 wε (Iε (fxk , Φk )(±t, .), Iε (gxl , Φl )(±t, .)), k l k l ′ ⊗ γ′ p ′ ′ çl . Forgetting the powers of ε factors, ^ ç with fxk = θΠk ρ^ ε,k and gx = θΠl ρ ⊗ γ q ε,l the involved Wigner transform integrals have the form Ú R6n k l ç ′ ′ ′ ′ iΨ1 (z,θ,z ,θ ,x,v)/ε dzdθdz ′ dθ′ dxdv, Tç ε κε (z, θ)Tε τε (z , θ )b1 (z, θ, z , θ , x, v)e ′ ′ (30) I with κε , τε = ε−1 uIε,γ , vε,γ and k, l = 0, ±1, or by expanding the FBI transforms Ú R8n ′ ′ ′ κε (w)τε (w′ )b2 (z, θ, z ′ , θ′ , x, v)eiΨ2 (w,w ,z,θ,z ,θ ,x,v)/ε dwdw′ dzdθdz ′ dθ′ dxdv. (31) Traditionally, this type of oscillating integrals is estimated by the stationary phase theorem. This method was successfully used in [16] for the computation of a Wigner measure for smooth data. The involved phase is complex and its Hessian restricted to the stationary set is non-degenerate in normal direction to this set. In our case however, the amplitude is not smooth because no such assumption was made on uIε and v Iε . So we can not estimate immediately the global integral (31) by the same techniques. One research lead is to resort to the stationary phase theorem with a complex phase depending on s ′ ′ ′ parameters for estimating R6n b2 (z, θ, z ′ , θ′ , x, v)eiΨ2 (w,w ,z,θ,z ,θ ,x,v)/ε dzdθdz ′ dθ′ dxdv. An alternative method was used in [89], where an integral of the form (30) associated to the Wigner transform for the Schrödinger equation with a WKB initial condition was simplified by elementary computations into an integral over R4n . However the method faced difficulties in deducing the exact relation between the Wigner measure of the solution and of the initial data. We adapt the result of [89] to our problem in section 3.1 and complete the analysis to prove the propagation of the microlocal energy density of uappr ε,γ along the flow in section 3.2. The proof is simple and elementary and the computations are made in an explicit way. Section 3.3 is devoted to the Wigner measures associated to the derivatives of the exact solution uε of (1). 3.1 - Wigner transform for Gaussian integrals 3.1 79 Wigner transform for Gaussian integrals l k depend on ε but they are uniformly bounded in L2 (R2n ) and and gt,x The functions ft,x their support is contained in a fixed compact. Slight modifications of the computations of [89] lead to the following more general result Lemma 3.1. Let fε and gε be sequences uniformly bounded in L2 (R2n ) and supported in compacts independent of ε. Let F be an open set containing suppfε ∪ suppgε and Φ, Ψ be phase functions in C ∞ (Rnx × F, C) satisfying i Φ(x, z, θ) = rΦ (z, θ) + θ · (x − z) + (x − z) · HΦ (z, θ)(x − z) for (x, z, θ) ∈ Rn × F, 2 Ψ(x, z ′ , θ′ ) = i rΨ (z ′ , θ′ ) + θ′ · (x − z ′ ) + (x − z ′ ) · HΨ (z ′ , θ′ )(x − z ′ ) for (x, z ′ , θ′ ) ∈ Rn × F, 2 where rΦ , rΨ ∈ C ∞ (F, R) and the matrices HΦ , HΨ ∈ C ∞ (F, Mn (C)) with positive definite real parts. Then for φ ∈ Cc∞ (F, R) Ú φ(s, σ)A(Φ, Ψ)(s, σ) < wε (Iε (fε , Φ), Iε (gε , Ψ)), φ >= R4n √ √ √ √ fε (s + εr, σ + εδ)gε∗ (s − εr, σ − εδ)eiΘε (Φ,Ψ)(s,σ,r,δ) drdδdsdσ + o(1), where 5n n 1 A(Φ, Ψ)(s, σ) = c2n 2 2 π 2 (det[HΦ (s, σ) + H̄Ψ (s, σ)])− 2 , and Θε (Φ, Ψ)(s, σ, r, δ) = rΦ (s + √ √ √ εδ)/ε − rΨ (s − εr, σ − εδ)/ε √ − 2σ · r/ ε + i(r, δ) · Q(HΦ (s, σ), H̄Ψ (s, σ))(r, δ). εr,σ + √ The matrix Q(HΦ (s, σ), H̄Ψ (s, σ)) and the square root are defined in Lemma 3.2. Proof. The proof is given in two steps. In a first time, the Fourier transform of a Gaussian type function is computed explicitly. Then, in a second time, a Gaussian approximation is used for several smooth functions appearing in the Wigner transform integral. For simplicity we denote u(x, z, θ) by u and u(x, z ′ , θ′ ) by u′ when integrating w.r.t. z, θ, z ′ , θ′ . We also omit the index ε in the notation of fε and gε . Step 1. Fourier transform. Firstly, we note that the Wigner transform at point (x, ξ) ∈ R2n may be written as wε (Iε (f, Φ), Iε (g, Ψ))(x, ξ) = 5n π −n c2n ε− 2 Ú R5n ′ ′ ′ ′ f g ∗ ′ eirΦ /ε−irΨ /ε+ix·(θ−θ )/ε+i(θ ·z −θ·z)/ε ′ ′ ′ Fv [e−(v+x−z)·HΦ (v+x−z)/(2ε)−(v−x+z )·H̄Ψ (v−x+z )/(2ε) ]((2ξ − θ − θ′ )/ε) dvdzdz ′ dθdθ′ . The Fourier transform of Gaussian functions’ product is given by the following Lemma, of which proof is postponed to the end of this section Wigner transforms and measures 80 Lemma 3.2. Let a, b ∈ Rd and M, N ∈ Md (C) symmetric matrices with positive definite real parts, then Fx [e−(x−a)·M (x−a)/2 e−(x−b)·N (x−b)/2 ](ξ) = d 1 (2π) 2 (det[M + N ])− 2 e−iξ·(b+a)/2 e−(b−a,ξ)·Q(M,N )(b−a,ξ)/4 , where Q(M, N ) is the symmetric symplectic matrix given by Q(M, N ) = A 2M (M + N )−1 N i(N − M )(M + N )−1 i(M + N )−1 (N − M ) 2(M + N )−1 B , and the square root is defined as explained in section 3.4 ofB [46]. Moreover, A Id Id Q(M, N )A(M, N ) = B(M, N ) with A(M, N ) = and B(M, N ) = −iN iM A B N M , and Q(M, N ) has a positive definite real part −iId iId ∗−1 Re Q(M, N ) = 2A(M, N ) A Hence wε (Iε (f, Φ), Iε (g, Ψ))(x, ξ) = n n c2n 2 2 π − 2 ε−2n Re N 0 0 Re M Ú B A(M, N )−1 . 1 ′ f g ∗ ′ (det[HΦ + H̄Ψ′ ])− 2 eirΦ /ε−irΨ /ε R4n i(θ+θ′ −2ξ)·(z−z ′ )/(2ε)+i(θ−θ′ )·x/ε+i(θ′ ·z ′ −θ·z)/ε e ′ ′ ′ ′ ′ e−(2x−z−z ,2ξ−θ−θ )·Q(HΦ ,H̄Ψ )(2x−z−z ,2ξ−θ−θ )/(4ε) dzdz ′ dθdθ′ . Making the change of variables √ √ √ √ (z, z ′ ) = (s + εr, s − εr) and (θ, θ′ ) = (σ + εδ, σ − εδ), √ √ √ √ and writing f+ for f (s + εr, σ + εδ) and g− for g(s − εr, σ − εδ), leads to n 5n wε (Iε (f, Φ), Iε (g, Ψ))(x, ξ) = c2n 2 2 π − 2 ε−n Ú 1 f+ g ∗ − (det[HΦ+ + H̄Ψ − ])− 2 R4n √ √ irΦ+ /ε−irΨ− /ε+2iδ·(x−s)/ ε−2iξ·r/ ε e e−(x−s,ξ−σ)·Q(HΦ+ ,H̄Ψ − )(x−s,ξ−σ)/ε drdδdsdσ. Step 2. Gaussian approximations. Taking the duality product of the Wigner √ transform with a test function φ ∈ C0∞ (F, R), and after setting (x′ , ξ ′ ) = (x−s, ξ−σ)/ ε, one has Ú √ √ 5n n < wε (Iε (f, Φ), Iε (g,Ψ)), φ >= c2n 2 2 π − 2 φ(s + εx′ , σ + εξ ′ )f+ g ∗ − R6n √ − 21 irΦ+ /ε−irΨ− /ε−2iσ·r/ ε+2i(x′ ,ξ ′ )·(δ,−r) (det[HΦ+ + H̄Ψ − ]) e −(x′ ,ξ ′ )·Q(HΦ+ ,H̄Ψ − )(x′ ,ξ ′ ) e dx′ dξ ′ drdδdsdσ. (32) Let ρ′ f and ρ′ g be cut-off functions supported in F s.t. ρ′ f ≡ 1 on a fixed compact containing suppf and ρ′ g ≡ 1 on a fixed compact containing suppg, and consider è bε : (x′ , ξ ′ , s, σ, r, δ) Ô→ φ(s + √ εx′ , σ + √ é ′ ′ ′ ′ εξ ′ ) − φ(s, σ) ρ′ f + ρ′ g − e−(x ,ξ )·Q(HΦ+ ,H̄Ψ − )(x ,ξ ) . 3.1 - Wigner transform for Gaussian integrals 81 The r.h.s. of (32) may then be written as < wε (Iε (f, Φ), Iε (g, Ψ)), φ >= n 5n c2n 2 2 π − 2 Ú 1 R6n φ(s, σ)f+ g ∗ − (det[HΦ+ + H̄Ψ − ])− 2 eirΦ+ /ε−irΨ− /ε−2iσ·r/ ′ 5n n +c2n 2 2 π − 2 Ú √ ε+2i(x′ ,ξ ′ )·(δ,−r) ′ ′ ′ e−(x ,ξ )·Q(HΦ+ ,H̄Ψ − )(x ,ξ ) dx′ dξ ′ drdδdsdσ 1 R4n (det[HΦ+ + H̄Ψ − ])− 2 f+ g ∗ − √ eirΦ+ /ε−irΨ− /ε−2iσ·r/ ε F(x′ ,ξ′ ) bε (−2δ, 2r, s, σ, r, δ)drdδdsdσ. (33) Leibnitz formula yields for α multi-index ∂xα′ ,ξ′ bε (x′ , ξ ′ ,s, σ, r, δ) = é è é è √ √ ′ ′ ′ ′ ρ′ f + ρ′ g − φ(s + εx′ , σ + εξ ′ ) − φ(s, σ) ∂xα′ ,ξ′ e−(x ,ξ )·Q(HΦ+ ,H̄Ψ − )(x ,ξ ) è é Ø √ √ |β| ′ ′ ′ ′ β Cβ,γ ε 2 ∂x,ξ +ρ′ f + ρ′ g − φ(s + εx′ , σ + εξ ′ )∂xγ′ ,ξ′ e−(x ,ξ )·Q(HΦ+ ,H̄Ψ − )(x ,ξ ) . β+γ=α,βÓ=0 √ √ √ √ As (s + εr, σ + εδ) varies in suppρ′ f and (s − εr, σ − εδ) varies in suppρ′ g , one can find by continuity a constant C > 0 s.t. Re Q(HΦ+ , H̄Ψ − ) ≥ CId on supp(ρ′ f + ρ′ g − ). Since (s, σ) and √ ε(r, δ) are bounded on supp(ρ′ f + ρ′ g − ), (34) it follows that there exists a constant C ′ > 0 s.t. √ ′ ′ ′ 2 |∂xα′ ,ξ′ bε (x′ , ξ ′ , s, σ, r, δ)| . εe−C (x ,ξ ) for all (x′ , ξ ′ , s, σ, r, δ), which leads to |F(x′ ,ξ′ ) bε (−2δ, 2r, s, σ, r, δ)| . √ ε(1 + (r, δ)2 )−n−1 for all (s, σ, r, δ). The second integral on the r.h.s. of (33) is then dominated by √ Ú ε R4n |f+ ||g− |(1 + (δ, r)2 )−n−1 drdδdsdσ. We deduce with Cauchy-Schwartz inequality w.r.t. s, σ that | < wε (Iε (f, Φ), Iε (g, Ψ)), φ > 5n n −c2n 2 2 π 2 Ú R4n 1 φ(s, σ)(det[HΦ+ + H̄Ψ − ])− 2 f+ g ∗ − √ eirΦ+ /ε−irΨ− /ε−2iσ.r/ ε e−(δ,−r)·Q(HΦ+ ,H̄Ψ − ) √ . εëf ëL2 (R2n ) ëgëL2 (R2n ) , −1 (δ,−r) drdsdδdσ| where we have replaced det Q(HΦ+ , H̄Ψ − ) by 1 since Q(HΦ+ , H̄Ψ − ) is symplectic. To go further, let us extend HΦ and HΨ as λHΦ + (1 − λ)Id and λHΨ + (1 − λ)Id by using a cut-off λ ∈ Cc∞ (R2n , [0, 1]) supported in F s.t. λ ≡ 1 on the compact set Wigner transforms and measures 82 suppρ′ f ∪ suppρ′ g ∪ suppφ. The extended matrices have positive definite real parts. The smoothness of these matrices implies by the mean value theorem and (34) that √ 1 1 |(det[HΦ+ + H̄Ψ − ])− 2 − (det[HΦ (s, σ) + H̄Ψ (s, σ)])− 2 | . ε|(r, δ)| on supp(φf+ g ∗ − ). Using the symplecticity and the symmetry of Q(HΦ+ , H̄Ψ − ), its inverse is −JQ(HΦ+ , H̄Ψ − )J. −1 Thus the quantity |e−(δ,−r)·Q(HΦ+ ,H̄Ψ − ) (δ,−r) − e−(r,δ)·Q(HΦ (s,σ),H̄Ψ (s,σ))(r,δ) | is dominated by è é |(r, δ) · Q(HΦ+ , H̄Ψ − ) − Q(HΦ (s, σ), H̄Ψ (s, σ)) (r, δ)| sup |e−s(r,δ)·Q(HΦ+ ,H̄Ψ − )(r,δ)−(1−s)(r,δ)·Q(HΦ (s,σ),H̄Ψ (s,σ))(r,δ) |. s∈[0,1] The positivity of Re Q(HΦ+ , H̄Ψ − ) and Re Q(HΦ (s, σ), H̄Ψ (s, σ)) and the mean value theorem for the matrix function Q(λHΦ + (1 − λ)Id, λH̄Ψ + (1 − λ)Id) give by (34) √ −1 2 |e−(δ,−r)·Q(HΦ+ ,H̄Ψ − ) (δ,−r) −e−(r,δ)·Q(HΦ (s,σ),H̄Ψ (s,σ))(r,δ) | . ε|(r, δ)|3 e−C(r,δ) √ √ √ √ for (s, σ) ∈ suppφ, (s + εr, σ + εδ) ∈ suppρ′ f and (s − εr, σ − εδ) ∈ suppρ′ g . It follows that | < wε (Iε (f, Φ), Iε (g, Ψ)), φ > 5n n −c2n 2 2 π 2 Ú 1 R4n φ(s, σ)(det[HΦ + H̄Ψ ])− 2 (s, σ)f+ g ∗ − √ eirΦ+ /ε−irΨ− /ε−2iσ·r/ ε−(r,δ)·Q(HΦ ,H̄Ψ )(s,σ)(r,δ) drdδdsdσ| √ . εëf ëL2 (R2n ) ëgëL2 (R2n ) . Now we give the proof of the Lemma 3.2. Proof. The matrix M + N has a positive definite real part and is then non-singular. By elementary calculus we have (x − a) · M (x − a) + (x − b) · N (x − b) = (b − a) · M (M + N )−1 N (b − a) + (x − (M + N )−1 (M a + N b)) · (M + N )(x − (M + N )−1 (M a + N b)). Thus, using the value of the Fourier transform of a Gaussian function (see Theorem 7.6.1 of [46]), it follows that d 1 e−iξ.(M +N ) −1 (M a+N b)−ξ·(M +N )−1 ξ/2 Fx [e−(x−a)·M (x−a)/2 e−(x−b)·N (x−b)/2 ](ξ) =(2π) 2 (det[M + N ])− 2 e−(b−a)·M (M +N ) −1 N (b−a)/2 . Writing M = 1/2(M + N ) + 1/2(M − N ) and N = 1/2(M + N ) − 1/2(M − N ), we get the expression with the matrix Q(M, N ) and the relation Q(M, N )A(M, N ) = B(M, N ). One can easily show that T B(M, N ) JB(M, N ) = A 0 i(M + N ) −i(M + N ) 0 B = A(M, N )T JA(M, N ), 3.2 - Wigner measure for superposed Gaussian beams 83 from which follows the symplecticity of Q(M, N ). We then write Q(M, N ) + Q(M, N ) è é = A(M, N )∗−1 A(M, N )∗ B(M, N ) + B(M, N )∗ A(M, N ) A(M, N )−1 , = A(M, N )∗−1 A(M, N )−1 , and obtain the value of Re Q(M, N ). In the remainder of this paper we fix t ∈ [0, T ] and apply Lemma 3.1 with F = B to and (fxk , Φk ) (ftk , Φk ) and (gtl , Φl ) for the Wigner transforms associated with vt± (±t, .), and (gxl , Φl ) for the Wigner transforms associated with vx± (±t, .). 3.2 Wigner measure for superposed Gaussian beams k l In this section we prove that the cross Wigner transforms wε (Iε (ft,x , Φk ), Iε (gt,x , Φl )) with o ∗ 2 appr k Ó= l, do not contribute to w[∂t uappr ε,γ (t, .)] + c Trw[∂x uε,γ (t, .)] in T Ω. We compute Θε (Φk , Φ√ k ) and A(Φk , Φk ) and analyze the transported FBI transforms at points (s ± √ εr, σ± εδ). This enables to complete the study of the Wigner measure for superposed Gaussian beams. I have infinitely small contributions in It was pointed out in A3’ that Tε uIε,γ , Tε vε,γ I have infinitely L2 (Rn × supp(1 − γ ′ )). Besides, Lemma B.2 shows that Tε uIε,γ , Tε vε,γ I 2 ′ n ′ I . small contributions in L (supp(1 − ρ ) × R ) because ρ ≡ 1 on suppuε,γ and suppvε,γ Therefore wε (Iε (ftk , Φk ), Iε (gtl , Φl )) ≈ A(Φk , Φl ) Ú 1 k c|σ|Πk pç ε,k 2 3 + c|σ|Πl pç̄ ε,l l 4 o eiΘε (Φk ,Φl ) drdδ in T ∗ Ω, − and a similar relation holds true for wε (Iε (fxk , Φk ), Iε (gxl , Φl )). We start by approaching (c(s)|σ|)+ (c(s)|σ|)− by c(s)2 |σ|2 in wε (Iε (ftk , Φk ), Iε (gtl , Φl )) wε (Iε (ftk , Φk ), Iε (gtl , Φl )) 2 2 ≈ A(Φk , Φl )c(s) |σ| Ú 1 k Πk pç ε,k 2 3 o ∗ + eiΘε (Φk ,Φl ) drdδ in T Ω, Πl l pç̄ ε,l 4 − (35) ∗ by σσ ∗ in wε (Iε (fxk , Φk ), Iε (gxl , Φl )) and σ+ σ− wε (Iε (fxk , Φk ), Iε (gxl ,Φl )) ≈ A(Φk , Φl )σσ ∗ Ú 1 Πk pç ε,k k 2 3 + Πl l pç̄ ε,l 4 o eiΘε (Φk ,Φl ) drdδ in T ∗ Ω. − (36) These approximations result from the following Lemma Lemma 3.3. Let Φ, Ψ and fε , gε satisfy the hypotheses of Lemma 3.1. If α and β are in C 1 (F, C) then wε (Iε (αfε , Φ), Iε (βgε , Ψ)) ≈ αβ̄wε (Iε (fε , Φ), Iε (gε , Ψ)) in F. Wigner transforms and measures 84 Proof. The proof relies on the use of Taylor’s formula on ρ′ f α and ρ′ g β̄, where ρ′ f and ρ′ g are the cut-offs used in the proof of Lemma 3.1 (supported in F and equal to 1 on suppfε and suppgε respectively). It follows by (35) and (36) that o c2 Tr wε (Iε (fxk , Φk ), Iε (gxl , Φl )) ≈ wε (Iε (ftk , Φk ), Iε (gtl , Φl )) in T ∗ Ω, which leads to wε (vt± (±t, .),vt± (±t, .)) ≈ c2 Tr wε (vx± (±t, .), vx± (±t, .)), o and wε (vt± (±t, .), vt∓ (∓t, .)) ≈ c2 Tr wε (vx± (±t, .), vx∓ (∓t, .)) in T ∗ Ω. (37) The standard estimate (see Proposition 1.1 in [39]) | < wε (aε , bε ), φ > | . ëaε ëL2 (Rn ) ëbε ëL2 (Rn ) , for aε , bε ∈ L2 (Rn ) and φ ∈ Cc∞ (R2n , R), (38) given in Lemma 2.7 to leads by using the approximations of the derivatives of uappr ε,γ 2 appr 4(wε [∂t uappr ε,γ (t, .)]+c Trwε [∂x uε,γ (t, .)]) ≈ wε [vt+ (t, .)] + c2 Trwε [vx+ (t, .)] + wε [vt− (−t, .)] + c2 Trwε [vx− (−t, .)] − wε (vt+ (t, .), vt− (−t, .)) + c2 Tr wε (vx+ (t, .), vx− (−t, .)) − wε (vt− (−t, .), vt+ (t, .)) + c2 Tr wε (vx− (−t, .), vx+ (t, .)) in R2n . o The cross terms between v + and v − cancel in T ∗ Ω from (37), leading to o 1 1 ∗ + − 2 appr (t, .)] + c Trw [∂ u (t, .)] ≈ (t, .)] + (−t, .)] in T Ω. w [v w [v wε [∂t uappr ε x ε ε ε,γ t t ε,γ 2 2 (39) Thus, we are left with the computation of the Wigner measure associated with vt+ , computations being also similar for vt− . One has wε [vt+ ] ≈ Ø k,l=0,1 c(s)2 |σ|2 A(Φk , Φl ) Ú R2n 3 1 k Πk pç ε,k Πl pç̄ ε,l l 4 2 + o eiΘε (Φk ,Φl ) drdδ in T ∗ Ω. (40) − o Moreover the inverse of the reflected/incident flow in T ∗ Ω is a reflected/incident flow {ϕtk }−1 = ϕ−t −k , k = 0, 1. o −t Thus for (s, σ) ∈ T ∗ Ω, at most one of rays x−t −k (s, σ) and x−l (s, σ) is in Ω. Consequently, the contribution of cross terms between different Gaussian beams in (40) vanishes in o T ∗ Ω, and we are left with the computation when ε goes to zero of each of the following two distributions k k ç µtε,k = c(s)2 |σ|2 wε (Iε (Πk pç ε,k , Φk ), Iε (Πk p ε,k , Φk )), k = 0, 1. (41) 3.2 - Wigner measure for superposed Gaussian beams k 85 k k t æ′ k ^ I I ^ æk k ε−1 T Remember that pç ε,k = a ε uε,γ + ak Tε vε,γ , so µε,k may be written as µtε,k = c2 (s)|σ|2 wε (Iε (Πk aæk k ε−1 Tε uIε,γ , Φk ), Iε (Πk aæk k ε−1 Tε uIε,γ , Φk )) I I + wε (Iε (Πk aæk k Tε vε,γ , Φk ), Iε (Πk aæk k Tε vε,γ , Φk )) I − ic(s)|σ|wε (Iε (Πk aæk k ε−1 Tε uIε,γ , Φk ), Iε (Πk aæk k Tε vε,γ , Φk )) o I + ic(s)|σ|wε (Iε (Πk aæk k Tε vε,γ , Φk ), Iε (Πk aæk k ε−1 Tε uIε,γ , Φk )) in T ∗ Ω, (42) In the remainder of this section we prove the following result Proposition 3.4. Let κε , τε be unifromly bounded sequences in L2 (Rn ). Then o wε (Iε (Πk aæk k Tε κε , Φk ), Iε (Πk aæk k Tε τε , Φk )) ≈ Π2k wε (κε , τε ) o{ϕtk }−1 in T ∗ Ω. Above ϕtk is extended outside B as the identity. Proof. Computation of the phase Θε (Φk , Φk ) and the amplitude A(Φk , Φk ). o We consider (s, σ) ∈ T ∗ Ω and start from √ Θε (Φk , Φk )(s, σ, r, δ) = −2σ · r/ ε + i(r, δ) · Q A B k ǣ k æ Λ (t, s, σ), Λ (t, s, σ) (r, δ). k k The particularAform of Λk (t) B= −iVkt (Ukt )−1 , see Lemma 2.6, induces a similar form for k ǣ æ k (t), Λ the matrix Q Λ k k (t) Q A B k ǣ k æ Λ (t), Λ (t) Ut k k where Utk and Vkt are the 2n × 2n matrices  ǣt k Uk  Utk =  ǣt Vk k  æt k U k Væt k k = −iVkt , k    and Vkt =  B Utk  ǣt k k Væt −Vk k ǣt k æt k −U Uk k   . Replacing Ukt and Vkt by their definitions links Utk and Vkt to the Jacobian matrix Fkt Vkt so that = æt k JF k A Q −Id Id iId iId A As ϕtk o ϕ−t −k = Id, one has and B k ǣ k æ Λ (t), Λ (t) k k = æt k J −iF k 3 A æt k J F æt k = −J F k k −Id Id iId iId 4−1 B , . k æt F −t = Id. F −k k t Combining this A relation withBthe symplecticity of Fk , one gets the following relation for k ǣ æ k (t), Λ the matrix Q Λ k (t) k A k ǣ k B æ (t), Λ (t) = (F −t )T F −t . Q Λ k k −k −k Wigner transforms and measures 86 Therefore √ −t Θε (Φk , Φk ) = −2σ · r/ ε + i[F−k (r, δ)]2 . Moving to the amplitude A(Φk , Φk ) = 5n n c2n 2 2 π 2 A æk det[Λ (23) and (24) k ǣ k (43) B− 1 2 + Λk ] , one gets by using Λk (t) + Λ̄k (t) = 2(Ūkt )−1T (Ukt )−1 . Hence A(Φk , Φk ) = æt k -- . -det U k - n c2n 22n π 2 Plugging the form of the indident and reflected amplitudes in Lemma 2.6 and using the (′ ) C 1 smoothness of ak on B yields by Lemmas 3.1 and 3.3 wε (Iε (Πk aæk k Tε κε Φk ), Iε (Πk aæk k Tε τε , Φk )) ≈ n c2n 22n π 2 Ú 3 Πk k Tç κ ε ε 4 + 3 4 çk Πk Tε τε e−i2σ·r/ √ −t (r,δ)]2 ε−[F−k − t drdδ =: Jε,k (κε , τε ). It remains to analyze the most difficult terms in the amplitude, which involve transported FBI transforms 3 Πk k Tç κ ε ε 4 + √ √ εδ) = [Πk Tε κε oϕ−t −k ](s+ εr, σ + 3 Πk Tç ε τε k 4 − = [Πk Tε κε oϕ−t −k ](s − √ εr, σ − √ εδ). Analysis of the transported FBI transforms −t 2n 2n k ∞ Let ϑ−t −k be a map of Cc (R , R ) that coincides with ϕ−k on Kz,θ (t) ∪ suppφ √ (see Theorem 1.4.1 of [46]) and use Taylor’s formula for this map to get for (s ± εr, σ ± √ k (t) and (s, σ) ∈ suppφ εδ) ∈ Kz,θ 1 2 √ √ x± = x−t ± ε ∂y x−t r ± ε ∂η x−t −k −k −k δ + εrε , ± 2 1 √ √ −t −t −t −t r ± ε ∂η ξ−k δ + εrεξ± , = ξ−k ± ε ∂y ξ−k ξ−k x−t −k ± with (rεx± , rεξ± )(s, σ, r,δ) = The change of variables Ú 1 1 2 √ 2 α (s, (r, σ) ± u (r, δ)α (1 − u)∂y,η ϑ−t ε δ) du. −k α! 0 |α|=2 A o Ø r′ δ′ B = −t (s, σ) F−k A r δ B t (κε , τε )(s, σ) is appropriate. in Jε,k Notice that for (s, σ) ∈ T ∗ Ω one has the following relations [59] −u −u −u x−u −k y (s, σ) · ξ−k (s, σ) − σ = 0 and x−k η (s, σ) · ξ−k (s, σ) = 0 for u ∈ R. In fact, one can show that the derivatives of the previous equations w.r.t. u are zero. Besides, the equalities clearly hold true at u = 0 for k = 0 and at u = Tk (s, σ) for k = ±1, as a consequence of (26). Hence, it follows that −t −t −t ′ σ · r = ξ−k (s, σ) · (∂y x−t −k (s, σ)r + ∂η x−k (s, σ)δ) = ξ−k (s, σ) · r , 3.2 - Wigner measure for superposed Gaussian beams 87 o which leads in T ∗ Ω to Ú √ √ ′ ′ ′ −t εr + εrεx+ , ξ−k + εδ ′ + εrεξ+ ) (Πk )+ Tε κε (x−t −k + √ ′ √ ′ ′ −t (Πk )− Tε τε (x−t εr + εrεx− , ξ−k − εδ ′ + εrεξ− ) −k − n t Jε,k (κε , τε ) =c2n 22n π 2 −t ′ e−2iξ−k ·r / where √ ε−r′ 2 −δ ′ 2 dr′ dδ ′ , ′ (rεx ′ , rεξ )(s, σ, r′ , δ ′ ) = (rεx , rεξ )(s, σ, r, δ). o Let φ be a test function of Cc∞ (R2n , R) supported in T ∗ Ω. We want to use the change of t (κε , τε ), φ >, so we extend ϕtk outside variables (s, σ) = ϕtk (y, η) when computing < Jε,k t B by the identity and still denote it ϕk , so that ϕtk is now a one to one map from R2n to ϕtk (R2n ). Then Πk oϕtk and φ oϕtk belong to Cc∞ (R2n , R) and are supported in B. Expanding the FBI transforms gives 3n n Ú t φ oϕtk (y, η) < Jε,k (κε , τε ), φ > = c4n 22n π 2 ε− 2 6n √ R √ (Πk oϕtk )(y + εr′ + εRεx+ , η + εδ ′ + εRεξ+ ) √ √ (Πk oϕtk )(y − εr′ + εRεx− , η − εδ ′ + εRεξ− ) κε (z)τ̄ε (z ′ )eiη·(2 ξ+ eiRε √ √ εr′ +εRεx+ −εRεx− −z+z ′ )/ε+iδ ′ ·(2y−z−z ′ +εRεx+ +εRεx− )/ ε √ √ √ ·(y+ εr′ +εRεx+ −z)−iRεξ− ·(y− εr′ +εRεx− −z ′ )−(y+ εr′ +εRεx+ −z)2 /(2ε) e−(y− √ √ εr′ +εRεx− −z ′ )2 /(2ε)−2iη·r ′ / ε−r ′ 2 −δ ′ 2 where dr′ dδ ′ dzdz ′ dydη, ′ (Rεx , Rεξ )(y, η, r′ , δ ′ ) = (rεx ′ , rεξ )(s, σ, r′ , δ ′ ). We perform the following changes of variables (x, u) = ( z + z′ √ z + z′ z − z′ , ) and y ′ = (y − )/ ε 2 ε 2 to obtain < t Jε,k (κε , τε ), φ >= n c4n 22n π 2 Ú R6n ε ε κε (x + u)τ̄ε (x − u)dε eiγε −iη·u dr′ dδ ′ dxdudy ′ dη, 2 2 where dε (x, y ′ ,η, r′ , δ ′ ) √ √ √ √ ′ ′ εy ′ , η)(Πk oϕk )(x + εy ′ + εr′ + εRεx+ , η + εδ ′ + εRεξ+ ) √ √ √ ′ ′ (Πk oϕk )(x + εy ′ − εr′ + εRεx− , η − εδ ′ + εRεξ− ), = φ oϕtk (x + γε (x, y ′ , η, r′ ,δ ′ , u) √ √ ′ ′ ′ ′ = η · (Rεx+ − Rεx− ) + δ ′ · (2y ′ + εRεx+ + εRεx− ) √ √ √ u ′ ′ + εRεξ+ · (y ′ + r′ + εRεx+ − ε ) 2 √ ξ− ′ √ x− ′ √ u 2 2 ′ ′ − εRε · (y − r + εRε + ε ) + ir′ + iδ ′ 2 √ √ √ √ ′ ′ + i(y ′ + r′ + εRεx+ − εu/2)2 /2 + i(y ′ − r′ + εRεx− + εu/2)2 /2, Wigner transforms and measures 88 and ′ (Rεx ′ , Rεξ )(x, y ′ , η, r′ , δ ′ ) = (Rεx , Rεξ )(x + √ εy ′ , η, r′ , δ ′ ). Notice that dε (x, y ′ , η, r′ , δ ′ ) converges when ε → 0 to d0 (x, η) = φ oϕtk (x, η)(Πk oϕk )2 (x, η). ± On the other hand, remainder terms√in the Taylor expansions of √ since εrx are the √ ′ ′ −t x−k (s ± εr, σ ± εδ) at order 2, rεx+ − rεx− is of order ε and so is Rεx+ − Rεx− , leading to 2 2 2 γε (x, y ′ , η, r′ , δ ′ , u) → γ0 (y ′ , r′ , δ ′ ) = 2δ ′ · y ′ + iy ′ + 2ir′ + iδ ′ . ε→0 One has |< Ú ε ε κε (x + u)τ̄ε (x − u)d0 eiγ0 e−iη·u dr′ dδ ′ dudy ′ dxdη| . 26 2 Ú 5Ú ε ε |κε |(x + u)|τε |(x − u)dx 2 2 Rn iγε iγ0 ′ sup-Fη [dε e − d0 e ](x, y , u, r′ , δ ′ , u)--dr′ dδ ′ dudy ′ . (44) t (κε , τε ), φ Jε,k > n −c4n 22n π 2 x Cauchy-Schwartz inequality w.r.t. dx insures that the bracket integral is less than ëκε ëL2 ëτε ëL2 . Let us examin the term Ú - - sup--Fη [dε eiγε − d0 eiγ0 ](x, y ′ , u, r′ , δ ′ , u)--dr′ dδ ′ dudy ′ . x For fixed y ′ , u, r′ , δ ′ , the functions dε and d0 are compactly supported w.r.t. (x, η) so sup|Fη [dε eiγε − d0 eiγ0 ](x, y ′ , u, r′ , δ ′ , u)| . sup|[dε eiγε − d0 eiγ0 ](x, y ′ , η, r′ , δ ′ , u)|. x (x,η) Note that |dε eiγε − d0 eiγ0 | is dominated by |dε − d0 | + |d0 ||eiγε −iγ0 − 1|. The convergence of dε when ε → 0 to its limit d0 is uniform w.r.t. (x, η) and so is the convergence of γε to γ0 on the support of d0 . Thus dε eiγε converges to d0 eiγ0 uniformly w.r.t. (x, η). It follows that sup|Fη [dε eiγε − d0 eiγ0 ](x, y ′ , u, r′ , δ ′ , u)| → 0 for every y ′ , u, r′ , δ ′ . ε→0 x On the other hand, successive integrations by parts give Ú Rn dε eiγε e−iη·u dη = (1 + u2 )−n Ú Rn 1 2 L dε eiγε e−iη·u dη, with L a differential operator w.r.t. η, of order 2n. Thus, 1 2 sup|Fη [dε eiγε ](x, y ′ , u, r′ , δ ′ , u)| . (1 + u2 )−n sup max |∂ηα dε eiγε (x, y ′ , η, r′ , δ ′ , u)|, (45) x (x,η)|α|≤2n √ √ for every y ′ , u, r′ , δ ′ . The quantities (x + εy ′ , η) and ε(r′ , δ ′ ) are bounded on the ′ ′ support of dε , so Rεx± , Rεξ± and their derivatives w.r.t. η are dominated by (r′ , δ ′ )2 and for every multiindex α, there exists C > 0 s.t. |∂ηα dε | ≤ C, |∂ηα γε | ≤ C(r′ , δ ′ )2 (1 + |η| + |δ ′ | + (r′ , δ ′ )2 √ √ √ √ ′ ′ + |y ′ + r′ + εRεx+ − εu/2| + |y ′ − r′ + εRεx− − εu/2|) 3.2 - Wigner measure for superposed Gaussian beams 89 for all (x, y ′ , η, r′ , δ ′ ) ∈ suppdε and u ∈ Rn . Thus, there exists C, C ′ > 0 s.t. ′ ′ ′ |∂ηα [dε eiγε ]| ≤ Ce−C (y +r + ′ ′ ≤ Ce−C (2y + √ √ √ ′ √ ′ √ εRεx+ − εu/2)2 −C ′ (y ′ −r′ + εRεx− + εu/2)2 −C ′ r ′ 2 −C ′ δ ′ 2 ′ √ ′ εRεx+ + εRεx− )2 −C ′ r ′ 2 −C ′ δ ′ 2 √ ′ for all (x, y ′ , η, r′ , δ ′ ) ∈ suppdε and u ∈ Rn . On the support of dε , εRεx± are dominated by |(r′ , δ ′ )|, which implies for some C0 > 0 that √ √ ′ ′ 2 (2y ′ + εRεx+ + εRεx− )2 ≥ 4y ′ − C0 |(r′ , δ ′ )||y ′ |. ′ ′ √ x+ ′ √ x− ′ 2 ′′ ′ 2 Hence, if |y ′ | ≥ C0 |(r′ , δ ′ )|, e−C (2y + εRε + εRε ) ≤ e−C y . Otherwise, ′′ ′ 2 ′′ ′ 2 ′′ ′ 2 ′ ′2 ′ ′2 e−C r −C δ ≤ e−C y −C r −C δ . In all cases, there exists C ′ , C ′′ > 0 s.t. |∂ηα [dε eiγε ]| ≤ C ′ e−C ′′ y ′ 2 −C ′′ r ′ 2 −C ′′ δ ′ 2 for every x, y ′ , η, r′ , δ ′ , u and ε ∈], ε0 ] with some ε0 > 0. Using this in (45) leads to sup|Fη [dε eiγε ](x, y ′ , u, r′ , δ ′ , u)| . (1 + u2 )−n e−Cy x ′ 2 −Cr ′ 2 −Cδ ′ 2 , and repeating the same arguments for sup|Fη [d0 eiγ0 ]| gives x sup|Fη [dε eiγε − d0 eiγ0 ](x, y ′ , u, r′ , δ ′ , u)| . (1 + u2 )−n e−Cy x ′ 2 −Cr ′ 2 −Cδ ′ 2 for every x, y ′ , η, r′ , δ ′ , u and ε ∈], ε0 ]. By the dominated convergence theorem, one obtains Ú sup|Fη [dε eiγε − d0 eiγ0 ](x, y ′ , u, r′ , δ ′ , u)|dy ′ dudr′ dδ ′ → 0. ε→0 x t (κε , τε ), one finally has by plugFrom the inequality (44) concerning the distribution Jε,k ging the expressions of d0 and γ0 Ú ε ε t 4 2n n 2 d0 κε (x + u)τ̄ε (x − u) < Jε,k (κε , τε ), φ >= cn 2 π 6n 2 2 R ′ ′ ′ 2iδ ′ ·y ′ −y ′ 2 −2r ′ 2 −δ ′ 2 −iη·u e dr dδ dxdudy dη + o(1). e Integration w.r.t. r′ , δ ′ , y ′ yields t < Jε,k (κε ,τε ), φ > = (2π)−n Ú é è ε ε Fη Π2k oϕtk φoϕtk (x, u)κε (x + u)τ̄ε (x − u)dxdu + o(1). 2 2 R2n The integral in the l.h.s. Π2k oϕtk φoϕtk . is exactly the Wigner transform of (κε , τε ) tested on One gets by using Proposition 3.4, Lemma B.6 and the expression (42) of µtε,k 1 2 o I µtε,k ≈ Π2k wε [vε,γ − ic|D|uIε,γ ] o{ϕtk }−1 in T ∗ Ω. Recalling the relation between the Wigner measure and the FBI transform (see Proposition 1.4 of [38]) Ú |Tε aε |2 θdydη → < w[aε ], θ > ε→0 for θ ∈ Cc∞ (R2n , R) and (aε ) uniformly bounded in L2 (Rn ), (46) Wigner transforms and measures 90 I − ic|D|uIε,γ ] ≈ 0 in (Ky × Kη )c or equivalently it follows that wε [vε,γ I k wε [vε,γ − ic|D|uIε,γ ] o{ϕtk }−1 ≈ 0 in (Kz,θ (t))c . k (t), one deduces Since Πk ≡ 1 on Kz,θ o I − ic|D|uIε,γ ] o{ϕtk }−1 in T ∗ Ω. µtε,k ≈ wε [vε,γ By summing over k = 0, 1 and letting ε → 0, we get the transport on the incident and I − ic|D|uIε,γ ] the reflected flows of µ[vε,γ Ø w[vt+ (t, .)] = k=0,1 o I µ[vε,γ − ic|D|uIε,γ ] o{ϕtk }−1 in T ∗ Ω, For u ∈ [−T, T ] and (y, η) ∈ Ky × (Rn \{0}), the incident and reflected flows are related to the broken bicharacteristic flow associated to −i∂t + c|D| as follows ϕub (y, η)    ϕu−1 (y, η) if u < T−1 (y, η) =  ϕu0 (y, η) if T−1 (y, η) < u < T1 (y, η)  u ϕ1 (y, η) if u > T1 (y, η). We define ϕub in (Ω\Ky ) × (Rn \{0}) by successively reflecting the rays at the boundary. We extend ϕub at times of reflections arbitrary. As only one incident/reflected ray can be in the interior of the domain at a fixed time φ oϕtb = Ø k=0,1 φ oϕtk in Ky × Rn \{0}. It follows that 1 I w[vt+ (t, .)] = µ[vε,γ − ic|D|uIε,γ ] o ϕtb 2−1 o in T ∗ Ω. The computations for vt− are similar. One has just to replace the index k = 1 by k k ç k = −1 and pç ε,k by q ε,k in equations (40), (41). Set I I ± I I Υ± ε = v ε ± ic|D|uε and Υε,γ = vε,γ ± ic|D|uε,γ . One gets 1 −t w[vt− (−t, .)] = w[Υ+ ε,γ ] o ϕb 2−1 o in T ∗ Ω. Plugging these results in the expression (39) of the scalar Wigner measure associated to uappr ε,γ leads to appr w[∂t uappr ε,γ (t, .)]+Trw[c∂x uε,γ (t, .)] 2 1 1 2−1 o 1 1 −t −1 − t ∗ = w[Υ+ + ] o ϕ w[Υ ] o ϕ in T Ω. ε,γ ε,γ b b 2 2 (47) 3.3 - Proof of the main theorem 3.3 91 Proof of the main theorem A consequence of the estimate (38) is | < w(aε , bε ), θ > | . lim ëaε ëL2 (Ω) lim ëbε ëL2 (Ω) , ε→0 (48) ε→0 o for aε , bε uniformly bounded in L2 (Rn ) and θ ∈ Cc∞ (T ∗ Ω, R). Using this estimate (38) on the difference between the derivatives of the exact and approximate solutions of the IBVP (1a)- (1b) with initial conditions (1c’), one deduces the measures associated to ∂t uε,γ and ∂x uε,γ and gets by (47) 2 1 1 2−1 o 1 1 −t −1 t w[∂t uε,γ (t, .)] + Trw[c∂x uε,γ (t, .)] = w[Υ+ + w[Υ− in T ∗ Ω. ε,γ ] o ϕb ε,γ ] o ϕb 2 2 Remark 3.5. Gaussian beams summation of first order beams allows to compute the Wigner measure for the solution of the IBVP (1), under hypothesis (A1)-(A3) on initial conditions. Summation of higher order beams may imply asymptotic formulas for the Wigner transform. Higher order terms in the Wigner transform’s expansion were studied for instance in [84] and [30] for WKB initial data. Let us now study the scalar Wigner measure for the problem (1), by making the I ) approach (uIε , vεI ). The contribution of the sets {η ∈ Rn , |η| ≥ r∞ /4} data (uIε,γ , vε,γ and {η ∈ Rn , |η| ≤ 4r0 } where γ Ó≡ 1 (remember the definition of γ in (14)) is controlled asymptotically by assumptions C2 and C3 respectively. Denote φt = φoϕtb , then φt ∈ Cc∞ (R2n , R). One has - 1 1 −t t < w[∂t uε (t, .)] + Trw[c∂x uε (t, .)], φ > − < w[Υ+ > − < w[Υ− ε ], φ ε ], φ > - ≤ 2 2 | < w[∂t uε (t, .)] − w[∂t uε,γ (t, .)], φ > | + n Ø b=1 | < w[c∂xb uε (t, .)] − w[c∂xb uε,γ (t, .)], φ > | 1 1 −t t > − < w[Υ− +| < w[∂t uε,γ (t, .)] + Trw[c∂x uε,γ (t, .)], φ > − < w[Υ+ ε,γ ], φ ε,γ ], φ > | 2 2 1 1 + −t − t > | + | < w[Υ− (49) + | < w[Υ+ ε,γ ] − w[Υε ], φ ε,γ ] − w[Υε ], φ > |. 2 2 We use (38) to get 1 + −t + + + | < w[Υ+ > | . lim ëΥ+ ε,γ ] − w[Υε ], φ ε,γ − Υε ëL2 (Rn ) lim ëΥε,γ ëL2 (Rn ) + ëΥε ëL2 (Rn ) ε→0 . ε→0 lim ëvεI ε→0 − I vε,γ ëL2 (Ω) + lim ëuIε − uIε,γ ëH 1 (Ω) . 2 ε→0 Similarly, by (48) | < w[∂t uε (t, .)] − w[∂t uε,γ (t, .)], φ > | . lim ë∂t uε (t, .) − ∂t uε,γ (t, .)ëL2 (Ω) (lim ë∂t uε (t, .)ëL2 (Ω) + lim ë∂t uε,γ (t, .)ëL2 (Ω) ), ε→0 ε→0 ε→0 and for b = 1, . . . , n | <w[∂xb uε (t, .)] − w[∂xb uε,γ (t, .)], φ > | . 3 4 lim ë∂xb uε (t, .) − ∂xb uε,γ (t, .)ëL2 (Ω) lim ë∂xb uε (t, .)ëL2 (Ω) + lim ë∂xb uε,γ (t, .)ëL2 (Ω) . ε→0 ε→0 ε→0 Wigner transforms and measures 92 The solution of the IBVP for the wave equation is given by a continuous unitary evolution group on the space H 1 (Ω, dx) × L2 (Ω, dx). Hence I ë∂t uε (t, .) − ∂t uε,γ (t, .)ëL2 (Ω) . ëvεI − vε,γ ëL2 (Ω) + ëuIε − uIε,γ ëH 1 (Ω) , I ë∂xb uε (t, .) − ∂xb uε,γ (t, .)ëL2 (Ω) . ëvεI − vε,γ ëL2 (Ω) + ëuIε − uIε,γ ëH 1 (Ω) , b = 1, . . . , n. We then have by (47) 1 1 −t t > − < w[Υ− < w[Υ+ ε ], φ ε ], φ > | . 2 2 I limëvεI − vε,γ ëL2 (Ω) + limëuIε − uIε,γ ëH 1 (Ω) . | < w[∂t uε ] + Trw[c∂x uε ], φ > − ε→0 (50) ε→0 We therefore need to estimate the difference between initial data (1c) and (1c’). We start by the initial speed. By the exponential decrease of Tε∗ γTε v Iε on the support of 1 − ρ (16), one has I ëvεI − vε,γ ëL2 (Ω) . ε∞ + ëv Iε − Tε∗ γTε v Iε ëL2 (Rn ) . Because Tε∗ is bounded on L2 (R2n ) → L2 (Rn ) and Tε∗ Tε = Id ëv Iε − Tε∗ γTε v Iε ëL2 (Rn ) ≤ ë(1 − χr∞ /2 )Tε v Iε ëL2 (R2n ) + ëχr∞ /2 χ4r0 Tε v Iε ëL2 (R2n ) . ① ② Firstly, writing the expression of the FBI transform given in Lemma B.1 as the Fourier transform of some auxiliary function, it follows by Parseval equality that ëε −n 4 (1 − χr∞ /2 (η)) Ú Fv Iε (ξ)eiξ·y−(η−εξ) n (2π)n ε− 2 n +(2π)n ε− 2 2 /(2ε) Ú |εξ|≤r∞ /4 Ú |εξ|≥r∞ /4 dξë2L2 (R2n ) = (1 − χr∞ /2 (η))2 |Fv Iε (ξ)|2 e−(η−εξ) 2 /ε (1 − χr∞ /2 (η))2 |Fv Iε (ξ)|2 e−(η−εξ) 2 /ε dξdη dξdη. The first integral in the r.h.s. is exponentially decreasing, which leads to lim ① . lim ε→0 ε→0 1Ú |εξ|≥r∞ /4 |Fv Iε (ξ)|2 dξ 21 2 . Secondly, as dist(suppvεI , supp(1 − ρ)) > 0, one gets ë(1 − ρ)Tε v Iε ëL2 (R2n ) ≤ e−C/ε by Lemma B.2 and thus ëχr∞ /2 (η)χ4r0 (η)Tε v Iε ëL2 (R2n ) . ε∞ + ëρ(y)χr∞ /2 (η)χ4r0 (η)Tε v Iε ëL2 (R2n ) . It results from the relation (46) applied to aε = vεI that (②)2 → < w[vεI ], ρ2 ⊗ χ24r0 χ2r∞ /2 > . ε→0 Because w[v Iε ] is a regular measure, assumption C3 yields ∀α > 0, ∃l0 (α) > 0 s.t. w[v Iε ]({|ξ| ≤ l0 (α)}) ≤ α. One deduces, for 4r0 ≤ l0 (α), that lim ② . ε→0 √ α, (51) 3.3 - Proof of the main theorem 93 which leads to limëv I ε→0 ε − I vε,γ ëL2 (Ω) . lim ε→0 AÚ |εξ|≥r∞ /4 |FvεI (ξ)|2 dξ B1 2 + √ α. Moving to the difference between uIε and uIε,γ in H 1 (Ω), we begin by estimating the spatial derivatives of the difference. It follows, by the formula of the inverse FBI transform’s derivative given in (17), that ∂xb uIε − ∂xb uIε,γ = ∂xb uIε − (∂xb ρ)Tε∗ γTε uIε − ρTε∗ γ∂yb Tε uIε . The term involving the derivative of ρ is exponentially decreasing by Lemma B.3. Since the FBI transform of a derivative is a derivative of the FBI transform by (17), one has to estimate ë∂xb uIε − ρTε∗ γTε ∂xb uIε ëL2 (Ω) . Employing the same previous techniques yields for b = 1, . . . , n limë∂xb uIε − ∂xb uIε,γ ëL2 (Ω) . lim ε→0 ε→0 AÚ |εξ|≥r∞ /4 |F∂xb uIε (ξ)|2 dξ if 4r0 ≤ lb (α) and w[∂xb uIε ]({|ξ| ≤ lb (α)}) ≤ α. Set r0 = B1 2 1 min l (α), 4 0≤b≤n b + √ α, then the Poincaré inequality yields the same bound for ëuIε − uIε,γ ëL2 (Ω) . Coming back to (50) we deduce that 1 1 −t t | < w[∂t uε (t, .)] + Trw[c∂x uε (t, .)], φ > − < w[Υ+ > − < w[Υ− ε ], φ ε ], φ > | 2 2 Ú è é 1 21 √ |F v Iε (ξ)|2 dξ 2 . α + lim ε→0 ε|ξ|≥r∞ /4 + n 1 Ø lim Ú ε→0 ε|ξ|≥r∞ /4 b=1 è é |F ∂xb uIε (ξ)|2 dξ 21 2 . (52) The assumption C2 of ε−oscillation means by definition that lim Ú ε→0 ε|ξ|≥r∞ /4 è é |F v Iε (ξ)|2 dξ → r∞ →+∞ lim Ú 0, ε→0 ε|ξ|≥r∞ /4 è é |F ∂xb uIε (ξ)|2 dξ → r∞ →+∞ 0 for b = 1, . . . , n. (53) Since the l.h.s. of (52) does not depend on α nor r∞ , one deduces by passing to the limits α → 0 and r∞ → ∞ that 2 1 1 2−1 o 1 1 −t −1 − w[∂t uε (t, .)] + Trw[c∂x uε (t, .)] = w[Υ+ + in T ∗ Ω. ] o ϕ w[Υ ] o ϕtb ε ε b 2 2 Proof of the relation between incident and reflected beams’ phases 94 A Proof of the relation between incident and reflected beams’ phases Let A(t, x, ξ) = ∂x r(x, ξ) + ∂ξ r(x, ξ)∂x2 ψinc (t, x) and B(t, x, ξ) = ∂x λ(x, ξ) + ∂ξ λ(x, ξ)∂x2 ψinc (t, x). One can dispose of a phase function θ ∈ C ∞ (Rt × Rnx , C) s.t. R−1 ∂x θ(t, r(x, ∂x ψinc )) ≍ t λ(x, ∂x ψinc ), (A.1) x=x0 if A(t, xt0 , ξ0t ) is non singular and R−2 B(t, x, ∂x ψinc )A(t, x, ∂x ψinc )−1 ≍ t A(t, x, ∂x ψinc )T −1 B(t, x, ∂x ψinc )T . x=x0 (A.2) From (10) one gets A(t, xt0 , ξ0t )(∂y xt0 + i∂η xt0 ) = ∂y xt1 + i∂η xt1 . Since ϕt1 is symplectic, the matrix A ∂y xt1 ∂η xt1 ∂y ξ1t ∂η ξ1t B is symplectic. This implies in par- ticular the relation ∂η ξ1t (∂y xt1 )T − ∂y ξ1t (∂η xt1 )T = Id, and the symmetry of ∂y xt1 (∂η xt1 )T . Thus, ker(∂η xt1 )T ∩ ker(∂y xt1 )T = {0} and at the same time, (∂y xt1 + i∂η xt1 )(∂y xt1 + i∂η xt1 )∗ = ∂y xt1 (∂y xt1 )T + ∂η xt1 (∂η xt1 )T . This proves that ∂y xt1 + i∂η xt1 is invertible and so is A(t, xt0 , ξ0t ). On the other hand, A Let M(x, ξ) = A A B B = A ∂x r ∂ξ r ∂x λ ∂ξ λ T [A B − B A] = A Id ∂x2 ψinc B . B ∂x r(x, ξ) ∂ξ r(x, ξ) ∂x λ(x, ξ) ∂ξ λ(x, ξ) T BA . Then A Id ∂x2 ψinc BT T M JM A Id ∂x2 ψinc B B 0 Id where is J = is the standard symplectic matrix. −Id 0 DsT1 JDs1 , the symplecticity of s1 leads to , Since MT JM = MT JM = J. Hence T T [A B − B A] = A Id ∂x2 ψinc BT J A Id ∂x2 ψinc B = 0, and the requirement (A.2) is fulfilled. Using the compatibility conditions é1 2ï dî α è ∂t,x f (t,x, ∂x ψinc (t, x)) t, xt0 dt 1 2 1 2 α α = ∂t ∂t,x [f (t, x, ∂x ψinc (t, x))] t, xt0 + ∂x ∂t,x [f (t, x, ∂x ψinc (t, x))] t, xt0 ẋt0 3.3 - Proof of the main theorem 95 on the maps (t, x, ξ) Ô→ ∂x θ(t, r(x, ξ)) and (x, ξ) Ô→ λ(x, ξ) yields recursively on |α| ≤ R−1 R−2 ∂t ∂x θ(t, r(x, ∂x ψinc ))+∂x2 θ(t, r(x, ∂x ψinc ))∂ξ r(x, ∂x ψinc )∂t ∂x ψinc ≍ t ∂ξ λ(x, ∂x ψinc )∂t ∂x ψinc . x=x0 On the other hand R−2 ∂x2 θ(t, r(x, ∂x ψinc )) ≍ t [BA−1 ](t, r(x, ∂x ψinc )). x=x0 Using (A.2), one gets ∂t ∂x θ(t, r(x,∂x ψinc )) R−2 ≍ [∂ξ λ(x, ∂x ψinc ) − (A−1T B T )(t, r(x, ∂x ψinc ))∂ξ r(x, ∂x ψinc )]∂t ∂x ψinc . x=xt0 Since T T A ∂ξ λ − B ∂ξ r = A Id ∂x2 ψinc BT T M JM it follows that A 0 Id B = Id, R−2 ∂t ∂x θ(t, r(x, ∂x ψinc ))A(t, x, ∂x ψinc ) ≍ t ∂t ∂x ψinc . x=x0 Setting ∂t θ(t, r(xt0 , ξ0t )) = ∂t ψinc (t, xt0 ) implies then that R−1 ∂t θ(t, r(x, ∂x ψinc )) ≍ t ∂t ψinc . (A.3) x=x0 Putting together (11),(A.1),(A.3) and the eikonal equation satisfied by ψinc , the phase θ satisfies R−1 p(r(x, ∂x ψinc ), ∂t θ(t, r(x, ∂x ψinc )), ∂x θ(t, r(x, ∂x ψinc ))) ≍ t 0. x=x0 In fact, the relation holds true also at order R. To see this, let u(t, x) = p(x, ∂t θ(t, x), ∂x θ(t, x)). The formula of composite functions’ high derivatives yields for |α| = R è é ∂xα u(t, r(x, ∂x ψinc (t, x))) = Ø ∂xβ u(t, r(x, ∂x ψinc (t, x)))vβ (t, x) + zα (t, x), |β|=R where zα depends on derivatives of u of order lower than R. By Remark 2.1, the terms ∂xβ u(t, xt1 ) involve partial derivatives of θ of order at most R, so one can substitute for them from (A.1) and (A.3) to obtain R p(r(x, ∂x ψinc ), ∂t θ(t, r(x, ∂x ψinc )), ∂x θ(t, r(x, ∂x ψinc ))) ≍ t 0. x=x0 (A.4) To compare time and tangential derivatives of θ and ψinc at (T1 , xT0 1 ), let us introduce a C ∞ parametrization of a neighbourhood U of xT0 1 in ∂Ω σ : N → Rn , Results related to the FBI and the Wigner transforms 96 where N is an open subset Rn−1 , σ(N ) = U and σ is a diffeomorphism from N to U. For x ∈ Rn close to xT0 1 , we may write x = σ(v̂) + vn ν(σ(v̂)), with v̂ ∈ N and vn ∈ R. Denote σ(v̂1 ) = xT0 1 and set θb (t, v̂) = θ(t, σ(v̂)) and ψinc b (t, v̂) = ψinc (t, σ(v̂)) the phases o at the boundary near xT0 1 . Since r(X, Ξ) = X for (X, Ξ) ∈ T ∗ Rn |∂Ω , it follows that r(σ(v̂), ∂x ψinc (t, σ(v̂))) ∞ ≍ (t,v̂)=(T1 ,v̂1 ) σ(v̂), which implies by (A.3) that ∂t θb R−1 ≍ (t,v̂)=(T1 ,v̂1 ) ∂t ψinc b . o Similarly λ(X, Ξ) = Ξ − 2(Ξ · ν(X))ν(X) for (X, Ξ) ∈ T ∗ Rn |∂Ω , leading to Dσ(v̂) · λ(σ(v̂), ∂x ψinc (t, σ(v̂))) ∞ ≍ (t,v̂)=(T1 ,v̂1 ) Dσ(v̂) · ∂x ψinc (t, σ(v̂)). Since ∂v̂ θb (t, v̂) = Dσ(v̂) · ∂x θ(t, σ(v̂)) and a similar relation holds true for ∂v̂ ψinc b , one gets from (A.1) ∂v̂ θb R−1 ≍ (t,v̂)=(T1 ,v̂1 ) ∂v̂ ψinc b . Hence θb and ψinc b have the same time and tangential derivatives at (T1 , v̂1 ) from order 1 to order R − 1. If we add to the relation (A.1) the condition θ(T1 , xT0 1 ) = ψinc (T1 , xT0 1 ), then the phase θ satisfies the same requirements that uniquely determine the reflected phase ψref for a fixed phase ψinc (see Remark 2.1 for the uniqueness the spatial derivatives involving a normal derivation being deduced from time and tangential derivatives). The two phases are thus equal on (t, xt1 ) up to the order R. B Results related to the FBI and the Wigner transforms Lemma B.1. For u in L2 (Rn ) n n Tε u(y, η) = ε− 4 cn (2π)− 2 Ú Rn Fu(ξ)eiξ·y−(η−εξ) 2 /(2ε) dξ. Proof. The equality is proven by Parseval formula. Lemma B.2. Lemma 2.4 of chapter 1 Let a be a positive real and G a measurable subset of Rn s.t. dist(G, K) ≥ a. If u ∈ L2 (Rnx ) is supported in K then n ë1G (y)Tε uëL2y,η = cn ε− 4 ë1G (y)u(x)e−(x−y) 2 /(2ε) ëL2xy . e−a 2 /(4ε) ëuëL2x . Proof. The proof consists of writing the FBI transform as the Fourier Transform of some auxiliary function and using Parseval equality. 3.3 - Proof of the main theorem 97 Lemma B.3. Let θ be a cut-off of Rn and u ∈ L2 (Rn ) compactly supported. If E is a measurable subset of Rn s.t. dist(E, suppu) > 0, then ëTε∗ θ(η)Tε uëL2 (E) . e−C/ε ëuëL2 (Rn ) . Proof. The kernel of Tε∗ θTε is ′ Kε (x, x ) = 3n ε− 2 c2n Ú ′ ′ 2 /(2ε)−(x−y)2 /(2ε) θ(η)eiη·(x−x )/ε−(y−x ) = ε−n (2π)−n Fθ( dydη x′ − x −(x′ −x)2 /(4ε) )e . ε Cauchy-Schwartz inequality yields ëTε∗ θTε uë2L2 (E) −n ≤ (2π) ëFθë2L2 (Rn ) Ú ′ |u(x′ )|2 1E (x)e−(x −x) 2 /(2ε) dxdx′ . e−C/ε ëuë2L2 (Rn ) . Lemma B.4. Let θ be a cut-off of Rn and u ∈ L2 (Rn ). If F is a measurable subset of Rn s.t. dist(F, suppθ) > 0, then ëTε Tε∗ θ(η)Tε uëL2 (Rn ×F ) . e−C/ε ëuëL2 (Rn ) . Proof. Tε∗ θTε u may be written as acting on Fu, using the expression of the FBI transform given in Lemma B.1 Tε∗ θTε u(x) = n ε− 2 c2n Ú Fu(ξ)eiξ·x Gε θ(εξ)dξ, where Gε denotes the operator defined by Ú Gε a(ξ) = Rn ′ 2 /ε a(ξ ′ )e−(ξ−ξ ) dξ ′ , for a ∈ L2 (Rn ). It follows that Tε Tε∗ θTε u(y, η) = 3n n ε− 4 c3n (2π) 2 Ú Fu(ξ)eiξ·y−(η−εξ) Ú |Fu(ξ)|2 1F (η)e−(η−εξ) 2 /(2ε) Gε θ(εξ)dξ, By Parseval equality, one has ëTε Tε∗ θ(η)Tε uë2L2 (Rn ×F ) = 3n ε− 2 c6n (2π)2n 2 /ε |Gε θ(εξ)|2 dξdη. 2 2 Let d = dist(F, suppθ). If dist(εξ, F ) ≥ d/2 then |1F (η)e−(η−εξ) /ε | ≤ e−C(η−εξ) /ε e−C/ε . If dist(εξ, F ) ≤ d/2 then dist(εξ, suppθ) ≥ d/2, and |Gε θ(εξ)|2 ≤ e−C/ε . Since Gε θ is bounded, integrating w.r.t. ξ and η ends the proof. Lemma B.5. Lemma 3.4 of chapter 1 ëε−1 Tε uIε,γ ëL2 (R2n ) . 1. Proof. Derivating (12) w.r.t. yb , 0 ≤ b ≤ n, yields 1 1 3n ε 2 ∂yb (Tε uIε,γ ) = iηb ε− 2 Tε uIε,γ − cn ε− 4 Ú Rn 1 uIε,γ (w)ε− 2 (yb − wb ) eiη.(y−w)/ε−(y−w) 2 /(2ε) dw. Results related to the FBI and the Wigner transforms 98 The l.h.s. is bounded in L2y,η because ∂yb (Tε uIε,γ ) = Tε (∂wb uIε,γ ). The second term of the r.h.s. is the Fourier transform of a bounded function in L2w , thus it can be estimated using Parseval equality. One gets ëε − 3n 4 Ú 1 Rn uIε,γ (w)ε− 2 (yb − wb ) eiη.(y−w)/ε−(y−w) 2 /(2ε) dwëL2y,η . ëuIε,γ ëL2w . 1 1 Thus ëε− 2 ηb Tε uIε,γ ëL2y,η . 1 and consequently ëε− 2 φ(η)Tε uIε,γ ëL2y,η . 1. A3′ yields 1 Hence ëuIε,γ ëL2 . √ ëε− 2 Tε uIε,γ ëL2y,η . 1. ε. Reproducing the same arguments on the following equality 3n ∂yb (Tε uIε,γ ) = iηb ε−1 Tε uIε,γ − cn ε− 4 Ú Rn 1 1 2 1 i 1 2 ε− 2 uIε,γ (w)ε− 2 (yb − wb ) e ε η.(y−w)− 2 (y−w) dw, leads to ëuIε,γ ëL2 (Rn ) . ε. Lemma B.6. Let aε and bε two sequences uniformly bounded in L2 (Rn ) and H 1 (Rn ) respectively. If ε−1 bε is uniformly bounded in L2 (Rn ), then wε (aε , |D|bε ) ≈ |ξ|wε (aε , ε−1 bε ) on Rn × (Rn \{0}). Proof. We use another expression of the Wigner transform using the Fourier transform. Let φ be a test function of Cc∞ (Rn × (Rn \{0}), R) and denote cε = |D|bε . Then Ú ε Fξ φ(x − v, v)aε (x)c̄ε (x − εv)dvdx. 2 R2n < wε (aε , cε ), φ >= (2π)−n Since Fξ φ is rapidly decreasing ε sup|Fξ φ(x − v, v) − Fξ φ(x, v)| . ε(1 + v 2 )−n−1 . 2 x By Cauchy-Schwartz inequality Ú 3 4 ε | Fξ φ(x − v, v) − Fξ φ(x, v) aε (x)c̄ε (x − εv)| . εëaε ëL2 ëcε ëL2 . 2 R2n It follows that < wε (aε , cε ), φ >= (2π)−n Ú R3n φ(x, ξ)e−iv·ξ aε (x)c̄ε (x − εv)dvdxdξ + o(1). Integrating w.r.t. v leads to < wε (aε , cε ), φ >= (2π)−n ε−n Ú R2n φ(x, ξ)e−ix·ξ/ε aε (x)F c̄ε (−ξ/ε)dxdξ + o(1), and replacing F c̄ε (−ξ/ε) by ε−1 |ξ|F b̄ε (−ξ/ε) ends the proof. 99 Chapter III Elasticity system : asymptotic solutions and Wigner measures Contents 1 Introduction 100 2 Gaussian beams for the elasticity equations 101 3 4 2.1 Longitudinal beams . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 2.2 Transversal beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 2.3 Reflection of a beam L . . . . . . . . . . . . . . . . . . . . . . . . . . 110 2.4 Reflection of a beam T . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Construction of the approximate solution 115 3.1 Gaussian beams summation . . . . . . . . . . . . . . . . . . . . . . . 115 3.2 Justification of the asymptotics . . . . . . . . . . . . . . . . . . . . . 117 Wigner transforms and measures 119 4.1 First order beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.2 Wigner measures for the asymptotic solution . . . . . . . . . . . . . 120 Introduction 100 1 Introduction In this chapter, we focus on the elasticity equations for x ∈ Ω a three dimensional bounded domain and t ∈ [0, T ] P uε = ρ∂t2 uε − ∂x (λdivuε ) − 3 Ø ∂xj (µ∂x (uε )j + µ∂xj uε ) = 0, (1a) j=1 subject to the initial conditions uε |t=0 = uIε , ∂t uε |t=0 = vεI , (1b) Buε |∂Ω = 0, (1c) and the boundary conditions where B is the Dirichlet or the Neumann operator. We assume that λ > 0, µ > 0 and µ Ó= λ + 2µ. We construct asymptotic solutions for the problem (1) and compute the associated microlocal energy density under suitable assumptions on the geometry of the domain and the initial data. We again use the Gaussian beams summation technique to produce our approximate solutions. Gaussian beams for the elasticity system have the form N Ø εj aj eiψ/ε , j=0 with vector amplitudes aj and a complex phase ψ. Individual beams in the whole space domain have been constructed by Ralston in [86], and reflection of these beams has been studied in [20, 79, 82]. Summation of Gaussian beams for the elasticity system has been investigated in [17]. As for the scalar wave equations, different techniques may be used to superpose an infinite number of beams. Here we use the same strategy we applied for the scalar wave equation in chapter 1, appealing to the FBI transforms to fulfill general initial data. Hypotheses similar to those we imposed on initial data for the scalar wave equation are assumed A1. uIε and vεI are uniformly bounded respectively in H 1 (Ω)3 and L2 (Ω)3 , A2. uIε and vεI are uniformly supported in a fixed compact set K ⊂ Ω, A3. ëTε uIε ëL2 (R3 ×Rηc )3 = O(ε∞ ) and ëTε vεI ëL2 (R3 ×Rηc )3 = O(ε∞ ), where Rη = {η ∈ R3 , r0 ≤ |η| ≤ r∞ }, 0 < r0 < r∞ . For our construction to work, we must avoid tangential rays. Since the elasticity equations have two different wave speeds, a ray that emanates from the domain and strikes the boundary at a critical angle gives birth to a tangential ray. So we add a further assumption A4. ëTε uIε ëL2 (T rc )3 = O(ε∞ ) and ëTε vεI ëL2 (T rc )3 = O(ε∞ ), 101 where T r ⊂ R2n denotes the set of points giving birth to rays having a transversal contact with the boundary with an angle smaller than the critical angle, after one or more reflections on [0, T ]. As regards the domain, we suppose that B2. No ray remains in a compact of R3 for growing times, B3. The boundary has no dead-end trajectories, that is infinite number of successive reflections cannot occur in a finite time. Under these assumptions on initial data and the domain Ω, we prove the following theorem Theorem 1.1. For any integer R ≥ 2, there is an asymptotic solution to (1) of the form qs k iψk (t,x,y,η,R)/ε dydη, uR ε (t, x) = R2n aε (t, x, y, η, R)e k where akε eiψk /ε are Gaussian beams and the summation over k is finite. uR ε is asymptotic to the exact solution of the IBVP (1) in the following sense Sup ëuR ε (t, .) − uε (t, .)ëH 1 (Ω)3 = O(ε R−1 2 ), t∈[0,T ] and Sup ë∂t uR ε (t, .) − ∂t uε (t, .)ëL2 (Ω)3 = O(ε R−1 2 ). t∈[0,T ] As an application of this construction we compute the microlocal energy density associated to the elasticity system [80] 3 λ µØ ρ Trw[∂xj uε + ∂x (uε )j ] + w[divuε ]. E = Trw[∂t uε ] + 2 4 j=1 2 (2) The techniques used to describe the Wigner measures are an adaptation of chapter 2. However a new difficulty arises due to the existence of two different families of rays. This chapter is organised as follows. Section 2 is devoted to the construction of individual beams for the elasticity equations and their reflection at the boundary. In section 3 we superpose an inifinite number of these beams to construct asymptotic solutions to the problem (1) and prove theorem 1.1. Computation of the Wigner measure is achieved in section 4 through the first order asymptotic solution. 2 Gaussian beams for the elasticity equations We search for a high frequency solution of (1a) under the form ωε = N Ø εj aj eiψ/ε . j=0 Applying the equations of elasticity to ωε gives a function of the same form P ωε = N +2 Ø j=0 εj−2 cj eiψ/ε , Gaussian beams for the elasticity equations 102 with cj = ij (Jaj + M aj−1 + N aj−2 ), J = ρ[c2L ∂x ψ∂x ψ · +c2T (|∂x ψ|2 Id − ∂x ψ∂x ψ·) − (∂t ψ)2 ], M = − 2ρ∂t ψ∂t + (λ + µ)∂x (∂x ψ·) + (λ + µ)∂x ψ(∂x ·) + 2µ(∂x ψ · ∂x ) − ρ∂t2 ψ + µ△ψ + ∂x λ(∂x ψ·) + (∂x µ·)∂x ψ + (∂x µ · ∂x ψ), (3) and N a matrix differential operator of order 2 that we won’t specify (see 2.4 in [79] for explicit expression). Above, cT and cL denote the speeds associated to the elasticity system µ λ + 2µ c2T = , c2L = , ρ ρ and a−1 = a−2 = aN +1 = aN +2 = 0. As in [79, 86], the construction of the Gaussian beams consists in making the terms cj vanish up to the order R − 2j on some fixed ray, where R is an integer larger than 1. The elasticity operator has two families of rays : the rays associated to the speed cT and those associated to the speed cL . In fact, the determinent principal symbol of P is ρ3 (c2L (x)|ξ|2 − τ 2 )(c2T (x)|ξ|2 − τ 2 )2 . Each ray is a curve (t, x±t ) where xt is the projection of a Hamiltonian flow (xt , ξ t ) either associated to the symbol cL (x)|ξ| or cT (x)|ξ|            dxtL ξLt t = c L (xL ) t , dt |ξL |       or      dξLt t t dt = −∂x cL (xL )|ξL |, dxtT ξTt t = c T (xT ) t , dt |ξT | (4) dξTt t t dt = −∂x cT (xT )|ξT |. In the remainder, we build Gaussian beams associated to rays propagating in the positive sense, that is (t, xtL,T ). We will deal with complex valued vectors, but we use the same definitions for the dot product and the vectorial product. A complex valued vector a is said to be a unit vector if a · a = 1. Its modulus |a| is a square root of a · a that will be each time specified. We begin by making c0 = Ja0 vanish up to order R on some ray (t, xt ) of one family k of rays or the other. Let us use the notation ≍ t introduced in chapter 2 to denote that x=x the spatial derivatives of the quantities at its left and at its right match up to the order k on (t, xt ). By considering c0 · ∂x ψ, one gets è é R ρ c2T |∂x ψ|2 − (∂t ψ)2 a0 ∧ ∂x ψ ≍ t 0. x=x Note that in general, if two functions a, b of C ∞ (Rt × Rnx , C) satisfy k ab ≍ t 0 and a(t, xt ) Ó= 0, x=x k then one can see recursively by using Leibnitz formulae that b ≍ t 0. So here either x=x R i) c2T |∂x ψ|2 − (∂t ψ)2 ≍ t 0, x=x 103 or R ii) a0 ∧ ∂x ψ ≍ t 0. x=x On the other hand, taking the dot product of c0 with ∂x ψ gives the relation è é R ρ c2L |∂x ψ|2 − (∂t ψ)2 (∂x ψ · a0 ) ≍ t 0, x=x which shows that in the first case one must have R i) ∂x ψ · a0 ≍ t 0, x=x while in the second case a non zero amplitude implies that R ii)c2L |∂x ψ|2 − (∂t ψ)2 ≍ t 0. x=x So we are lead in each case to an eikonal equations on the phase, coupled with an information on the direction of the first amplitude. As usual, each eikonal equation with speed cα , α = L, T , is solved on the rays corresponding to this speed. Henceforth, the phases and amplitudes will be indexed by L or T to refer to which family of rays they are associated. To summarize, one can make cL,T vanish up to order R on some 0 ray (t, xtL,T ) by imposing respectively one of the following conditions R R x=xL x=xL c2L |∂x ψL |2 − (∂t ψL )2 ≍ t 0 and aL0 ∧ ∂x ψL ≍ t 0, or (5) R R x=xT x=xT c2T |∂x ψT |2 − (∂t ψT )2 ≍ t 0 and aT0 · ∂x ψT ≍ t 0. The previous two constraints lead respectively to Gaussian beams of type L or longitudinal beams and Gaussian beams of type T or transversal beams. The associated phases ψL and ψT are thus constructed in the same way as the phase has been in section 2 of chapter 1. From each eikonal equation results systems of ODEs on the spatial derivatives of the associated phase up to the order R on the ray. In particular, ∂t ψL (t, xtL ) = −cL (xtL )|ξLt |, ∂x ψL (t, xtL ) = ξLt , ∂t ψT (t, xtT ) = −cT (xtT )|ξTt |, ∂x ψT (t, xtT ) = ξTt , (6) (7) Then we expand the phases for all (t, x) ∈ Rn+1 as follows ψL,T (t, x) = Ø 1 (x − xtL,T )α ∂xα ψL,T (t, xtL,T ). α! |α|≤R The equations on the amplitudes in (5) may be written R R x=xL x=xT aL0 ≍ t sL0 pL for a beam L, aT0 ≍ t bT0 for a beam T. Here pL is a fixed complex unit vector satisfying pL ∧ ∂x ψL = 0 and pL · ∂x ψL = |∂x ψL |, the modulus obtained by continuity from |ξLt | for x = xtL . The same properties are satisfied by pT with the phase ψT and bT0 denotes an orthogonal vector to p. Remark 2.1. The constraints on the amplitudes at order R require the derivatives of the phases at order R + 1 on the associated rays. A similar observation was made By Norris in [79] for R = 2. He deduced that one must compute third order derivatives of the phase in order to obtain . This gap is proper to the elasticity equations and does not appear for the scalar wave equation. Gaussian beams for the elasticity equations 104 To make the terms cL,T for j > 0 vanish, let us analyse the action of the operator j M on vectors colinear and orthogonal to ∂x ψL,T . In the following we omit the indexes L and T in the phases and the vectors p when computations hold true for both of them. We start by computing M (ap), for a scalar a ∈ C. Focusing on the second and third operators in the sum (3), we note that ∂x (∂x ψ · ap) = ∂x (|∂x ψ|)a + |∂x ψ|∂x a, ∂x ψ∂x · (ap) = a∂x ψ∂x · p + ∂x ψ(∂x a · p). p · ∂x (|∂x ψ|) + |∂x ψ|∂x · p = △ψ, and thus (λ + µ)p · (∂x |∂x ψ| + ∂x · p∂x ψ) = (λ + µ)△ψ. Therefore, the dot product of p with the second and third terms of M (ap) is (λ + µ)p · [∂x (∂x ψ · ap) + ∂x ψ∂x · (ap)] = (λ + µ)△ψa + 2(λ + µ)∂x ψ · ∂x a. (8) On the other hand, since p · ∂x p = 0 and ∂x ψ · ∂x (ap) = (∂x ψ · ∂x a)p + ∂x p∂x ψa, one gets 2µp · [∂x ψ · ∂x (ap)] = 2µ(∂x ψ · ∂x a), (9) which is the dot product of p with the forth term in M (ap). Let LT,L (ψ) = −2ρ∂t ψ∂t + 2ρc2T,L ∂x ψ∂x , PT,L = ∂t2 − ∂x · (c2T,L ∂x ) and βT,L (ψ) = −ρPT,L ψ + c2T,L (∂x ρ · ∂x ψ). Taking the dot product of M (ap) with p, (8) and (9) give M (ap) = [(LL (ψ) + βL (ψ))a] p + T⊥ a, with T⊥ a differential operator of order 1 satisfying p · T⊥ = 0. Moving to vectorial amplitudes b s.t b · ∂x ψ = 0, one has M b = LT (ψ)b + (λ + µ)∂x ψ∂x · b + (∂x µ · b)∂x ψ + βT (ψ)b, which may be written M b = (LT (ψ) + βT (ψ))b + që b, with që a differential operator of order 1, colinear to p. Let us decompose each amplitude tangentially and orthogonally to pL,T as L,T aL,T = sL,T j j pL,T + bj . To make cL,T vanish up to the order R − 2, one plugs one of the forms of the amplitude 1 aL,T chosen in (5). 0 2.1 - Longitudinal beams 2.1 105 Longitudinal beams For a beam L, one obtains R −icL1 ≍ t ρ(c2T − c2L )|∂x ψL |2 bL1 + (LL (ψL ) + βL (ψL ))sL0 pL + T⊥ sL0 . x=xL (10) R−2 Imposing cL1 ≍ t 0 determines sL0 and bL1 up to order R − 2 at x = xtL . Indeed, starting x=xL by taking the dot product of cL1 with pL , one gets an evolution equation on sL0 (equation (3.3) in [79]) R−2 (LL (ψL ) + βL (ψL ))sL0 ≍ t 0. (11) x=xL Compare this equation with the one satisfied by the amplitude of a first order beam for the scalar wave equation (see (7) p.65). For a non constant density, it has an extra term c2L (∂x ρ · ∂x ψL ). By writing ∂x ψL (t, xtL ) = ξLt and ∂t ψL (t, xtL ) = −cL (xtL )|ξL |, one gets (LL (ψL )f ) |(t,xtL ) = −2ρ(xtL )∂t ψL (t, xtL ) 2 d 1 f (t, xtL ) for f ∈ C ∞ (Rt × R3x , C), dt (12) so order 0 of (11) gives the transport equation C D 2 1 PL ψL (t, xtL ) d d 1 L s0 (t, xtL ) = − ln(ρ(xtL )) sL0 (t, xtL ). t t dt 2 cL (xL )|ξL | dt (13) 1 The amplitude of a first order beam for the scalar wave equation is det(xt y + ixt η )]− 2 up to a coefficient (see (25) p.73). Thus sL0 (t, xtL ) = sL0 (0, x0L ) C D− 1 ρ(xtL ) det(∂y xtL + i∂η xtL ) ρ(x0L ) 2 . Above, the square root is obtained by continuity from 1 at t = 0. This result is similar to the formulae (3.28) in [79]. For |α| ≥ 1, ∂xα LL (ψL ) = LL (ψL )∂xα + RLα (14) with RLα a differential operator of order less than |α|. By (12), it follows that [∂xα (LL (ψL )f )] |(t,xtL ) = −2ρ(xtL )∂t ψL (t, xtL ) 2 d 1 α ∂x f (t, xtL ) + (RLα f )|(t,xtL ) dt Thus, (11) gives at order 0 < k ≤ R−2 a non-homogeneous transport equation 1 equation 2 t α L . To summarize, the spatial derivatives of sL0 up to order R − 2 on ∂x s0 (t, xL ) (|α|=k) on (t, xtL ) are uniquely determined by (11) given their values on (0, x0L ). We observe a gap in the order up to which the derivatives of bLa and sL0 are determined on the ray. So if aL0 (t, x) is chosen to be polynomial on x − xtL , the degree of this polynom has to be at least R, thus the spacial derivatives of sL0 on the ray at orders R − 1 and R are needed. This requirement defracts from the results obtained for the beams’ first amplitudes in the case of the scalar wave equation. So we choose ∂xα sL0 (0, x0L ) for |α| ≤ R − 2 arbitrary permutable families, ∂xα sL0 (t, xtL ) for t ∈ R, R − 2 < |α| ≤ R arbitrary smooth permutable families, Gaussian beams for the elasticity equations 106 and set 1 2 1 (x − xtL )α ∂xα sL0 pL (t, xtL ). α! |α|≤R aL0 (t, x) = χd (x − xtL ) Ø (15) R−2 Next, using the evolution equations (11), the equation cL1 ≍ t 0 becomes by (10) x=xL R−2 ρ(c2T − c2L )|∂x ψL |2 bL1 ≍ t −T⊥ sL0 . x=xL One obtains then bL1 up to order R − 2 on (t, xtL ) by plugging the value of sL0 ’s spatial derivatives on the ray. Similar results were obtained by [79] (see equations (6.6), (6.7)). Note that the derivatives of sL0 at order R − 1 on the ray are needed, since T⊥ is of order 1. For 1 ≤ j ≤ N , the amplitudes aLj2are computed recursively as follows. Let us assume 1 have been determined by the vanishing of that aL0 , . . . , aLj−1 and ∂xα bLj (t, xtL ) |α|≤R−2j cL0 , . . . , cLj on the ray up to R, . . . , R − 2j under the choice of ∂xα sLk (0, x0L ) for |α| ≤ R − 2k − 2 arbitrary permutable families, ∂xα sLk (t, xtL ) for t ∈ R, R − 2k − 2 < |α| ≤ R − 2k arbitrary smooth permutable families, and the expansion of the amplitudes beyond the rays as 1 2 1 (x − xtL )α ∂xα sLk pL + bLk (t, xtL ), α! |α|≤R−2k Ø aLk (t, x) = χd (x − xtL ) for k = 0, . . . , j − 1. By using the eikonal equation on ψL R (−i)j+1 cLj+1 ≍ t ρ(c2T − c2L )|∂x ψL |2 bLj+1 x=xL è é + (LL (ψL ) + βL (ψL ))sLj pL + T⊥ sLj + (LT (ψL ) + βT (ψL ))bLj + që bLj + N aLj−1 . Making the dot product of cLj+1 with pL vanish up to the order R − 2j − 2 gives a non-homogeneous evolution equation on sLj (LL (ψL ) + βL (ψL ))sLj + që bLj + [(LT (ψL ) + βT (ψL ))bLj + N aLj−1 ].pL R−2j−2 ≍ x=xtL 0. (16) This equation determines ∂xα sLj (t, xtL ) given ∂xα sLj (0, x0L ) for |α| = 0, . . . , R − 2j − 2. Once chosen ∂xα sLj (0, x0L ) for |α| ≤ R − 2j − 2 arbitrary permutable families, ∂xα sLj (t, xtL ) for t ∈ R, R − 2j − 2 < |α| ≤ R − 2j arbitrary smooth permutable families, we set aLj (t, x) = χd (x − xtL ) Ø |α|≤R−2j 1 2 1 (x − xtL )α ∂xα sLj pL + bLj (t, xtL ), α! 2.2 - Transversal beams 107 The spatial derivatives of bLj+1 on the ray are then fully determined up to the order R−2j−2 R − 2j − 2 by the equation cLj+1 ρ(c2T − c2L )|∂x ψL |2 bLj+1 ≍ x=xtL 0 which becomes é è R−2j−2 ≍ t −T⊥ sLj − (Id − pL pL ·) (LT (ψL ) + βT (ψL ))bLj + N aLj−1 . x=xL The system is closed, since bLN +1 = 0 and the equation cLN +1 R−2N −2 ≍ x=xtL 0 determines the spatial derivatives of sLN on the ray up to the order R − 2N − 2. Notice that R − 2N − 2 is positive by the relation between R and the number of amplitudes N (see Remark 2.1 in chapter 1). We end the construction of the beams’ amplitudes by expanding aLN for all (t, x) ∈ Rn+1 as Ø aLN (t, x) = χd (x − xtL ) |α|≤R−2N 2.2 2 1 1 (x − xtL )α ∂xα sLN pL + bLN (t, xtL ). α! Transversal beams The analysis for beams T is similar. One has −icT1 R ≍t x=xT ρ(c2L − c2T )|∂x ψT |2 sT1 pT + (LT (ψT ) + βT (ψT ))bT0 + që bT0 . (17) Since pT · bT0 = 0, it follows that pT · ∂t bT0 = −∂t pT · bT0 and pT · ∂x bT0 = −∂x pT · bT0 . Thus pT · LT (ψT )bT0 = −(LT (ψT )pT ) · bT0 , (18) R−2 which leads, by using (Id − pT pT ·)cT1 ≍ t 0, to the following equation x=xT R−2 (LT (ψT ) + βT (ψT )) bT0 + pT (LT (ψT )pT ) · bT0 ≍ t 0. x=xT (19) Let us write, for j = 0, . . . , N , bTj = rjT qjT , with qjT a unit vector. We first consider the projection of (19) on q0T and then on pT ∧ q0T . Since (LT (ψT )q0T ) · q0T = 0, the dot product of (19) with q0T may be written R−2 (LT (ψT ) + βT (ψT ))r0T ≍ t 0. (20) x=xT Thus we dispose of an evolution equation on r0T that determines its spatial derivatives up to order R − 2 on the ray. In particular, replacing ∂x ψT (t, xtT ) by ξTt and ∂t ψT (t, xtT ) by −cT (xtT )|ξT | and using a similar relation to (12) with indexes T give at order 0 the following transport equation C D 2 d 1 T 1 PT ψT (t, xtT ) d r0 (t, xtT ) = − ln(ρ(xtT )) r0T (t, xtT ). t t dt 2 cT (xT )|ξT | dt (21) Gaussian beams for the elasticity equations 108 Thus r0T (t, xtT ) C D− 1 ρ(xtT ) = det(∂y xtT + i∂η xtT ) ρ(x0T ) 2 , where the square root is defined by continuity from 1 at t = 0. r0T is computed up to the order R − 2 on the ray as sL0 and its spatial derivatives on the ray up to this order are fully determined by their initial values. On the other hand, taking the dot product of (19) with pT ∧ q0T leads to R−2 (LT (ψT )q0T ) · (pT ∧ q0T ) ≍ t 0, (22) x=xT since r0T (t, xtT ) Ó= 0. As q0T is a unit vector orthogonal to pT , one can prove that this equation fully determines the spatial derivatives of q0T up to order R − 2 on the ray given their initial values. We show this recursively on k = 0, . . . , R − 2. At order 0, equation (22) reads C D 2 2 d 1 T d 1 T q0 (t, xtT ) = q0 (t, xtT ) · pT (t, xtT ) pT (t, xtT ), dt dt which is equivalent to the system of ODEs C D 2 2 d 1 d 1 T q0 (t, xtT ) = − pT (t, xtT ) pT (t, xtT ) · q0T (t, xtT ). dt dt q0T (t, xtT ) is thus determined modulo its initial value. Now suppose ∂xβ q0T (t, xtT ) determined for |β| ≤ k. Since q0T is a unit vector orthogonal to pT , ∂xα q0T (t, xtT ) · q0T (t, xtT ) and ∂xα q0T (t, xtT ) · pT can be determined for |α| ≤ k + 1 by derivating q0T · q0T and q0T · pT w.r.t. x. The remaining unknowns are thus ∂xα q0T (t, xtT ) · (pT (t, xtT ) ∧ q0T (t, xtT )) for |α| = k + 1. Using (12) and (14), the equation (22) gives at order k + 1 é 1 2 d è α T ∂x q0 (t, xtT ) · (pT (t, xtT ) ∧ q0T (t, xtT )) + Zα q0T |(t,xtT ) = 0, |α| = k + 1, dt with Zα a differential operator of order |α|. The latter equations provide the projections of ∂xα q0T (t, xtT ) on pT (t, xtT ) ∧ q0T (t, xtT ) under the knowledge of their initial values. Remark 2.2. A useful property of q0T is determined by using the transverse vectors et1 , et2 defined by et1 = cos θt nt − sin θt bt , et2 = sin θt nt + cos θt bt , s where nt and bt are the ray xtT normal and binormal, θt = ss0 τ (s′ )ds′ + θ0 , τ (s′ ) the ray det det torsion and s the arc length. Indeed dt1 and dt2 have nonzero projections only on the é è d q T (t, xt ) · et = d q T (t, xt ) · et + q T (t, xt ) · d et is vector pT (t, xtT ). It follows that dt T T 0 T 0 1 1 dt 0 dt 1 é è d t t T t t T zero as well as dt q0 (t, xT ) · e2 and q0 (t, xT ) remains constant in the basis (e1 , et2 ) of the hyperplane pT (t, xtT )⊥ (see [79], section 4). 2.2 - Transversal beams 109 We choose q0T (0, x0T ) an arbitrary unit vector orthogonal to ξT0 , ξ0 ∂xα r0T (0, x0T ) for |α| ≤ R − 2 and ∂xβ q0T (0, x0T ) · ( T0 ∧ q0T (0, x0T )) for 1 ≤ |β| ≤ R − 2 |ξT | arbitrary permutable families, ξt ∂xα r0T (t, xtT ) and ∂xα q0T (t, xtT ) · ( Tt ∧ q0T (t, xtT )) for t ∈ R, R − 2 < |α| ≤ R arbitrary |ξT | smooth permutable families, (23) and set Ø 1 (x − xtT )α ∂xα bT0 (t, xtT ). α! |α|≤R aT0 (t, x) = χd (x − xtT ) (24) Finally, the spatial derivatives of sT1 on the ray are obtained up to order R − 2 by plugging the values of r0T and q0T in (17). For 0 ≤ j ≤ N , the eikonal equation on ψT implies (−i)j+1 cTj+1 R ≍t x=xT (c2L − c2T )|∂x ψT |2 sTj+1 pT [LL (ψT ) + βL (ψT )]sTj pT + që bTj + (LT (ψT ) + βT (ψT ))bTj + T⊥ sTj N aTj−1 . (25) 2 1 T T t T Assume that a0 , . . . , aj−1 and sj (t, xT ) have been determined by the vanishing + + |α|=R−2j cT0 , . . . , cTj on the ray up to the orders R, . . . , R − 2j, under similar choices to (23) of and the expansion of the amplitudes beyond the rays as 1 2 1 (x − xtT )α ∂xα sTk pT + bTk (t, xtT ), α! |α|≤R−2k Ø aTk (t, x) = χd (x − xtT ) for k ≤ j − 1. Since pT · bTj = 0, a similar equation to (18) gives, by using (Id − pT pT ·)cTj+1 (LT (ψT ) + βT (ψT )) bTj + [(LT (ψT )pT ) · bTj ]pT + T⊥ sTj + (Id − pT pT ·)N aTj−1 R−2j−2 ≍ x=xtT R−2j−2 ≍ x=xtT 0 0. Writing bTj = rjT qjT with qjT a unit vector leads to ODEs on the spatial derivatives of rjT and qjT up to order R − 2j − 2 by taking the dot product of the previous equation with qjT and pT ∧ qjT . We set aTj (t, x) = χd (x − xtT ) Ø |α|≤R−2j 2 1 1 (x − xtT )α ∂xα sTj pT + bTj (t, xtT ). α! Next, the derivatives of sTj+1 on the ray up to order R − 2j − 2 are obtained by plugging the value of bTj in (25) and solving cTj+1 · pT (c2L − c2T )|∂x ψT |2 sTj+1 R−2j−2 ≍ x=xtT 0 R−2j−2 ≍ x=xtT − [LL (ψT ) + βL (ψT )]sTj − pT · që bTj + [(LT (ψT )pT ) · bTj ] − pT · N aTj−1 . Gaussian beams for the elasticity equations 110 The system is closed, since sLN +1 = 0 and the equation cTN +1 R−2N −2 ≍ x=xtT 0 determines the spatial derivatives of bTN on the ray up to the order R − 2N − 2. We end the construction of the beams’ amplitudes by expanding aTN for all (t, x) ∈ Rn+1 as aTN (t, x) = χd (x − xtT ) Ø |α|≤R−2N 2.3 1 2T 1 (x − xtT )α ∂xα sTN pT + bTN (t, xtT ). α! Reflection of a beam L The reflection of a beam L gives rise to two reflected beams : a beam L and a beam T . After one reflection, we search for a solution of the form ωεL + Ref ωεL = j Ø iψLL /ε iψLT /ε εj aLj eiψL /ε + εj aLL + εj aLT , j e j e j=0 where the index LL denotes the reflected beam L and the index LT the reflected beam T. To fulfill the boundary condition, the reflected phases must have the same time and tangential derivatives as the incident phase at the instant tL and the point xtLL = xtLL of reflection, and this up to the order R. The phase ψLL is thus constructed like the reflected phase for the scalar wave equation associated to the symbol cL |ξ| (chapter 1 p. 35). L The phase ψLT is associated to some reflected bicharacteristic ϕLT satisfying xtLT = Since ψLT is solution of the eikonal equation with speed cT , one has xtLL . t t ∂t ψLT (t, xtLT ) = −cT (xtLT )|ξLT |, ∂x ψLT (t, xtLT ) = ξLT . To make the first order time and tangential derivatives of ψLT and ψL (see equation (6)) match at (tL , xtLL ), the following conditions on ϕLT at t = tL must be fulfilled tL cT (xtLL )|ξLT | = cL (xtLL )|ξLtL |, tL tL ξLT − (ξLT · ν(xtLL ))ν(xtLL ) = ξLtL − (ξLtL · ν(xtLL ))ν(xtLL ), tL tL where ξLtL = ξLtL and ξLT = ξLT . The further requirement tL ξLT · ν(xtLL ) ≤ 0 insures that (t, xtLT ) is a reflected ray entering the domain for t = tL + 0. Introducing the angles θ0 and θ2 as in fig.2 and the coefficient κ = ccTL leads to tL |ξLT | = κ(xtLL )|ξLtL |, sin θ2 = κ−1 (xtLL ) sin θ0 , π θ2 ≤ . 2 Since κ > 1, it follows that θ2 < θ0 and 3 tL ξLT = ξLtL − |ξLtL | cos θ0 + ñ 4 κ2 (xtLL ) − sin2 θ0 ν(xtLL ). 2.3 - Reflection of a beam L 111 Figure 2: Reflection of a wave L The reflected amplitudes are determined by the boundary condition. Let mB be the order of the boundary operator, that is mB = 0 or 1 for a Dirichlet or a Neumann problem. We adapt the method [85] p. 224 for a vector beam. To do this we introduce vector amplitudes d−mB +j defined on the boundary ∂Ω by B(ωεL + Ref ωεL ) = + + 1 ε−mB dL−mB + · · · + εN dLN eiψL /ε 1 iψLT /ε N LT . ε−mB dLT −mB + · · · + ε dN e 1 2 2 N LL iψLL /ε ε−mB dLL −mB + · · · + ε dN e 2 tL LT We impose on dL−mB +j +dLL −mB +j +d−mB +j to vanish on (tL , xL ) up to the order R−2j −2 for j = 0, . . . , N . For j = 0, . . . , N and for α = L, T , we write the reflected amplitudes Lα with bLα = sLα as aLα j · pLα = 0. If j = 0, one uses the form of the reflected j pLα + bj j LT amplitudes aLL and a to get 0 0 LT LT sLL 0 b(x, ∂x ψLL )pLL + b(x, ∂x ψLT )r0 q0 R−2 ≍ t (t,x′ )=(tL ,xLL ) −sL0 b(x, ∂x ψL )pL , (26) where b(x, ξ) denotes the principal symbol of B on the boundary. Here, we use the notation k ≍ (t,x′ )=(ta ,xtaa ) to denote that both sides have the same time and tangential derivatives at the instant ta and the boundary point xtaa up to order k. In the following, we prove that tL b(xtLL , ξLT )|p⊥ tL LT (tL ,xL ) is invertible, (27) tL tL tL tL b(xtLL , ξLL )pLL (tL , xtLL ) is not contained in the image of p⊥ LT (tL , xL ) by b(xL , ξLT ), (28) Gaussian beams for the elasticity equations 112 tL tL where ξLL = ξLL . Since q0LT is a unit vector orhtogonal to pLT , it follows that equation LL (26) determines time and tangential derivatives of sLL and q0LT on (tL , xtLL ) up to 0 , r0 order R − 2. For a Dirichlet problem, (27) and (28) are immediate since b = Id. For a Neumann problem, we examin its principal symbol b(x′ , ξ) = f (x′ ) [λ (ν(x′ )ξ·) + µ(ν(x′ ) · ξ)Id + µ (ξν(x′ )·)] , for (x′ , ξ) ∈ ∂Ω × Rn , where f is a positive function. Thus det b(x′ , ξ) is zero iff ξ · ν(x′ ) = 0 or |ξ|2 = (1 + κ(x′ )2 )(ξ · ν). Excluding vectors ξ tangential to the boundary, the kernel of b(x′ , ξ) is in the worst case a vector space of dimension one spanned by ξ − 2(ξ · ν)ν(x′ ). So for vectors ξ non tangential to ∂Ω, b(x′ , ξ)|ξ⊥ is invertible and (27) holds true. On the other hand, one has 2 1 b(x′ , ∂x ψL(L) )pL(L) = f (x′ ) λ|∂x ψL(L) |ν + 2µ|∂x ψL(L) |(pL(L) · ν)pL(L) , 1 2 b(x′ , ∂x ψLT )q0LT = f (x′ ) µ|∂x ψLT |(pLT · ν)q0LT + |∂x ψLT |µ(q0LT · ν)pLT . Let p′L denote the tangential part of pL and fix the orientation of q0LT (tL , xtLL ) as in fig. 2 . tL tL Consider the projections of the vectors b(xtLL , ξLL )pLL (tL , xtLL ) and b(xtLL , ξLT )pLT (tL , xtLL ) on ν(xtLL ) and p′L (xtLL ), and note that 1 (ν · b(x′ , ∂x ψLL )pLL ) |(t,x′ )=(tL ,xtL ) = |ξLtL |f (xtLL ) λ(xtLL ) + 2µ(xtLL ) cos2 θ0 L and (p′L · b(x′ , ∂x ψLL )pLL ) |(t,x′ )=(tL ,xtL ) = −(f µ)(xtLL )|ξLtL | sin 2θ0 2 L are of different signs, while 1 2 ν · b(x′ , ∂x ψLT )q0LT |(t,x′ )=(tL ,xtL ) = − (f µκ) (xtLL )|ξLtL | sin 2θ2 and 1 p′L · b(x ′ , ∂x ψLT )q0LT 2 L |(t,x′ )=(tL ,xtL ) = − (f µκ) (xtLL )|ξLtL | cos 2θ2 L have the same sign. This proves (28). Thus, at order 0, projections of (26) on ν(xtLL ) and p′L (tL , xtLL ) lead to the system tL tL tL LT L − cos θ0 sLL 0 (tL , xL ) + sin θ2 r0 (tL , xL ) = − cos θ0 s0 (tL , xL ), tL tL tL LT L sin θ0 sLL 0 (tL , xL ) + cos θ2 r0 (tL , xL ) = − sin θ0 s0 (tL , xL ), for a Dirichlet problem, and 1 2 tL tL tL LT λ(xtLL ) + 2µ(xtLL ) cos2 θ0 sLL 0 (tL , xL )− (µκ) (xL ) sin 2θ2 r0 (tL , xL ) = 2 1 − λ(xtLL ) + 2µ(xtLL ) cos2 θ0 sL0 (tL , xtLL ), tL tL tL LT −µ(xtLL )|ξLtL | sin 2θ0 sLL 0 (tL , xL ) − (µκ) (xL ) cos 2θ2 r0 (tL , xL ) = − µ(xtLL )|ξLtL | sin 2θ0 sL0 (tL , xtLL ), for a Neumann problem. One obtains the ratios sLL cos(θ0 + θ2 ) 0 , (tL , xtLL ) = L s0 cos(θ0 − θ2 ) r0LT sin 2θ0 , (tL , xtLL ) = − L s0 cos(θ0 − θ2 ) 2.3 - Reflection of a beam L 113 for a Dirichlet problem, and sLL sin 2θ0 sin 2θ2 − κ2 cos2 2θ2 tL 0 , ) = (t , x L L sL0 sin 2θ0 sin 2θ2 + κ2 cos2 2θ2 r0LT κ2 sin 2θ0 cos 2θ2 tL ) = (t , x L L sL0 sin 2θ0 sin 2θ2 + κ2 cos2 2θ2 for a Neumann problem. These are the same ratios as plane waves ([1], p. 176-177). Under the hypothesis of non tangential contact of xtL with the boundary, the spacial derivatives at (tL , xtLL ) are deduced from the time and tangential derivatives at this t α LT t point. So ∂xα sLL on the 0 (t, xLL ) and ∂x b0 (t, xLT ) are uniquely determined 2 associated 1 β LL t β LT t LT rays. We then fix the values of ∂x s0 (t, xLL ) and ∂x b0 (t, xLT ) · q0 ∧ pLT (t, xtLT ) for |β| = R − 1, R as arbitrary smooth functions of t ∈ R and set 2 1 1 t (x − xtLL )α ∂xα sLL 0 pL (t, xLL ), α! |α|≤R t aLL 0 (t, x) = χd (x − xLL ) Ø Ø 1 t (x − xtLT )α ∂xα bLT 0 (t, xLT ). α! |α|≤R t aLT 0 (t, x) = χd (x − xLT ) at (tL , xtLL ) up to order and aLT For 1 ≤ j ≤ N time and tangential derivatives of aLL j j and R − 2j − 2 are determined recursively. Indeed, assume that for k = 0, . . . , j, aLL k tL have been determined at (t , x ) up to the orders R − 2k − 2 by the vanishing aLT L L k LT up to the same order at this point. Assume that the + d of dL−mB +k + dLL −mB +k −mB +k LT derivatives of sLL and b at orders R − 2k − 1 and R − 2k on the assocaited rays have k k LL LT been fixed. bj+1 and sj+1 are thus known at (tL , xtLL ) up to the order R − 2j − 2, by the construction of the amplitudes of beams L and T achieved in section 2. One has LT dL−mB +j+1 +dLL −mB +j+1 + d−mB +j+1 = LL ′ LT LT L b(x′ , ∂x ψL )aLj+1 + b(x′ , ∂x ψLL )[sLL j+1 pLL + bj+1 ] + b(x , ∂x ψLT )[sj+1 pLT + bj+1 ] + gj+1 , L depends only on the amplitudes r ≤ j. The constraint where gj+1 LT dL−mB +j+1 + dLL −mB +j+1 + d−mB +j+1 R−2j−4 ≍ t (t,x′ )=(tL ,xLL ) 0 may be written as ′ LT ′ LT sLL j+1 b(x , ∂x ψLL )pLL + rj+1 b(x , ∂x ψLT )qj+1 R−2j−4 ≍ t (t,x′ )=(tL ,xLL ) ′ L LT − b(x′ , ∂x ψL )aLj+1 − b(x′ , ∂x ψLL )bLL j+1 − sj+1 b(x , ∂x ψLT )pLT − gj+1 . By the same arguments as for the first amplitudes, this equation determines the time tL LT LT and tangential derivatives of sLL j+1 , rj+1 and qj+1 up to the order R − 2j − 4 at (tL , xL ). We end the construction by expanding the reflected amplitudes as Ø t aLL j (t, x) = χd (x − xLL ) |α|≤R−2j t aLT j (t, x) = χd (x − xLT ) Ø |α|≤R−2j for j = 1, . . . , N . 2 1 1 t (x − xtLL )α ∂xα sLL p L (t, xLL ), j α! 1 t (x − xtLT )α ∂xα bLT j (t, xLT ), α! Gaussian beams for the elasticity equations 114 Figure 3: Reflection of a wave T 2.4 Reflection of a beam T Reflection of a beam T gives rise in general to a beam L and a beam T . We search for a solution of the form ωεT + Ref ωεT = j Ø εj aTj eiψT /ε + εj aTj L eiψT L /ε + εj aTj T eiψT T /ε , j=0 where the index T L denotes a reflected beam L and the index T T a reflected beam T . The construction is similar to the reflection of a beam L. The reflected phase ψT T is built as the reflected phase for the scalar wave equation associated to the symbol cT (x)|ξ|. The construction of the phase ψLT leads to the equations cL (xtTT )|ξTtTL | = cT (xtTT )|ξTtT |, ξTtTL − (ξTtTL · ν(xtTT ))ν(xtTT ) = ξTtT − (ξTtT · ν(xtTT ))ν(xtTT ), which are completed with the requirement ξTtTL · ν(xtTT ) ≤ 0. One gets |ξTtTL | = κ−1 (xtTT )|ξTtT |, sin θ1 = κ(xtTT ) sin θ0 , π θ1 ≤ . 2 where the angles θ0 and θ1 are specified in fig.3. For θ1 to be a real valued angle, θ0 3.1 - Gaussian beams summation 115 must be smaller than a critical angle [1] θc = arcsin(κ−1 (xtTT )), (29) for which ξTtTL is tangential to the boundary. In the sequel we assume that this critical angle is not reached. For a = T T, T L, we define pa the unit vectors associated to ∂x ψa . For 0 ≤ j ≤ N , the reflected amplitudes are written aaj = saj pa + baj with qja a unit vector orthogonal to pa . We mimic the computation of the reflected amplitudes achieved in the previous paragraph, replacing the indexes L, LL and LT by T , T L and T T respectively. 3 Construction of the approximate solution In this section, we build uR ε , the approximate solution up to order O(ε the asymptotics. 3.1 R−1 2 ) and justify Gaussian beams summation The summation process based on the FBI technique consists of integrating over the phase space domain Gaussian beams microlocalized at t = 0 near (y, η), after weighting them by some quantities related to the initial data’s FBI transforms at point (y, η). The very first step is then to generalize the notion of a FBI transform to vector functions. This is done naturally by setting for u ∈ L2 (R3 )3 3n Tε u(y, η) = cn ε− 4 Ú R3 u(x)eiη·(y−x)/ε−(y−x) 2 /(2ε) dx. (30) One gets properties similar to the scalar case. In particular, Tε is an isometry from L2 (R3 )3 → L2 (R3 × R3 )3 and Tε∗ Tε = Id, (31) where Tε∗ is the adjoint of Tε and satisfies Tε∗ f (x) = cn ε − 3n 4 Ú R6 iη·(x−y)/ε−(x−y)2 /(2ε) f (y, η)e dydη =: cn ε − 3n 4 Ú R6 f (y, η)eiφ0 (x,y,η)/ε dydη. Next we select beams with phases that coincide at t = 0 with φ0 i ψ(0, x) = η · (x − y) + (x − y)2 . 2 (32) We refer to the dependence of such beams on the starting point and direction of their associated rays. We thus denote ωεL (t, x, y, η) an L-beam associated with the bicharacteristic (xtL , ξLt ) s.t. (x0L , ξL0 ) = (y, η), and use a similar notation for a T -beam. Hence, we consider only bicharacteristics satisfying x0L = x0T = y and ξL0 = ξT0 = η. Let us now examine the amplitudes of these beams at t = 0. The first term in the Taylor series of aL0 (0, x) near y is sL0 (0, y) η . As regards aT0 (0, x), it is a unit vector |η| Construction of the approximate solution 116 orthogonal to η multiplied by r0T (0, y). We shall thus decompose the FBI transforms |η| on the basis ( η , e01 , e02 ) and fit each component by using three appropriate elementary |η| solutions based on Gaussian beams. So we aim to construct solutions ι1ε , ι2ε , ι3ε that equal at t = 0 respectively η eiφ0 /ε , e01 eiφ0 /ε and e02 eiφ0 /ε modulo residues of order O(εp ) |η| with p sufficiently large. For any two beams ωεL (., y, η) and ωεT (., y, η) we may write ωεL (0, x, y, η) + ωεT (0, x, y, η) = N Ø è é εj aLj (0, x, y, η) + aTj (0, x, y, η) eiφ0 (x,y,η)/ε . j=0 (33) We write rjT qjT =: rjS qS + rjZ qZ for j = 0, . . . , N where (pT , qS , qZ ) is a fixed orthonormal ξt basis that coincides on (t, xtT ) with the basis ( Tt , et1 , et2 ) . One has by (15) and (24) |ξT | é è 1 (x − y)α ∂xα sL0 pL + r0S qS + r0Z qZ (0, y). α! |α|≤R aL0 (0, x) + aT0 (0, x) = χd (x − y) Ø We set (sL0 , r0S , r0Z )(0, y) equal to (1, 0, 0) for the construction of ι1ε , (0, 1, 0) for the construction of ι2ε or (0, 0, 1) for the construction of ι3ε . We then impose that    sL0    ∂xα M  r0S  (0, y) = 0 for 1 ≤ |α| ≤ R, r0Z (34) where M is the matrix (pL , qS , qZ ). Since the vectors pL , qS and qZ form at point (0, y) an orthonormal basis, M (0, y) is invertible and the previous equations determine the spatial derivatives of sL0 , r0S and r0Z at (0, y) up to order R. One gets aL0 (0, x) + aT0 (0, x) = χd (x − y) C sL0 (0, y) η + r0S (0, y)e01 + r0Z (0, y)e02 |η| D é è 1 (x − y)α ∂xα sL0 pL + r0S qS + r0Z qZ (0, y). α! |α|=R−1 + χd (x − y) R Ø Recursively, for j = 1, . . . , N , we impose that é è è é ∂xα sLj pL + rjS qS + rjZ qZ (0, y) = −∂xα bLj + sTj pL (0, y) for |α| ≤ R − 2j − 2. (35) The r.h.s. is known by the construction of the amplitudes aLk and aTk for k ≤ j − 1. Therefore, hypothesis (34) and (35) lead to è é η ωεL (0, x) + ωεT (0, x) = χd (x − y) sL0 (0, y) + r0S (0, y)e01 + r0Z (0, y)e02 eiφ0 (x,y,η)/ε . |η| We thus succeeded in building the elementary solutions ι1ε , ι2ε and ι3ε as a sum of two ′ suitable beams of type L and T . Similar ideas give elementary solutions ιjε , j = 1, 2, 3, s.t. their time derivatives at t = 0 approach respectively ε−1 η eiφ0 /ε , ε−1 e01 eiφ0 /ε and |η| ε−1 e02 eiφ0 /ε modulo small residues. Indeed, under a suitable choice of the derivatives of ′ ′ ′ sLj , rjS and rjZ up to the order R − 2j 1 2 η i∂t ψL aL0 + i∂t ψT aT0 (0, y) equals or e01 or e02 |η| 1 2 ∂xα ∂t ψL aL0 + ∂t ψT aT0 (0, y) = 0 for 1 ≤ |α| ≤ R, 1 2 ∂xα i∂t ψL aLj + ∂t aLj−1 + i∂t ψT aT0 + ∂t aTj−1 (0, y) = 0 for 0 ≤ |α| ≤ R − 2j and 1 ≤ j ≤ N, 3.2 - Justification of the asymptotics 117 one obtains è 1 ∂t ωεL (0, x) + ∂t ωεT (0, x) = ε−1 χd (x − y) sL0 (0, y) + N +1 Ø Ø εj−1 j=0 |α|=R−2j+1 2 η + r0S (0, y)e01 + r0Z (0, y)e02 |η| é (x − y)α resαj eiφ0 (x,y,η)/ε Without loss of generality, one can choose T sufficiently small so that at most one reflection occurs for rays originating from Ky × (Rn \{0}) for some compact Ky ⊂ Ω containing K. Let ρ′ be a cut-off of C0∞ (R3 , [0, 1]) supported in Ky and satisfying ρ′ (y) = 1 if dist(y, K) < ∆ for a small ∆ > 0, (36) and γ ′ a cut-off of C0∞ (R3 , [0, 1]) supported in a compact Kη ⊂ R3 \{0} s.t. γ ′ ≡ 1 on Rη . We search for an approximate solution such as uR ε (t, x) = + 5 2 η Ø cn − 3n +1 Ú 1 Ø ε 4 Ref k ι1ε (t, x) + Ref k (ι1ε (−t, x)) 2 |η| k=0,1 k=0,−1 1Ø + Ref k 3 ιε (t, x) + + ′ Ref k ι1ε (t, x) − Ref k 2′ ιε (t, x) k=0,1 + 1Ø k=0,1 Ref k 2 (ι3ε (−t, x)) k=0,−1 k=0,1 1Ø Ø + k=0,1 51 Ø 2 Ref k (ι2ε (−t, x)) e01 k=0,−1 k=0,1 1Ø Ø Ref k ι2ε (t, x) + Ref k 3′ ιε (t, x) − − Ø ′ ′ Ø 2 Ref k (ι2ε (−t, x)) k=0,−1 Ref k 2 ′ (ι3ε (−t, x)) k=0,−1 6 1 2 Ref k (ι1ε (−t, x)) k=0,−1 Ø e02 2 · ε−1 Tε uIε ρ′ ⊗ γ ′ 1 η cL |η| 1 0 e cT 1 6 A B 1 1 0 e2 · i Tε vεI ρ′ ⊗ γ ′ dydη, cT |η| ′ where ιjε and ιjε are the elemntary solutions constructed previously as sums of two beams and Ref denotes the reflections of each one of these beams. 3.2 Justification of the asymptotics In this section we prove theorem 1.1. The initial boundary value problem (1) is well posed and moreover we dispose of the following energy estimate as a consequence of e.g. [48] R supëuR ε (t, .) − uε (t, .)ëH 1 (Ω)3 + supë∂t uε (t, .) − ∂t uε (t, .)ëL2 (Ω)3 . [0,T ] [0,T ] ë∂t uR ε (0, .) I − vεI ëL2 (Ω)3 + ëuR ε (0, .) − uε ëH 1 (Ω)3 R + supëP uR ε (t, .)ëL2 (Ω)3 + ëBuε ëH s ([0,T ]×∂Ω)3 , [0,T ] where s = 23 for a Dirichlet problem and 1/2 for a Neumann problem. We estimate each term of the r.h.s. of the previous inequality. The computations are based on the results established in section 3 of chapter 1. Construction of the approximate solution 118 We start by estimating the error in the initial conditions. The phases ψL and ψT satisfy by construction the fundamental estimate Im ψ(t, x) & |x − xt |2 for |x − xt | ≤ d. At t = 0, all of the rays created by reflection at the boundary are in the exterior of the domain and the contribution of the reflected beams to uR ε is then exponentially 2 3 3 decreasing in L (R ) as explained in chapter 1 p.39. In the expression of uR ε , only elementary solutions weighted by Tε uIε thus have a non negligeable contribution initially. Hence, Ú è é η η 2 R − 3n 4 · +e01 e01 · +e02 e02 · Tε uIε eiη·(x−y)/ε−(x−y) /(2ε) dydη uε (0, x) = cn ε |η| |η| 3n + c n ε− 4 N Ø R−2j Ø εj j=0 |α|=R−2j−1 Ú (x − y)α resα Tε U I eiη·(x−y)/ε−(x−y) 2 /(2ε) dydη + O(ε∞ ). Moreover, at t = 0, the contribution of x − y ∈ / suppχd is then exponentielly decreasing. The estimates established in chapter 1 for similar quantities yield I uR ε |t=0 = uε + O(ε R−1 2 ) in L2 (Ω)3 . Similar arguments show that I ∂xb uR ε |t=0 = ∂xb uε + O(ε and 3n − 4 uR ε (0, x) = cn ε + cn ε − 3n 4 Ú è N Ø j=0 R−1 2 ) in L2 (Ω)3 , b = 1, 2, 3, é η η 2 · +e01 e01 · +e02 e02 · Tε uIε eiη·(x−y)/ε−(x−y) /(2ε) dydη |η| |η| ε R−2j Ø j |α|=R−2j−1 Ú (x − y)α resα Tε U I eiη·(x−y)/ε−(x−y) I ∂t uR ε |t=0 = vε + O(ε R−1 2 2 /(2ε) dydη + O(ε∞ ). ) in L2 (Ω)3 . To estimate the interior equation, we note that by construction, P uR ε is a sum of terms of the form Ú − 3n −1+j 4 ε cj eiψ/ε fε dydη, R−2j where fε denotes some projection of ε−1 Tε uIε or i/(c|η|)Tε vεI and cj ≍ t 0, vanishing for x=x |x − xt | ≥ d. For k = 1, 2, 3, one may then write {P uR ε }k as a sum of terms of the form ε +j−1 − 3n 4 Ú eαj (x − xt )α eiψ/ε fε dydη, with |α| = R − 2j + 1 and eαj vanish for |x − xt | ≥ d. Since fε is uniformly bounded in L2 (R3 ), the operators Oα introduced in chapter 1 give the following estimate ë which leads to Ú 3n eαj (x − xt )α eiψ/ε fε dydηëL2 (Ω) . ε 4 + supëP uR ε (t, .)ëL2 (Ω)3 . ε R−1 2 |α| 2 , . [0,T ] The estimate of the boundary conditions is similar to the scalar case. One obtains ëBuR ε ëH s ([0,T ]×∂Ω)3 . ε R+1 −mB −s 2 . 4.1 - First order beams 4 119 Wigner transforms and measures In this section we compute the scalar Wigner measure e defined in (2) for initial data satisfying A1-A4 by using the first order approximate solution obtained as a summation . of first order beams. We denote henceforth this asymptotic solution by uappr ε 4.1 First order beams We give explicit expressions for the beams’ phases and the first term in the amplitudes when R = 2. In this case, the phases are quadratic on (x − xt ) and may be written as 1 21 2−1 1 ψ(t, x) = ξ t · (x − xt ) + (x − xt ) · ∂y ξ t + i∂η ξ t ∂y xt + i∂η xt (x − xt ). 2 Above the considered phases and flows may be incident or reflected ones. Only the first terms in the Taylor expansion of the first amplitudes a0 near the . The constraints on the beams used in the construction of the ray contribute to uappr ε elementary solutions lead to ξLt , |ξLt | aS0 (t, xtT ) = aT (t)et1 , aZ0 (t, xtT ) = aT (t)et2 , aL0 (t, xtL ) = aL (t) where aL and aT denote the quantities associated to the flows ϕL et ϕT defined by D− 1 C ρ(xt ) det(∂y xt + i∂η xt ) a(t) = ρ(y) 2 . After reflection, the beams of type T are projected on et1 and et2 . We then obtain the scalar amplitudes r0ὰS and r0ὰZ defined as t LS t t LZ t t bLT 0 (t, xLT ) = r0 (t, xLT )e1 LT + r0 (t, xLT )e2 LT , t ὰS t t ὰZ t t bὰT 0 (t, xὰT ) = r0 (t, xὰT )e1 T T + r0 (t, xὰT )e2 T T , ὰ = S, Z, We next define the coefficients of reflection linking the scalar amplitudes of the reflected beams to the one of the incident beam at the instant of reflection    + + + 0 0 sL0 (tL , xtLL ) RLL RSL RZL   + + +  0 r0S (tT , xtTT ) 0 =  RLS RSS RZS   tT + + + Z RLZ RSZ RZZ 0 0 r0 (tT , xT )   tT tT tL ZL SL sLL 0 (tL , xL ) s0 (tT , xT ) s0 (tT , xT )   LS t t t  r0 (tL , xLL ) r0SS (tT , xTT ) r0ZS (tT , xTT )  r0LZ (tL , xtLL ) r0SZ (tT , xtTT ) r0ZZ (tT , xtTT ) Thus, the scalar reflected amplitudes satisfy at any time t + + t ὰL t sLL 0 (t, xLL ) = RLL aLL (t), s0 (t, xὰL ) = RὰL aT L (t), + aLT (t), r0ὰά (t, xtὰά ) = Rὰ+ά aT T (t), r0Lά (t, xtLά ) = RLS where aLL , aLT , aT L and aT T denote the scalar amplitudes associated to the flows ϕLL , ϕLT , ϕT L and ϕT T respectively and ὰ, ά = S, Z. We also define RL+ = RS+ = RZ+ = 1 to get similar expressions for the incident amplitudes. For beams assocaited to flows that propagate in the positive sens, we associate reflection coefficients with an exponent −. Wigner transforms and measures 120 4.2 Wigner measures for the asymptotic solution as ωεα,± with We denote the Gaussian beams used in uappr ε α ∈ I = {L, S, Z}, for the incident beams and α ∈ R = {LL, LS, LZ, SL, SS, SZ, ZL, ZS, ZZ}, for the reflected beams. For α ∈ I ∪ R, we use the following notations ὰ := α, ά := α if α ∈ I and α = ὰά if α ∈ R, [α] denotes the index of the associated flow, that is [L] = L, [S] = [Z] = T and for ὰά ∈ R, [ὰά] = [ὰ][ά], α þ is the initial unit vector associated with ωεα,± . It may be η/|η|, e01 or e02 . For example, for a beam ωεLS , that is α = LS, we have ὰ = L, ά = S, [α] = LT can then be and α þ = e01 . Time and spatial derivatives of the asymptotic solution uappr ε written using the integrals 3n Iα± (ε, fε , v)(t, x) = cn ε− 4 +1 s ′ ′ t −1 t −1 R6 (ρ ⊗ γ ) o{ϕ[α],± } (z, θ)fε o{ϕα,± } (z, θ) t −1 iψ (t,x,{ϕ[α],± (z,θ)} )/ε v(z, θ)dzdθ, (Rα± aα )(t, {ϕt[α],± }−1 (z, θ))e α where v is a vector smooth on {ϕtα,± }−1 (Ky × Kη ) and fε is uniformly bounded in L2 (R6 )3 , and the functions γεL = (Tε uIε + η i i Tε vεI ) · , (γεS , γεZ ) = (Tε uIε + Tε vεI ) · (e01 , e02 ), cL |η| |η| cT |η| κLε = (Tε uIε − i η i Tε vεI ) · , (κSε , κZε ) = (Tε uIε − Tε vεI ) · (e01 , e02 ). cL |η| |η| cT |η| In fact, as proven for the scalar wave equation in chapter 2, one has in L2 (Ω)3 , uniformly for t ∈ [0, T ] √ 2∂t uappr = vt+ + vt− + O( ε), ε with vt+ = −iIL+ (ε, γεL , cL η) − iIS+ (ε, γεS , cL |η|e01 ) − iIZ+ (ε, γεZ , cL |η|e02 ) + + + (ε, γεZ , cL |η|e02 ) (ε, γεS , cL |η|e01 ) − iIZL −iILL (ε, γεL , cL η) − iISL + + + −iILS (ε, γεL , cT η) − iISS (ε, γεS , cT |η|e01 ) − iIZS (ε, γεZ , cT |η|e02 ) + + + (ε, γεZ , cT |η|e02 ) (ε, γεS , cT |η|e01 ) − iIZZ −iILZ (ε, γεL , cT η) − iISZ q α), Iα+ (ε, γεὰ , c[ά] |η|þ = −i (37) α∈I∪R and vt− = i Ø α∈I∪R Likewise, for b = 1, 2, 3 Iα− (ε, κὰε , c[ά] |η|þ α). √ = vb+ + vb− + O( ε), 2∂xb uappr ε (38) 4.2 - Wigner measures for the asymptotic solution with vb+ = + + + = 121 η IL+ (ε, γεL , ηb |η| ) + IS+ (ε, γεS , ηb e01 ) + IZ+ (ε, γεZ , ηb e02 ) η + + + (ε, γεZ , ηb e02 ) ) + ISL (ε, γεS , ηb e01 ) + IZL ILL (ε, γεL , ηb |η| η + + + ILS (ε, γεL , ηb |η| ) + ISS (ε, γεS , ηb e01 ) + IZS (ε, γεZ , ηb e02 ) η + + + (ε, γεZ , ηb e02 ) ) + ISZ (ε, γεS , ηb e01 ) + IZZ ILZ (ε, γεL , ηb |η| q þ ), iIα+ (ε, γεὰ , ηb α (39) α∈I∪R and Ø vb− = iIα− (ε, κὰε , ηb α þ ). α∈I,R 3×3 matrix (v1+ , v2+ , v3+ ) and compute the Wigner measures associated We denote 3 q to Trwε [vt± ], Trwε [vb± + Vx± · kb ], wε [ vb± · kb ], as well as the cross measures involving b=1 b=1 (vt+ , vt− ) and (vb+ , vb− ) for b = 1, 2, 3. Here kb denotes the vector of R3 s.t. (kb )j = δbj . Vx+ the 3 q One needs then to estimate β̀ ὰ , d)), wε (Iαp (ε, ̟ε,p , b), Iβq (ε, τε,q for p, q = ±, ̟ε,+ , τε,+ = γε , ̟ε,− , τε,− = κε , α, β ∈ I ∪ R and b, d vector functions smooth respectively on {ϕt[α],± }−1 (Ky × Kη ) and {ϕt[β],± }−1 (Ky × Kη ). The previous analysis carried in chapter 2 for similar quantities associated to the scalar wave equation leads to s p,q iΘε,α,β (x,ξ,r,δ) ὰ β̀ < wε (Iαp (ε, ̟ε,p , b), Iαq (ε, τε,q , d)), ψ >= R12 ψ(x, ξ)b(x, ξ)d∗ (x, ξ)Ap,q α,β (x, ξ)e √ √ 1 ὰ (ρ′ ⊗ γ ′ Rαp ρ 2 ) o{ϕt[α],p }−1 (x, ξ)ρ−1 (x)̟ε,p o{ϕt[α],p }−1 (x + εr, ξ + εδ) √ √ 1 β̀ (ρ′ ⊗ γ ′ Rβq ρ 2 ) o{ϕt[β],q }−1 (x, ξ)τ̄ε,q o{ϕt[β],q }−1 (x − εr, ξ − εδ)dxdξdrdδ + o(1). Let 1 1 1 2 Eα+ (x, ξ) = ρ− 2 (x) ρ′ ⊗ γ ′ Rα+ ρ 2 γεὰ o{ϕt[α],+ }−1 (x, ξ), 1 1 1 2 Eα− (x, ξ) = ρ− 2 (x) ρ′ ⊗ γ ′ Rα− ρ 2 κὰε o{ϕt[α],− }−1 (x, ξ), 1 +,+ t −1 −1 ′ ′ + 2 Fα,β (x, ξ) = A+,+ α,β (x, ξ)ρ (x)(ρ ⊗ γ Rα ρ ) o{ϕ[α],+ } (x, ξ) ′ (ρ ⊗ γ ′ 1 Rβ+ ρ 2 ) o{ϕt[β],+ }−1 (x, ξ) γ¯ε β̀ o{ϕt[β],+ }−1 (x − √ εr, ξ − √ Ú +,+ R6 eiΘε,α,β (x,ξ,r,δ) γεὰ o{ϕt[α],+ }−1 (x + √ εr, ξ + √ εδ) εδ)drdδ, +,− −,− and define similarly Fα,β and Fα,β with terms γεὰ κ¯ε β̀ and κὰε κ¯ε β̀ . We may write the previous equation in the sense of distributions as p,q ὰ β̀ wε (Iαp (ε, ̟ε,p , b), Iβq (ε, τε,q , d)) ≈ Fα,β bd∗ , and the trace measure satisfies then p,q ∗ ὰ β̀ , b), Iβq (ε, τε,q , d)) ≈ Fα,β d b. Trwε (Iαp (ε, ̟ε,p We get Trwε [vt+ ] ≈ Ø α,β∈I∪R +,+ þ c[ά] c[β́] |η|2 Fα,β (þ α.β). (40) Wigner transforms and measures 122 The terms coming from the cross Wigner measure between beams with different directions vanish and only the terms satisfying α þ = βþ contribute to Trw[vt+ ]. On the other n hand, for s ∈ Ω, σ ∈ R \{0}, t ∈ [0, T ] and α ∈ R, only one of the points xtὰ (s, σ) and xtα (s, σ) may be in Ω. It follows that +,+ +,+ Fὰ,α = Fα, ὰ = 0. All in all, the cross terms that have a non-zero contribution to Trw[vt+ ] are associted to reflected beams having the same direction. One has therefore 1 ρTrwε [vt+ ] ≈ |η|2 c2T |ES+ |2 + c2T |EZ+ |2 +c2T Ø α,β∈R þ 0 ,e0 α þ =β=e 1 2 +,+ Fα,β + c2L |EL+ |2 + c2L Ø 2 +,+ . Fα,β α,β∈R þ η α þ =β= |η| From (39), one gets vb+ + Vx+ · kb = Ø iIα+ (ε, γεὰ , ηb α þ +α þ b η). (41) α∈I∪R It follows, by using approximation (40), that the Wigner transform Trwε [vb+ + Vx+ · kb ] is a sum of terms of the form +,+ Fα,β (ηb α þ +α þ b η)(ηb βþ T + βþb η T ), α, β ∈ I ∪ R, modulo a vanishing residue. Since 3 Ø Tr[(ηb α þ +α þ b η)(ηb βþ T + βþb η T )] = b=1 3 Ø b=1 (ηb2 δαβ + ηb βþb δαL + ηb α þ b δβL + α þ b βþb |η|2 ) = 2|η|2 (δαβ + δLα δLβ ), the cross terms between beams of different directions do not contribute to 3 q Trwε [vb+ + Vx+ · kb ] and one obtains b=1 3 Ø b=1 Trwε [vb+ + Vx+ · kb ]  ≈ 2|η|2  Ø α=S,Z  |Eα+ |2 + 2|EL+ |2  + 2|η|2 3 Ø +,+ Fα,β +2 α,β∈R þ 0 ,e0 α þ =β=e 1 2 Ø α,β∈R þ η α þ =β= |η| 4 +,+ . Fα,β Finally, 3 Ø b=1 vb+ · kb = Ø Iα+ (ε, γεὰ , iη T α þ) = α∈I∪R Ø η α þ = |η| Iα+ (ε, γεὰ , i|η|), (42) which implies, using approximation (40), that wε [ 3 Ø b=1 vb+ · kb ] ≈ |η|2 |EL+ |2 + |η|2 Ø α,β∈R þ η α þ =β= |η| +,+ Fα,β . (43) 4.2 - Wigner measures for the asymptotic solution 123 By replacing the speeds cT and cL by their values, it follows that ρTrwε [vt+ ] è ≈ |η|2 c2T |ES+ |2 + c2T |EZ+ |2 + c2T ≈ Ø α,β∈R þ 0 ,e0 α þ =β=e 1 2 +,+ + c2L |EL+ |2 + c2L Fα,β Ø +,+ Fα,β α,β∈R þ η α þ =β= |η| é 3 3 Ø µØ Trwε [vb+ + Vx+ · kb ] + λwε [ vb+ · kb ]. 2 b=1 b=1 (44) The analysis for vt− and vb− , b = 1, 2, 3, is similar. The only differences are that the Hamiltonian flows propagate in the negative sense and that γεὰ is replaced by κὰε . One has Ø vb− + Vx− · kb ≈ iIα− (ε, κὰε , ηb α þ +α þ b η), (45) α∈I∪R and 3 Ø b=1 vb− · kb ≈ Ø Iα− (ε, κὰε , iη T α þ) = α∈I∪R Ø η α þ = |η| Iα− (ε, γεὰ , i|η|). (46) One may write a similar equation to (44) for vt− ρTrwε [vt− ] 2 ≈ |η| ≈ 3 c2T |ES− |2 + c2T |EZ− |2 + Ø c2T −,− Fα,β 3 q b=1 + c2L Ø −,− Fα,β α,β∈R þ η α þ =β= |η| 3 3 Ø µØ Trwε [vb− + Vx− · kb ] + λwε [ vb− · kb ]. 2 b=1 b=1 It remains to estimate the cross terms Trwε (vt+ , vt− ), and wε ( + α,β∈R þ 0 ,e0 α þ =β=e 1 2 c2L |EL− |2 vb+ · kb , ρTrwε (vt+ , vt− ) 2 ≈ −|η| 3 3 q b=1 3 q b=1 (47) Trwε (vb− + Vx− · kb , vb− + Vx− · kb ) vb− · kb ). By (37) and (38) and approximation (40), one gets +,− c2T FS,S + +,− c2T FZ,Z + Ø c2T +,− Fα,β + +,− c2L FL,L + c2L α,β∈R þ 0 ,e0 α þ =β=e 1 2 Ø α,β∈R þ η α þ =β= |η| +,− Fα,β 4 On the other hand, equations (41) and (45) lead to 3 Ø b=1 Trwε (vb+ + Vx+ · kb , vb− + Vx− · kb ) ≈ 2|η|2 1 Ø 2 +,− +,− + 2|η|2 Fα,α + 2FL,L α=S,Z è Ø +,− Fα,β +2 α,β∈R þ 0 ,e0 α þ =β=e 1 2 Ø α,β∈R þ η α þ =β= |η| and (42) and (46) imply wε ( 3 Ø b=1 vb+ · kb , 3 Ø b=1 4 +,− vb− · kb ) ≈ |η|2 FL,L + |η|2 Ø α,β∈R þ η α þ =β= |η| +,− Fα,β . é +,− , Fα,β . Wigner transforms and measures 124 Hence, the cross terms have a zero contribution to the measure E t and one has 1 E t = (E+t + E−t ), 2 where E−t 1 = |η|2 c2T 2 3 Ø 1 α=S,Z ρ|γεα |2 2 o{ϕtα,+ }−1 3 1 2 2 1 L 22 + |η| cL ρ|γε | o{ϕtL,+ }−1 + 2 Ø + +,+ Fα,β 4 −,− Fα,β 4 þ 0 ,e0 α þ =β=e 1 2 α,β∈R Ø +,+ Fα,β þ η α þ =β= |η| α,β∈R 4 , and E+t 1 = |η|2 c2T 2 3 Ø 1 α=S,Z ρ|καε |2 2 o{ϕtα,− }−1 31 2 1 + |η|2 c2L ρ|γεL |2 o{ϕtL,− }−1 + 2 Ø + Ø þ 0 ,e0 α þ =β=e 1 2 α,β∈R 4 −,− . Fα,β þ η α þ =β= |η| α,β∈R For α ∈ I, {ϕtα,± }−1 = ϕtα,∓ . For ὰά ∈ R, the inverse of ϕtὰά,± is ϕtάὰ,∓ . Moreover, the flows ϕT and ϕT T keep cT |η| invariant and the flows ϕL and ϕLL keep cL |η| invariant, while for ϕT L and ϕLT one has cL (xT L (y, η))|ξT L (y, η)| = cT (y)|η| and cT (xLT (y, η))|ξLT (y, η)| = cL (y)|η|. It follows that 2E−t = + + + B-2 A -√ √ γεS -- o{ϕtT,+ }−1 + | λ + 2µ|η|γεL |2 o{ϕtL,+ }−1 - µ|η| Z γ A ε BA B-2 + + -√ RSS RZS γεS -- µ|η| - o{ϕtT T,+ }−1 + + RSZ RZZ γεZ -2 A B + -√ RLS L−1 t - λ + 2µ|η| γ + ε -- o{ϕLT,+ } RLZ A B-2 -2 -√ -√ 1 2 γεS -+ + + Lt −1 λ + 2µ|η|R γ o{ϕ } + - µ|η| RSL RZL LL ε T L,+ γεZ - −,− 2 2 −,− + c2T |η|2 FLT,T T + cL |η| FT L,LL . o{ϕtLL,+ }−1 (48) A similar result can be established for E+t . In order to understand the transported terms, let us write the Helmholtz decomposition of the initial conditions as uIε = fε + Ψε , vεI = gε + Θε with fε = ∂x aε , gε = ∂x bε , divΨε = 0 and divΘε = 0. Since ëTε ∂x u − iηTε uëL2 (R2n ) . √ εëuëL2 (Rn ) for u ∈ H 1 (Rn ), one has ë(Id − √ ηη t )Tε gε ëL2 (R6 )3 . ε. 2 |η| (49) 4.2 - Wigner measures for the asymptotic solution 125 On the other hand −iεTε divΘε = η · Tε Θε + Ú i 1 2 (x − y) · Θε (x)e ε η.(y−x)− 2ε (y−x) dx. Since Θε ∈ L2 (Ω)3 , the integral term is of order √ ëη · Tε Θε ëL2 (R6 ) . ε in L2 (R6 ). Thus √ ε. The same arguments hold true for ε−1 Tε uIε because ε−1 uIε ∈ L2 (Ω)3 , leading to ë(Id − √ √ ηη t )Tε fε ëL2 (R6 )3 . ε and ëη · Tε Ψε ëL2 (R6 ) . ε. 2 |η| The Helmholtz decomposition of the initial data implies then a decomposition of their FBI transforms tangentially and orthogonally to η as follows 1 2 Tε ε−1 fε + i(cL |η|)−1 gε =γεL √ η + O( ε), |η| 2 1 √ and Tε ε−1 Ψε + i(cT |η|)−1 Θε = γεS e01 + γεZ e02 + O( ε), in L2 (R3 )3 . We deduce in the sens of measures that |γεL |2 ≈ |Tε (ε−1 fε + i(cL |η|)−1 gε ) |2 , |γεS |2 + |γεZ |2 ≈ |Tε (ε−1 Ψε + i(cT |η|)−1 Θε ) |2 . By Lemma B.1 in chapter 2, Tε |D| ≈ |η|Tε and crude computations show that Tε ϑ ≈ ϑTε for functions ϑ ∈ C ∞ . One obtains the following approximations in the sens of measures 2 √ ρgε − i λ + 2µε−1 |D|fε |2 , 1√ 2 1 2 √ µ|η|2 |γεS |2 + |γεZ |2 ≈ |Tε ρΘε − i µε−1 |D|Ψε |2 . (λ + 2µ)|η|2 |γεL |2 ≈ |Tε 1√ Similar results can be established for κε with a change of signs. Since the FBI transform is related to the Wigner measure (see Lemma 1.2 of [38]) in that |Tε zε |2 ≈ w[zε ] for zε uniformly bounded in L2 (Rn ), the first six be written as the trace of transported Wigner measures of √ terms of (48) may √ √ √ ρgε − i λ + 2µ|D|fε and ρΘε − i µ|D|Ψε . The remaing terms in (48) are however troublesome. Indeed, they exhibit cross quantities γεὰ o{ϕtα,+ }−1 γ¯ε β̀ o{ϕtβ,+ }−1 with α Ó= β, which quantities can not be interpreted as Wigner measures. However, if one assumes one additionnal hypothesis D1. The Wigner measures associated to fε and gε are zero, or D2. The Wigner measures associated to Ψε and Θε are zero, which is a commonly assumed hypothesis when studying the Wigner measure for a system with two wave speeds at the boundary (see [14] for a similar hypothesis), then Wigner transforms and measures 126 the cross terms between γεL and (γεS , γεZ ) vanish. For example under D1, (48) may be written in terms of transported initial Wigner measures as √ √ −1 t 2E−t = Trw[ ε ] o {ϕT,+ } 2 1 ρΘε − i µ|D|Ψ √ √ + Tr RT+T · RT+T w[ ρΘε − i µ|D|Ψε ] o{ϕtT T,+ }−1 1 2 √ √ + Tr RT+L · RT+L w[ ρΘε − i µ|D|Ψε ] o{ϕtT L,+ }−1 , (50) with η T ) , |η| + 0 + 0 + 0 + 0 = (RSS e1 + RSZ e2 )(e01 )T + (RZS e1 + RZZ e2 )(e02 )T . + 0 + 0 RT+L = (RSL e1 + RZL e2 )( and RT+T Above, the exponent T denotes transposition. A similar expression can be obtained for E+t √ √ t −1 2E+t = Trw[ ε ] o {ϕT,− } 2 1 ρΘε + i µ|D|Ψ √ √ + Tr RT−L · RT−L w[ ρΘε + i λ + 2µ|D|Ψε ] o{ϕtT L,− }−1 (51) 2 1 √ √ − − −1 t + Tr RT T · RT T w[ ρΘε + i µ|D|Ψε ] o{ϕT T,− } . ± 2 ± 2 ± 2 One can prove a similar result under D2, with coefficients (RLL ) and (RLT ) = (RLS ) + ± 2 (RLZ ) . This leads to our final theorem Theorem 4.1. Suppose that the initial conditions satisfy assumtions A1-A2 and D1 or D2. Assume the following further assumptions C1. The Wigner measures of v Iε and ∂xb uIε , b = 1, 2, 3, are unique, C2. v Iε and ∂xb uIε , b = 1, 2, 3 are ε-oscillatory (see equation (53), chapter 2), C3. The Wigner measures of v Iε and ∂xb uIε , b = 1, 2, 3 do not charge the set R3 × {ξ = 0}, C4. The Wigner measures of v Iε and ∂xb uIε , b = 1, 2, 3 do not charge T r. 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