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Dans cette thèse, nous étudions la contrôlabilité et la stabilisation pour des équation hyperboliques et dispersives.

La première partie de cette thèse est consacrée à la stabilisation du système de Stokes hyperbolique. La propagation des singularités pour le système de Stokes semi-classique est établie dans chapitre 1. La preuve repose sur la stratégie de Ivrii et Melrose-Sjöstrand. Cependant, par rapport à l'opérateur de Laplace, la difficulté est causée par la pression qui a un effet non trivial pour les solutions concentrées au bord. Nous utilisons la paramétrix des solutions près d'un point elliptique ou hyperbolique. En suite, on traite les solutions concentrées près de l'ensemble «glancing» par une décomposition micro-locale. L'effet de la pression est alors bien contrôlé grâce à la géométrie. Finalement on utilise un argument récurrence pour terminer la preuve. Par conséquent, nous prouvons la stabilisation du système de Stokes hyperbolique dans le chapitre 2 sous la condition de contrôle géométrique sur le support de l'amortissement.

La deuxième partie est consacrée à la contrôlabilité et la stabilisation de l'équation de Kadomtsev-Petviashvili ( KP en bref). Dans le chapitre 3, en utilisant l'analyse semiclassique, nous avons prouvé la contrôlabilité verticale pour des données dans L 2 (T). De plus, un résultat négatif concernant la contrôlabilité horizontale est aussi obtenu. Dans le chapitre 4, nous considérons la contrôlabilité de l'équation de KP-I linéaire. C'est un modèle intéressant dans lequel la vitesse de groupe peut être dégénéré. Plus général, on a obtenu le plus petit ordre requis pour assurer l'observabilité des équations de KP-I fractionnaire linéaire. Finalement dans le chapitre 5, nous avons montré la contrôlabilité et la stabilisation des 'equations de KP-II et 5KP-II avec grandes données initiales dans l'espace de Sobolev, si la donnée initiale satisfait certaines hypothèses de compacité partielles. Ceci généralise la contrôlabilité des solutions de KP-II avec données petites dans le chapitre 3. i

Les objectives de cette thèse sont les problèmes de contrôlabilité et stabilisation pour des équations hyperboliques et dispersives. Ces équations peuvent être considérées comme les systèmes dynamiques de la forme

∂U ∂t + AU = N (U ), (0.1.1)
où l'opérateur linéaire A est anti-symmétrique sur certain espace hibertien H, tandis que le terme non-linéaire N (U ) est assez special pour que le système (0.1.1) possède des quantités conservées, comme la masse et l'énergie, etc. Cela constitue une grande famille de modèles mathématiques découlant de la physique et des sciences de l'ingénieur, qui décrivent les phénomènes de propagation des ondes.

Contrôlabilité, Observabilité et Stabilisation

Le problème de contrôle s'écrit sous la forme abstraite suivante:

   ∂U ∂t + AU = N (U ) + Bf, t ∈ [0, T ] U (0) = U 0 ∈ H (0.1.2)
où H est un espace hibertien sur lequel le demi-groupe engendré par A est bien défini, B est l'opérateur de contrôle avec un certain contrôle f à choisir. Pour le problème de contrôle interne, Bf est supporté dans un ouvert et le cas le plus fréquent est B = 1 ω .

La question de la contrôlabilité exacte est de savoir si étant donné un état initial U 0 au temps t = 0 et un état final U 1 au temps t = T , on peux trouver un contrôle f ∈ L 2 ([0, T ]; ;H) tel que la solution de (0.1.2) satisfait à U (T ) = U 1 . En particulier, lorsque l'état final est donné nul, nous parlons de contrôlabilité exacte à zéro.

Pour le système linéaire

   ∂U ∂t + AU = Bf, t ∈ [0, T ] U (0) = U 0 ∈ H (0.1.3)
La méthode HUM de Jacques-Louis Lions [START_REF] Lions | On some hyperbolic equations with a pressure term, partial differential equations and related subjects: Proceedings of the conference dedicated do l. nirenberg[END_REF] ramène la contrôlabilité exacte à zéro à la validité d'une inégalité d'observabilité pour le système adjoint. Plus précisément, pour le système adjoint

   ∂U ∂t -A * U = 0, t ∈ [0, T ] U (T ) = U T ∈ H, (0.1.4)
la contrôlabilité exacte à zero est equivalent à l'inégalité d'observabilité

U (0) 2 H ≤ C T 0 B * U (t) 2 H dt.
Pour les systèmes réversibles, on a A * = -A (typiquement les équations hyperboliques et dispersives que l'on va considérer dans cette thèse), et donc la contrôlabilité exacte à zero implique la contrôlabilité exacte.

La stabilisation est une autre terminologie liée à la contrôlabilité. Il s'agit de remplacer le contrôle Bf dans (0.1.2) ou (0.1.3) par un retour qui ne dépend que de la solution.

Stabilisation signifie que l'on peux utiliser le retour pour stabiliser le système dans le sens que la solution converge vers certain équilibre. Ce problème est pratique dans des domaines de l'ingénieur, car en réalité, les équilibres sont souvent instables. Dans de nombreux cas, les retours servent de termes d'amortissement qui décroissent l'énergie des systèmes.

Quelques résultats sur des équations linéaires typiques

Dans cette section, nous allons rappeler quelques résultats sur la contrôlabilité de trois types d'équations d'évolution linéaires.

Equation de la chaleur

Equation de la chaleur ∂ t u -∆u = 0 est une EDP de type parabolique qui décrit la distribution de la chaleur (ou variation de témperature) dans une région au fil du temps. La propriété de dissipation nous permet d'obtenir la contrôlabilité exacte à zéro en temps T > 0 quelconque par un contrôle supporté dans un ouvert ω ⊂ Ω quelconque.

Théorème 0.1 (Lebeau-Robbiano [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF],Fursikov-Imanuvilov [START_REF] Vladimirovič | Controllability of evolution equations[END_REF])

Soit Ω un domaine borné régulier de R d et ω ⊂ Ω un ouvert non vide. Soient T > 0 et il existe un contrôle g ∈ L 2 ([0, T ]; L 2 (M )), supporté par [0, T ]×ω, tel que la solution unique de ∂ 2 t u -∆u = g, (u, ∂ t u)| t=0 = (u 0 , u 1 ) satisfait (u, ∂ t u)| t=T = (0, 0). De plus, pour l'équation d'ondes amorties ∂ 2 t u -∆u + a(x)∂ t u = 0, avec l'amortissement 0 ≤ a ∈ C(M ) et ω = {x : a(x) > 0},on a stabilisation uniforme

(u(t), ∂ t u(t)) H 1 ×L 2 ≤ C 0 e -γt (u(0), ∂ t u(0)) H 1 ×L 2 .
Ce résultat est initialement montré par la propagation du front d'onde. La preuve peut être simplifiée en utilisant la mesure de défaut introduit indépendamment dans [START_REF] Gérard | Microlocal defect measures[END_REF], [START_REF] Tartar | H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations[END_REF]. Il y a plein de travaux pour les contrôlabilité et stabilisation dans ce cadre d'analyse microlocale. Par exemple, le meilleur taux de décroissance exponentielle en fonction du spectre et des moyennes de a(x) sur les géodésiques de M est obtenu par Lebeau dans [START_REF] Lebeau | Equation des ondes amorties[END_REF], la stabilisation uniforme pour un amortissement peu-régulier a(x) est obtenu dans Burq-Gérard [START_REF] Burq | Stabilisation of wave equations on the torus with rough dampings[END_REF], le meilleur taux de décroissance polynomial est obtenu dans [START_REF] Nalini Anantharaman | Sharp polynomial decay rates for the damped wave equation on the torus[END_REF] lorsque la condition de contrôle géométrique n'est pas satisfaite, et les contrôlabilités et stabilisations des autres systèmes hyperboliques (le système de Lamé, l'équation d'ondes couplé) sont obtenus dans Burq-Lebeau [START_REF] Burq | Mesures de défaut de compacité, application au systeme de lamé[END_REF], Dehman-Raymond [START_REF] Dehman | Exact controllability for the lamé system[END_REF], Dehman-Le Rousseau-Léautaud [START_REF] Dehman | Controllability of two coupled wave equations on a compact manifold[END_REF], etc. Dans le chapitre 2, nous considérons la stabilisation d'un système de Stokes hyperbolique. Ce système est lié au système de Lamé au point de vue de la modélisation. Il mélange le phénomène de la propagation d'ondes et le problème aux limites d'opérateur elliptique.

Équation de Schrödinger

L'équation de Schrödinger i∂ t u = (-∆u + V (x))u joue un rôle fondamental en mécanique quantique, comme l'équation de Newton en mécanique classique. La solution u s'interprète comme la fonction d'onde d'un système quantique.

Le résultat le plus général sur le contrôle de l'équation de Schrödinger libre est du à Lebeau.

Théorème 0.3 (Lebeau [START_REF] Lebeau | Contrôle de l'équation de schrödinger[END_REF]) Soit M une variété riemannienne compacte et ω ⊂ M satisfaisant la condition de contrôle géométrique. Alors pour T > 0 et u 0 ∈ L 2 (M ) quelconques, il existe un contrôle g ∈ L 2 ([0, T ]; L 2 (M )), supporté dans [0, T ] × ω tel que la solution de i∂ t u + ∆u = 1 ω g, u(0) = u 0 satisfait u(T ) = 0.

Dans ce résultat, la condition de contrôle géométrique n'est pas nécessaire. En raison de la vitesse infinie de propagation des équations dispersives, la condition géométrique sur ω pour avoir le contrôlabilité dépend des propriétés géométriques de la variété sous-jacente. Principalement, si les trajectoires sont instables(exemple typique: le tore), on pourrait avoir l'observabilité de la solution de l'équation de Schrödinger dans un ouvert ω non vide quelconque. En revanche, si les trajectoires sont stables (exemple typique: la sphère), il y a certains ouverts dans lesquels on ne pourrait pas avoir l'observabilité en temps fini. Ceci est due à la concentration à l'équateur des fonctions propres de hautes fréquences. La compréhension de l'effet de la géométrie sur la dispersion et l'observabilité est beaucoup plus compliquée. Le résultat suivant semble un point de départ de cette direction.

Théorème 0.4 (Jaffard [START_REF] Jaffard | Contrôle interne exact des vibrations d'une plaque rectangulaire[END_REF],Burq-Zworski [START_REF] Burq | Control for schrödinger operators on tori[END_REF]) Soit M = T 2 . La conclusion du Théorème 0.3 est vraie pour un ouvert ω ⊂ T 2 non-vide quelconque.

La contrôlabilité exacte de l'équation de Schrödinger avec un potentiel V ∈ C ∞ (T 2 ) est aussi obtenu dans [START_REF] Burq | Control for schrödinger operators on tori[END_REF] en utilisant la méthode de forme normale semi-classique. Ce résultat est amélioré pour V ∈ L ∞ dans Anantharaman-Macià [START_REF] Anantharaman | Semiclassical measures for the schrödinger equation on the torus[END_REF] comme une conséquence de la structure fine de la mesure semi-classique. Dans Bourgain-Burq-Zowrski [START_REF] Bourgain | Control for schrödinger operators on 2-tori: rough potentials[END_REF], le résultat dans [START_REF] Burq | Control for schrödinger operators on tori[END_REF] est amélioré au cas V ∈ L 2 (T 2 ) par une méthode différente à l'aide de la dispersion.

Quelques résultats pour les équations des ondes et les équations dispersives nonlinéaires

Jusqu'à présent, Il semble que l'on ne s'ait pas sorti d'un petit voisinage des équations linéaires dans le domaine de l'analyse des EDP. Les méthodes pour traiter les problèmes nonlinéaires sont extrêmement limitées. Néanmoins, de nombreux phénomènes de physique modélisés par EDPs sont, soit avec de bonnes structures nonlinéaires, soit une petite pertubation des équations linéaires. Sous certain niveau de non-linéarité, ces EDPs se comprennent efficacement.

Beaucoup d'EDPs dispersives sont semi-linéaires, dans le sens où elles peuvent être résolues par la formule d'itération de Duhamel U (t) = e -tA U 0 + t 0 e -(t-t )A N (U (t ))dt dans des espaces fonctionnels adaptés. En effet, la résolution dépend de la propriété dispersive du groupe linéaire e -tA ainsi que de la non-linéairité que l'on peut autoriser. D'un côté, l'analyse harmonique nous fournit divers types d'estimations de e -tA avec lesquelles on a des flexibilités pour résoudre les équations dans les espaces fonctionnels plus petits. D'autre part, les non-linéarités raisonnables doivent respecter la règle: grande non-linéarité pour des données petites(sur-critique) tandis que non-linéarité petite pour des données grandes (sous-critique). D'un point de vue heuristique, les hautes fréquences de la solution des équations sous-critiques se comportent comme les équations linéaires à mesure que le temps passe. Ceci est le point clé pour avoir une bonne théorie de Cauchy.

La théorie du contrôle dépasse la théorie de Cauchy. Une fois qu'on a établi le caractère bien-posé et la contrôlabilité pour la partie linéaire, la contrôlabilité des équations nonlinéaires avec données initiales petites est prèsque gratuite. Ce type de résultat de s'appele contrôlabilité locale, ce qui est souvent suivi d'un argument de point fixe. Pour obtenir la contrôlabilité pour des données grandes, une façon possible est la stratégie de «stabilisation-contrôlabilité locale» que nous allons expliquer. Cette méthode s'effectue en deux étapes. La première étape consiste à établir la contrôlabilité locale et la deuxième étape consiste à obtenir la stabilisation pour les équations nonlinéaires. Alors le contrôle désiré est la combinaison d'un contrôle de retour et un contrôle local. Le désavantage de cette méthode est aussi claire. Pour réaliser le contrôle, on a besoin d'attendre longtemps, ce qui n'est pas pratique dans le domaine de l'ingénierie.

Dans ce qui suit, nous allons présenter des résultats sur la stabilisation des 'equations des ondes et dispersives. Théorème 0.5 (Dehman-Lebeau-Zuazua [START_REF] Dehman | Stabilization and control for the subcritical semilinear wave equation[END_REF],Dehman-Gérard [START_REF] Dehman | Stabilization for the nonlinear Klein Gordon equation with critical exponent[END_REF]) Soient a(x) ≥ c 0 > 0 pour |x| ≥ R et 1 < p ≤ 5. Étant donnée E 0 > 0, il existe des constantes C > 0, γ > 0, telles que la solution u de

∂ 2 t u -∆u + a(x)∂ t u + u + |u| p-1 u = 0, (t, x) ∈ R + × R 3 (u, ∂ t u)| t=0 = (u 0 , u 1 ) satisfait E[u](t) ≤ Ce -γt E[u](0), ∀t ≥ 0,
à condition que E[u](0) ≤ E 0 , où la fonctionnelle d'énergie est

E[u](t) = 1 2 R 3 (|∇ t,x u(t, x)| 2 + |u(t, x)| 2 )dx + 1 p + 1 R 3 |u(t, x)| p+1 dx.
Dans ce théorème, le cas d'énergie sous-critique 1 < p < 5 est résolu dans [START_REF] Dehman | Stabilization and control for the subcritical semilinear wave equation[END_REF] tandis que le cas critique p = 5 est obtenu dans [START_REF] Dehman | Stabilization for the nonlinear Klein Gordon equation with critical exponent[END_REF] en utilisant la méthode de décomposition en profils introduit initialement par Bahouri-Gérard [START_REF] Bahouri | High frequency approximation of solutions to critical nonlinear wave equations[END_REF] pour les équations des ondes. La généralisation de ce résultat à une variété compacte riemannienne avec certaines conditions supplémentaires est due à Laurent [START_REF] Laurent | On stabilization and control for the critical klein-gordon equation on a 3-d compact manifold[END_REF], voir aussi Joly-Laurent [START_REF] Joly | Stabilization for the semilinear wave equation with geometric control condition[END_REF] pour une méthode différente.

Pour l'équation de Schrödinger nonlinéaire sur une variété compacte, les contrôlabilités et stabilisations sont initialement obtenues sous certaines hypothèses raisonnables. Plus précisément, on a Théorème 0.6 (Dehman-Gérard-Lebeau [START_REF] Dehman | Stabilization and control for the nonlinear schrödinger equation on a compact surface[END_REF]) Soient M une variété riemannienne compacte de dimension 2(sans bord), 0 ≤ a ∈ C ∞ (M ) et ω = {x : a(x) > 0}. Supposons que ω satisfait la condition de contrôle géométrique et la propriété de prolongement unique pour certain opérateur de Schrödinger linéaire particulier. Alors, étant donnée R 0 > 0, il existe constantes C > 0, γ > 0, telles que la solution u de

i∂ t u + ∆u -a(x)(1 -∆) -1 a(x)∂ t u = |u| p-1 u satisfait E[u](t) ≤ Ce -γt E[u](0), ∀t ≥ 0 à condition que u(0) H 1 ≤ R 0 , où la fonctionnelle d'énergie est E[u](t) = 1 2 M |∇ x u(t, x)| 2 + 1 p + 1 M |u(t, x)| p+1 dx.
Pour NLS cubique en dimension 1, la stabilisation dans L 2 est obtenu dans Laurent [START_REF] Laurent | Global controllability and stabilization for the nonlinear schrödinger equation on an interval[END_REF]. L'analogie du Théorème 0.6 en dimension 3 est obtenue dans Laurent [START_REF] Laurent | Global controllability and stabilization for the nonlinear schrödinger equation on some compact manifolds of dimension 3[END_REF].

Un autre exemple typique d'équation dispersive est l'équation de Korteweg-de Vries(KdV bref). C'est un modèle mathématique pour décrire les surfaces d'eau peu profonde dans un canal. L'équation de KdV ∂ t u + ∂ 3

x u + u∂ x u = 0 a de riches propriétés mathématiques. En particulier, c'est un système intégrable avec une infinité de lois de conservation. Concernant la stabilisation de KdV, on a le résultat suivant:

Théorème 0.7 (Laurent-Rosier-Zhang [START_REF] Laurent | Control and stabilization of the korteweg-de vries equation on a periodic domain[END_REF]) Il existe γ > 0 de sorte que pour tout R 0 > 0, il existe une constante C > 0 telle que pour u 0 ∈ L 2 0 (T) avec u 0 L 2 ≤ R 0 , la solution u de

∂ t u + ∂ 3 x u + u∂ x u = Au
avec donnée initiale u 0 satisfait u(t) L 2 ≤ Ce -γt u 0 L 2 , ∀t ≥ 0, où l'opérateur A est définie par

Af (x) = a(x) f (x) - T a(x )f (x )dx
pour certaine fonction a ∈ C ∞ (T) fixé avec T a = 1.

Dans la deuxième partie de cette thèse, nous allons montrer quelques résultats sur la contrôlabilité et la stabilisation de l'équation de Kadomtsev-Petviashvili equation, ce qui est la généralisation de KdV en dimension 2. 0.2 Résultats principaux obtenus par l'auteur 0.2.1 Partie I La motivation des résultats dans la première partie de cette thèse est double. Considérons le système de contrôle de Stokes

     ∂ t v -∆v + ∇p = f 1 ω dans R × Ω, div v = 0 dans R × Ω, v = 0 sur R × ∂Ω, v(0, x) = v 0 ∈ H, (0.2.1) où H = {u ∈ L 2 (Ω) d : div u = 0, u • ν| ∂Ω = 0}
et ν(x) est la normale sortent unitaire de Ω au point x ∈ ∂Ω. Il est bien connu (voir par exemple [START_REF] Fernández-Cara | Local exact controllability of the navier-stokes system[END_REF]) que pour tout v 0 ∈ H, (0.2.1) est contrôlable exacte à zéro à temps T > 0 quelconque. De plus, on a

f 1 ω L 2 (Ω) ≤ C(T ) v 0 L 2 (Ω) .
Une question naturelle consiste à le coût de contrôle C(T ). D'après le «Control Transmutation Method»(CTM) introduit par Miller [START_REF] Miller | The control transmutation method and the cost of fast control[END_REF], la contrôlabilité exacte à zéro du système de Stokes hyperbolique

   ∂ 2 t u -∆u + ∇p = f 1 ω in R × Ω, div u = 0 in R × Ω, u = 0 on R × ∂Ω, (0.2.2)
à un certain temps T 0 > 0 entraîne que l'ordre de coût de contrôle du système de Stokes (0.2.1) est e C T quand T → 0, comme dans le cas de l'équation de la chaleur. Le système de Stokes hyperbolique ((0.2.2) avec f = 0) est intéressant en soi. Il est lié à un modèle de l'élasticité pour des milieux incompressibles. Plus précisément, il découle du système de Lamé dans la théorie de l'élasticité linéaire lorsque un paramètre tend vers l'infini ( [START_REF] Lions | On some hyperbolic equations with a pressure term, partial differential equations and related subjects: Proceedings of the conference dedicated do l. nirenberg[END_REF]).

Le problème que nous allons considérer dans cette thèse est la stabilisation du système de Stokes hyperbolique

     ∂ 2 t u -∆u + ∇p + a(x)∂ t u = 0 dans R × Ω, div u = 0 dans R × Ω, u = 0
sur R × ∂Ω, (u(0, x), ∂ t u(0, x)) = (u 0 , v 0 ) ∈ V × H, (0.2.3) où V = {u ∈ H 1 0 (Ω) d : div u = 0} . Dans (0.2.3), l'amortissement a ∈ L ∞ (Ω) est satisfait a(x) ≥ 0, pour tout x ∈ Ω.

Si u = u(t, x) est une (assez lisse) solution de ce système, on définit son énergie

E[u](t) = 1 2 Ω (|∂ t u(t, x)| 2 + |∇u(t, x)| 2 )dx, ∀t ∈ R.
Lorsque il n'y a pas d'amortissement, c'est-à-dire a ≡ 0, l'énergie est conservée, alors qu'en général elle est décroissante:

dE[u] dt = - Ω a(x)|∂ t u(t, x)| 2 dx ≤ 0.
D'après un argument standard, (voir Chapitre 2 pour plus de détails), le taux de décroissance exponentielle de E[u](t) est équivalent à l'inégalité d'observabilité

E[v](0) ≤ C T 0 T 0 0 ω a(x)|∂ t v(t, x)| 2 dxdt (0.2.4)
pour le système de Stoke hyperbolique original

     ∂ 2 t v -∆v + ∇p = 0 dans R × Ω, div v = 0 dans R × Ω, v = 0 sur R × ∂Ω, (v(0, x), ∂ t v(0, x)) = (v 0 , v 1 ) ∈ V × H, (0.2.5)
Ce système est bien-posé et la régularité cachée ∂ ν u| ∂Ω ∈ L 2 (∂Ω) est vraie si Ω est étoilé (voir [START_REF] Lions | On some hyperbolic equations with a pressure term, partial differential equations and related subjects: Proceedings of the conference dedicated do l. nirenberg[END_REF]). Notons que ω = {x : a(x) > 0}, lorsque ω est intersection de Ω et d'un voisinage de ∂Ω dans R d , l'observabilité (0.2.4) est établie dans [START_REF] Chaves-Silva | A hyperbolic system and the cost of the null controllability for the stokes system[END_REF] par la méthode de multiplicateur élémentaire . Ceci permet d'obtenir (voir [START_REF] Chaves-Silva | A hyperbolic system and the cost of the null controllability for the stokes system[END_REF]) l'ordre de coût du contrôle du système de Stokes, si Ω est étoilé. Par rapport à l'équation des ondes, il est naturel de s'interroger si (0.2.4) est varie lorsque ω satisfait la condition de contrôle géométrique. Dans le chapitre 2 de cette thèse, nous allons donner une réponse affirmative. Théorème 0.8 Soit Ω ⊂ R d est un domaine lisse, borné et étoilé. On suppose que les géodésiques de Ω n'ont pas de contact d'ordre infini avec ∂Ω. a ∈ C(Ω) est une fonction non-negative support dont le support satisfait la condition de contrôle géométrique. Alors, il existe les constantes C 0 et α telle que pour tout (v 0 , v 1 ) ∈ V × H, la solution v(t) de (0.2.5) satisfait (0.2.4).

Par rapport au problème aux limites pour l'équation des ondes étudié dans [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF], la difficulté est la présence du terme ∇p dont le comportement est assez compliqué au bord. On se ramène à l'observabilité du système de Stokes semi-classique.

Une réduction heuristique est la suivante. D'après l'équation, on note que ∇p ∈ H -1 loc (R × Ω) et pour t fixé, p(t, •) est une fonction harmonique dans Ω. Micro-localement près de la variété caractéristique τ 2 = |ξ| 2 , on a 1 τ ∇ x 1 τ p ∈ L 2 (Ω), où p est la transformation de Fourier en temps t. Choisissons la paramètre semi-classique h = 1 τ 1 et prenons la transformation de Fourier en variable de temps de l'équation, on a (-h 2 ∆ -1) u + h∇ x q = 0, h div u = 0 (0.2.6) avec q = 1 τ p. Notons que la solution ( u, h -1 q) est une fonction propre du système de Stokes statique -∆v + ∇P = λ 2 v, div v = 0 avec valeur propre λ = h -1 . Plus général, nous considérons le système

   -h 2 ∆v -v + h∇q = f dans Ω h div v = 0 in Ω v| ∂Ω = 0 (0.2.7)
avec f ∈ H petite au certain sens. Il est prouvé dans le chapitre 2 que le Théorème 0.8 est une conséquence de l'observabilité pour le système (0.2.7):

v L 2 ≤ C a 1/2 v L 2 + 1 h f L 2 (0.2.8)
Pour montrer (0.2.8), nous allons établir un théorème de propagation des singularités dans le chapitre 1, dans le cadre de la mesure semi-classique, qui est intéressant en soi.

Théorème 0.9

Soit Ω ⊂ R d est un domaine lisse, borné et étoilé. On suppose que les géodésiques de Ω n'ont pas de contact d'ordre infini avec ∂Ω. Supposons que (u k ) et une suite de solutions du problème de quasi-mode (0.2.7) avec paramètre semi-classique h = h k . Supposons que f k ∈ H, f k L 2 (Ω) = o(h k ) et u k convergent faiblement vers 0 dans L 2 (Ω). Soit µ est une mesure semi-classique associé à une sous-suite de la paire (u k , h k ), alors suppµ est invariante le long du flot de Melrose-Sjöstrand.

Nous faisons quelques remarques sur ce résultat. Premièrement, la mesure µ est à valeur matricielle hermitienne, et on n'a pas obtenu l'invariance de la mesure µ mais celle de supp(µ). Deuxièmement, notre hypothèse que u k tendent vers 0 dans L 2 (Ω) est naturelle, car chaque suite de la fonction propre de l'opérateur de Stokes est un exemple important.

La démonstration est adaptée de la stratégie de V.Ivrii et Melrose-Sjöstrand [START_REF] Richard | Singularities of boundary value problems[END_REF]. Nous allons donner le schéma comme suit:

• Propagation à l'intérieur: Pour un opérateur pseudo-différentiel A à support compact dans Ω × R d , l'intégration par parties et le calculs symbolique impliquent

i h [-h 2 ∆ -1, A]u, u L 2 (Ω) = o(1), h → 0.
• On construit une paramètrix standard et montre que le support de µ ne contient pas de point de la région elliptique E ⊂ T * ∂Ω.

• On utilise la structure de l'équation et on montre que le support de µ se propage le long du rayon brisé .

• On utilise la méthode de commutateur positif pour traiter le cas où les rayons sont proche de G. D'une part, pour un opérateur A bien choisi , on a i h Im((-h 2 ∆ -1)u, Au) L 2 = (B 0 u, u) L 2 où l'opérateur B a un symbole principal réel positif. D'autre part, en utilisant l'équation, on a, après micro-localisation, que

i h Im((-h 2 ∆ -1)u, Au) L 2 = (B 1 q, u) L 2 + o(1), h → 0.
Le but est de montrer que (B 1 q, u) L 2 = o [START_REF] Alinhac | Opérateurs pseudo-différentiels et théorème de Nash-Moser[END_REF]. Pour ce faire, on fait une microlocalisation de la solution selon la distance à la surface «glancing» G que leur traite séparément. Pour la partie proche de G, on utilise le fait que la pression p décroît rapidement dans le sens transverse tandis que les solutions, après micro-localisation, ne peuvent pas se concentrer trop près du bord. Pour la partie un peu loin de G, elle est bien contrôlée grâce à la géométrie et la récurrence.

Partie II

La deuxième partie de cette thèse est consacrée à la contrôlabilité de l'équation de Kadomtsev-Petviashvili(KP bref) ∂ x (∂ t u + ∂ 2 x u + u∂ x u) + λ∂ 2 y u = 0 (0.2.9) où λ = ±1. Cette équation est nommée après Boris Borisovich Kadomtsev et Vladimir Iosifovich Petviashvili. Elle décrit la propagation des ondes nonlinéaires en eau peu profonde. Elle modélise une onde d'eau de grande longueur d'onde avec force de rappel faiblement nonlinéaire et dispersion en fréquences. Dans le cas où la tension superficielle est faible par rapport à la force gravitationnelle, on utilise λ = 1(KP-II) ; dans le cas contraire, on utilise λ = -1(KP-I). En raison de l'asymétrie en variables x et y, les ondes décrites par l'équation de KP se comportent séparément dans la direction de propagation (direction x) et la direction transversale (direction y). L'oscillation en direction y tend à être plus douce (une petite déviation).

Ces équations se posent naturellement dans l'étude de la stabilité transversale des solitons d'équation de Kortewed de-Vries (KdV) et son analogie d'ordre supérieur. Ils sont largement étudiés dans des contextes nombreux et divers ( [START_REF] Bb Kadomtsev | On the stability of solitary waves in weakly dispersing media[END_REF], [START_REF] Molinet | Global well-posedness for the kp-ii equation on the background of a non-localized solution[END_REF], [START_REF] Rousset | Transverse nonlinear instability for twodimensional dispersive models[END_REF]).

Pour des distributions avec moyenne horizontale nulle, c'est-à-dire T u(x, •)dx = 0, on peut réécrire (0.2.9) sous la forme non-locale

∂ t u + ∂ 3 x u + u∂ x u ± ∂ -1
x ∂ 2 y u = 0. Nous allons étudier le problème de contrôlabilité pour l'équation de KP avec données périodiques et nous organisons le problème de contrôle comme

∂ t u + ∂ 3 x u + u∂ x u ± ∂ -1 x ∂ 2 y u = Gf.
L'opérateur G est construit afin de maintenir la moyenne horizontale nulle. Plus précisément, pour un domaine de contrôle vertical de la forme ω = (a, b) × T, on fixe une fonction réelle non-négative g ∈ C 2 c (ω) avec T g = 1. Puis, on définit le contrôle par G(h)(x, y) = G ⊥ (h)(x, y) := g(x) h(x, y) -

KP-II linéaire

Dans le chapitre 3, nous allons montrer la contrôlabilité exacte de l'équation de KP-II linéaire lorsque le domaine de contrôle est une bande verticale. Ce type de contrôle n'est pas artificiel puisque l'équation de KP modélise l'onde d'eau en dimension 2 dans un canal étroit, et la direction de propagation d'onde doit être proche de l'axe de x en réalité.

Théorème 0.10 Étant données T > 0, u 0 , u 1 ∈ L 2 0 (T), il existe un contrôle f ∈ L 2 ((0, T ); L 2 (T)), tel que la solution u de

∂ t u + ∂ 3 x u + ∂ -1 x ∂ 2 y u = Gf, u| t=0 = u 0 avec G = G ⊥ satisfait u| t=T = u 1 .
Lorsque le domaine de contrôle ω est une bande horizontale, on ne peut pas avoir la contrôlabilité de l'équation de KP-I linéaire et KP-II linéaire, pour T > 0 quelconque. Proposition 0.1 Supposons que ω = T × (a, b). Pour T > 0 quelconque, la contrôlabilité déquation de KP-I linéaire et KP-II linéaire n'est pas vraie.

En utilisant l'estimation bilinéaire de KP-II introduit par J.Bourgain dans [START_REF] Bourgain | On the cauchy problem for the kadomstev-petviashvili equation[END_REF], la contrôlabilité locale en temps petit de l'équation de KP-II est aussi obtenue dans le chapitre 3.

Théorème 0.11 Soit T > 0. Il existe r > 0 telle que pour toute u 0 , u 1 ∈ L 2 0 (T 2 ) ayant u 0 L 2 (T 2 ) ≤ r et u 1 L 2 (T 2 ) ≤ r, il existe un contrôle f ∈ L 2 ((0, T ); L 2 0 (T 2 ), tel que la solution u de

∂ t u + ∂ 3 x u + ∂ -1 x ∂ 2 y u + u∂ x u = Gf, u| t=0 = u 0 avec G = G ⊥ satisfait u| t=T = u 1 .
L'esquisse de la démonstration du Théorème 0.10 se déroule comme suit. D'après la méthode HUM standard, on se ramène à montrer l'observabilité suivante

u(0) 2 L 2 ≤ C T T 0 Gu(t) 2 L 2 dt (0.2.12)
pour toute solution u de l'équation de KP-II linéaire. On remarque que le symbole principal de l'équation est p + = k 3 -m 2 k . Contrairement à l'équation des ondes et à l'équation de Schrödinger, il n'y a pas d'homogénéité entre les variables x et y. Nous séparons les fréquences en plusieurs régimes:

• |D x | 2 ∼ |D y | ∼ 1
h 2 , la vitesse de groupe en direction de x est ∼ 1

h 2 • |D x | 2 ∼ 1 h 2 1
|D y | ∼ 1 h 2 , la vitesse de groupe en direction de x est ∼ 1

h 2 1 . • 1 |D x | 2 ∼ 1 h 2 1
|D y | ∼ 1 h 2 , la vitesse de groupe en direction de x est ∼ h 1 h 2 .

• 1 ∼ |D x | 2 |D y | ∼ 1 h 2 . • 1 ∼ |D x | 2 ∼ |D y | ∼ 1.
Pour les premiers trois régimes, on utilise la forte propagation en direction x. Pour le quatrième régime, on utilise la dispersion en direction y. Le cinquième régime se traite par la compacité .

KP-I Linéaire

Dans le chapitre 4, nous étudions la contrôlabilité de l'équation de KP-I linéaire

∂ t u + ∂ 3 x u -∂ -1 x ∂ 2 y u = 0 (0.2.13)
où le domaine de contrôle ω = (a, b)×T est une bande verticale. Contrairement à KP-II, la vitesse de groupe de (0.2.13) s'annule quelque part car la symbole L -= p -= k 3 + m 3 k nést pas monotone en variable k. Malgré cette dégénérescence, la dispersion forte en direction de x nous permet d'obtenir Théorème 0.12 Étant données

T > 0 et u 0 ∈ L 2 0 (T 2 ), u 1 ∈ L 2 0 (T 2 ), il existe un contrôle f ∈ L 2 ((0, T ); L 2 (T 2
)) tel que la solution u de l'équation

∂ t u + ∂ 3 x u -∂ -1 x ∂ 2 y u = Gf, u| t=0 = u 0 (0.2.14) satisfait u| t=T = u 1 .
Afin de comprendre l'effet entre la dispersion et la dégénérescence, nous étudions aussi la validité de l'observabilité pour des équations de KP-I linéaire fractionnaire de la forme

∂ t u -|D x | α ∂ x u -∂ -1 x ∂ 2 y u = 0. (0.2.15)
Nous allons montrer la dichotomie suivante qui donne le taux de dispersion nécessaire pour avoir l'observabilité (ou également, la contrôlabilité exacte).

Théorème 0.13 • Si α ≥ 1, alors pour T > 0 quelconque, il existe C T > 0, telle que

u(0) 2 L 2 (T 2 ) ≤ C T T 0 T 2 |Gu(t, x, y)| 2 dxdydt
est vraie pour toute la solution u de (0.2.15).

• Si 0 < α < 1, alors pour T > 0 quelconque, il existe une suite de solutions (u n ) de (0.2.15), telle que

lim n→∞ T 0 T 2 |Gu n (t, x, y)| 2 dtdxdy u n (0) 2 L 2 (T 2 ) = 0.
On esquisse le point clé de la démonstration du Théorème 0.13. Pour se ramener au problème unidimensionnel, on est obligé de traiter les cinq régimes comme ceux pour KP-II. La seul différence intervient dans le régime

|D x | ∼ |D y | 2 α+2 ∼ 1 h , puisque le symbole p = |ξ| α ξ + 1 ξ a deux points critiques ξ ± 0 = ± 1 α+1 1 α+2
. Un effort supplémentaire est nécessaire pour l'analyse près de ces deux points.

On note φ

α (ξ) = |ξ| α ξ + 1 ξ , u = e i ξ 0 /h x v, alors h α+1 D t v -Φ(hD x )v = 0, (0.2.16) avec Φ(ξ) = φ(ξ + h ξ 0 /h ). Le point critique de Φ près d'origine est σ h = ξ 0 -h ξ 0 /h ∈ (0, h). De plus, Φ(σ h ) = Φ (σ h ) = 0, Φ (σ h ) = 0. Remplaçant v → v exp iΦ(σ h )t h α+1
, on peut supposer que Φ(σ h ) = 0. Donc l'observabilité de u en fréquences localisées à ξ 0 est équivalent à l'observabilité de v en fréquences localisées à σ h ≈ 0. D'un point de vue heuristique, l'équation de KP-I linéaire se ramène à (0.2.16)

h α+1 D t v -Φ (0) 2 h 2 D 2 x v = O(h) par développement de Taylor, où v se localize en fréquence |D x | 1 h . Proposition 0.2 Soit 0 ≤ α < 1.
Étant donnée T > 0, il existe une suite w n de solutions de l'équation

h α+1 n D t w n + (h n D x ) 2 w n = 0, telle que T 0 T |g(x)w n (t, x)| 2 dxdt w n (0) 2 L 2 → 0, n → ∞.

D'après cette propriété,

• Si 0 ≤ α < 1, l'observabilité localisée n'est pas vraie pour tout T > 0. Ceci dit que la dégénérescence bat la dispersion.

• Si α = 1, la dispersion est la même que pour l'équation de Schrödinger. Si α > 1, l'effet de dispersion est plus fort que l'équation de Schrödinger, donc dans ces deux cas, l'observabilité localisé est vraie pour tout T > 0. Ceci dit que la dispersion bat la dégénérescence.

KP-II avec donnée grande

Dans le chapitre 5, nous étudions la contrôllabilité et la stabilisation pour des équations de types de KP-II avec données initiales grandes. Les résultats principaux dans ce chapitre sont les suivants.

Théorème 0.14

Soient s > 0, R 0 > 0. Il existe C > 0, γ > 0 et δ 0 > 0 tels que u(t) H s (T 2 ) ≤ Ce -γt u 0 H s (T 2 ) , ∀t > 0
pour toute la solution u de l'équation de KP-II amortie

∂ t u + ∂ 3 x u + ∂ -1 x ∂ 2 y u + u∂ x u = -G * Gu avec donné initiale qui satisfait u 0 H s (T 2 ) ≤ R 0 , u 0 H -1,0 (T 2 ) ≤ δ 0 .
En combinant la stabilisation et la contrôlabilité locale établies dans le chapitre 3, on déduit la contrôlabilité suivante

Corollaire 0.1 Soient s > 0, R 0 > 0. Il existe T > 0, δ 0 > 0, telles que si u 0 , u 1 ∈ H s 0 (T 2 ) satisferait u 0 H s (T 2 ) ≤ R 0 , u 1 H s (T 2 ) ≤ R 0 , u 0 H -1,0 (T 2 ) ≤ δ 0 , u 1 H -1,0 (T 2 ) ≤ δ 0 , il existe un contrôle Gh avec f ∈ L 2 ((0, T ); H s (T 2 )) tel que l'équation ∂ t u + ∂ 3 x u + ∂ -1 x ∂ 2 y u + u∂ x u = Gf admet une solution u ∈ C([0, T ]; H s 0 (T 2 )) qui satisfait u(0) = u 0 , u(T ) = u 1 .
On fait quelques remarques sur ces résultats. Premièrement, la condition u 0 H -1,0 ≤ δ 0 1 sera appelée l'hypothèse de petitesse en basse fréquences horizontales(en variable x). On a besoin de cette petitesse parce que dans notre preuve, nous ne somme pas capable de montrer l'observabilité pour le régime dans lequel les fréquences en x sont bornées tandis que les fréquences en y sont hautes. Essentiellement, dans ce régime, l'équation peut être considérée comme un système de Schrödinger nonlinéaire couplé en variable y. Dans ce cas, la propagation en direction x est négligeable et la propagation en variable y ne nous aide pas pour avoir l'observabilité dans une bande verticale. Dans le chapitre 3, on a utilisé l'effet de dispersion (l'inégalité d'Ingham) en variable y pour surmonter cette difficulté. Cependant, puisque H -1 2 ,0 est l'espace fonctionnel critique de KP-II par rapport au changement d'échelle, l'effet nonlinéaire ne pourrait plus être ignoré comme un système de Schrödinger nonlinéaire en y. Donc éliminer l'hypothèse de petitesse en basse fréquences horizontales est un problème compliqué.

On doit également mentionner que la régularité dont on a besoin pour obtenir la stabilisation d'équation de KP-II est H s avec s > 0. Ceci est dû à la difficulté de prouver la propagation de petitesse en basse fréquences dans L 2 . Dans le cadre général, afin de montrer la stabilisation, on a besoin de la propriété de semi-groupe nonlinéaire afin d'itérer. Dans chaque étape d'itération, il faut assurer la petitesse en basse fréquences horizontales pour continuer. Contrairement à la norme L 2 , les basse fréquences, ou en équivalence, la norme de Sobolev avec indice negatif en x, n'est pas conservée le long du flot nonlinéaire. Donc un autre problème naturel apparaît: Problème 0.1 Est-t-il existe s < 0 telle que pour tout T > 0, toute solution globale u de l'équation de KP-II satisfait sup

t∈[0,T ] u H s,0 (T 2 ) ≤ C(T, u(0) H s,0 (T 2 ) )?
Toute réponse positive à ce problème avec une borne plus faible C T ( u(0) L 2 (T 2 ) ) à droite pourrait améliorer le théorème 0.14 à s ≥ 0. On croit que ce problème 0.1 est intéressant en soi car il est relié à l'existence de solution faible de l'équation de KP-II au-dessous de L 2 (T 2 ). Les analogies de ces problèmes dans le cadre de l'équation de Schrödinger nonlinéaire, l'équation de KdV (sans amortissement) ont bien été étudié, voir par exemple [START_REF] Koch | Conserved energies for the cubic nls in 1-d[END_REF], [START_REF] Killip | Low regularity conservation laws for integrable pde[END_REF] et leurs bibliographies.

Quelques problèmes ouverts liés à cette thèse

Nous fournirons quelques problèmes ouverts liés à cette thèse.

Concernant le système de Stokes hyperbolique, on conjecture que la contrôlabilité exacte de (0.2.2) à temps T > T GCC est vraie. Ici T GCC est le temps avant lequel toute géodésique généralisée de la vitesse 1 rencontre l'ouvert ω. Problème 0.2 Soit Ω ⊂ R d est un domaine lisse, borné et étoilé. On suppose que les géodésiques de Ω n'ont pas de contact d'ordre infini avec ∂Ω. Soit ω ⊂ Ω un ouvert non vide qui satisfait la condition de contrôle géométrique. Supposons que T > T GCC . Est-ce que pour tout

(u 0 , u 1 ) ∈ V × H, il existe un contrôle f ∈ L 2 ([0, T ]; L 2 (Ω)), supporté par [0, T ] × ω, tel que la solution unique de (0.2.2) satisfait (u, ∂ t u)| t=T = (0, 0) ?
Une autre direction intéressante est la contrôlabilité des équations dispersives «dégénérées». Le mot dégénéré peut signifier plusieurs choses. L'équation de KP-I linéaire est un exemple dont l'ensemble de points critiques de symbole est 3ξ 2 -η 2 ξ 2 = 0 . Un autre exemple est l'opérateur hypoelliptique, typiquement l'opérateur de Grushin ∆ G = ∂ 2

x + x 2 ∂ 2 y . La contrôlabilité de l'équation de la chaleur avec l'opérateur de Grushin a attiré des intérêts récemment , voir par exemple [START_REF] Cannarsa | Heat equation on the heisenberg group: Observability and applications[END_REF], [START_REF] Koenig | Non-null-controllability of the grushin operator in 2d[END_REF].

Problème 0.3 Trouver des conditions sur ω pour lesquelles on a la contrôlabilité exacte pour l'équation de Grushin Schrödinger

i∂ t u + ∆ G u = 1 ω f.
Concernant les équations de KP, en plus du problème mentionné précédemment, il est aussi intéressant de généraliser le résultat sur KP-I linéaire au cas non-linéaire. Problème 0.4 Obtenir la contrôlabilité et la stabilisation pour l'équation de KP-I avec des données périodique.

Part I

Semiclassical analysis of Stokes-system and application

Chapter 1

Semi-classical propagation of singularity for Stokes system

Introduction

Let Ω ⊂ R d be a smooth bounded, star-shaped domain. Consider the eigenvalue problem of Stokes equation

   -∆u k + ∇P k = λ 2 k u k in Ω div u k = 0 in Ω u k | ∂Ω = 0 (1.1.1) where u k ∈ (H 2 (Ω)) d ∩ V , u k L 2 = 1, are R d -valued normalized eigenfunctions and V = {u ∈ (H 1 0 (Ω)) d : div u = 0}.
We collect several facts which are well-known in functional analysis:

• u k forms a orthonormal basis of

H = {u ∈ (L 2 (Ω)) d : div u = 0, u • ν| ∂Ω = 0} The canonical projector Π : (L 2 (Ω)) d → H is called Leray projector. • The pressure P k ∈ L 2 (Ω)/R satisfies Ω P k = 0. • ∇u k 2 L 2 = λ 2 k , u k H 2 ≤ Cλ 2 k , ∇P k L 2 ≤ Cλ 2 k , P k L 2 ≤ Cλ 2 k .
We rephrase the system (1.1.1) by semi-classical reduction. Taking h k = λ -1 k and q k = λ -1 k P k , dropping the sub-index, we obtain the following h-dependent quasi-mode system

   -h 2 ∆u -u + h∇q = f in Ω h div u = 0 in Ω u| ∂Ω = 0 (1.1.2)
with the following conditions:

u L 2 = 1, h∇u L 2 = O(1), h 2 ∇ 2 u L 2 = O(1), h∇q L 2 = O(1), f ∈ H, f L 2 = o(h).
When h is small, the corresponding solution u = u(h) can be interpreted as high-frequency quasi-mode as its mass, i.e., the L 2 -norm, is essentially concentrated on the frequency scale h -1 .

Before stating the main result, it is worth mentioning the eigenvalue problem of Laplace operator in semi-classical version:

-h 2 ∆u -u = 0 in Ω u| ∂Ω = 0. (1.1.3)
One method to capture the high-frequency behavior of the solutions of (1.1.3) is to use semiclassical defect measure associated to a bounded sequence (u k ) of L 2 (Ω) and to a sequence of positive scales h k converging to zero. This measure is aimed to describe quantitatively the oscillations of (u k ) at the frequency scale h -1 k . More precisely, for any bounded sequence (w k ) of L 2 (R d ), there exists a subsequence of (w k ) and a non-negative Radon measure µ on T * R d such that for any a(x, ξ) ∈ S(R 2d ),

lim k→∞ (a(x, h k D x )w k |w k ) L 2 (R d ) = µ, a .
When Ω is a bounded domain, the precise definition of defect measure corresponding to the boundary value problem will be described later.

Let us mention that a counterpart of semi-classical defect measure, micro-local defect measure, was introduced by P.Gérard [START_REF] Gérard | Microlocal defect measures[END_REF] and L.Tartar [START_REF] Tartar | H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations[END_REF] independently. These objects are widely used in the study of control and stabilization, scattering theory and quantum ergodicity, see for example [START_REF] Burq | Mesures de défaut de compacité, application au systeme de lamé[END_REF], [START_REF] Burq | Semi-classical estimates for the resolvent in nontrapping geometries[END_REF], [START_REF] Gérard | Ergodic properties of eigenfunctions for the dirichlet problem[END_REF].

In the context of semi-classical defect measure, the classical theorem of Melrose-Sjöstrand about propagation of singularity for hyperbolic equation can be rephrased as follows:

Theorem 1.1 ([26])
Assume that Ω is a smooth, bounded domain with no infinite order of contact on the boundary. Suppose µ is the semi-classical defect measure associated to the pair

(u k , h k ) where (u k ) is a sequence of solutions to (1.1.3) (with h = h k ) which are bounded in L 2 (Ω).
Then µ is invariant under the Melrose-Sjöstrand flow.

We will give the precise definition of Melrose-Sjöstrand flow and the associated concept of the order of contact in the second section. Intuitively, these flows are the generalization of geometric optics. No infinite order of contact means that the trajectory of the flow can not tangent to the boundary with an infinite order.

The main result of this paper is as follows.

Theorem 1.2

Assume that Ω is a smooth, bounded star-shaped domain with no infinite order of contact on the boundary. Suppose (u k ) is a sequence of solutions to the quasi-mode problem (1.1.2)

with semi-classical parameters h = h k . Assume that f k ∈ H, f k L 2 (Ω) = o(h k )
and u k converges weakly to 0 in L 2 (Ω). Assume that µ is any semi-classical measure associated to some subsequence of (u k , h k ), then supp(µ) is invariant under Melrose-Sjöstrand flow.

We make some comments about the result. Firstly, the measure µ is Hermitian matrixvalued, and we have no information on the precise propagation for µ except for supp(µ). Secondly, since the eigenfunctions of Stokes operator converge weakly to 0 in L 2 (Ω), our results includes this special case.

An application of Theorem 1.2 is the stabilization of a hyperbolic Stokes system under the geometric control condition. Let us consider the following hyperbolic-Stokes equation with damping

     ∂ 2 t u -∆u + ∇p + a(x)∂ t u = 0 in R × Ω, div u = 0 in R × Ω, u = 0 on R × ∂Ω, (u(0, x), ∂ t u(0, x)) = (u 0 , v 0 ) ∈ V × H, (1.1.4)
The energy

E[u](t) = 1 2 Ω (|∂ t u| 2 + |∇u| 2 )dx
is dissipative. This is the main objective in the next Chapter.

Let us describe briefly our strategy for the proof of Theorem 1.2. The pressure term q is harmonic and in heuristic, it can only have the influence to the solution near the boundary. We will prove that the measure is propagated in the same way as Laplace quasi-mode (semi-classical analogue of wave equation) along the rays inside the domain. When a ray touches the boundary, we need a more careful analysis between the wave-like propagation phenomenon and the impact of pressure. It is difficult to get a simple propagation formula near the boundary, comparing to the treatment of quasi-mode problem of Laplace operator as in [START_REF] Burq | Mesures de défaut de compacité, application au systeme de lamé[END_REF], [START_REF] Gérard | Ergodic properties of eigenfunctions for the dirichlet problem[END_REF]. We partition the phase space into elliptic region E, hyperbolic region H and glancing surface G. It turns out that there is no singularity accumulated near elliptic region. For the hyperbolic region, the propagation argument is also standard, with an additional treatment when the incidence of the ray is right. Near the glancing surface, we will follow the arguments of V.Ivrii's and Melrose-Sjöstrand. The main difference is that we will encounter two new cross terms essentially of the form (q|u) L 2 after certain microlocalization. To overcome this difficulty, we further micro-localize the solution according to the distance to the glancing surface G and treat them separately. For the part nearing G, we use the fact that the pressure decays fast away from the boundary while the solution can not concentrate too much near the boundary, provided that it is micro-localized close enough to the glancing surface. For the part away from G, it can be well-controlled by geometric considerations.

Preliminary

Notations

We will sometimes drop the sub-index k for a sequence of functions (u k ) and semi-classical parameters h k . In this circumstance, the notion u X = O(1), o(1) as h → 0 should be understood as

u k X = O(1), o(1) as k → ∞ (thus h k → 0) up to certain subsequence.
As in the introduction, we always use

V = {u ∈ H 1 0 (Ω) N : div u = 0} and H = {u ∈ L 2 (Ω) N : div u = 0, u • ν| ∂Ω = 0}.
In this paper we always use ν to denote the outward normal vector on ∂Ω.

For a manifold M , we let T M be its tangent bundle and T * M be the cotangent bundle with canonical projection π : T M ( or T * M ) → M.

We will identify system (1.1.2) as a system on differential form

   h 2 ∆ H u -u + hdq = f in Ω hd * u = 0 in Ω u| ∂Ω = 0 (1.2.1)
where the unknown u ∈ Λ 1 (Ω) is 1-form, and

d : Λ p (Ω) → Λ p+1 (Ω), d * : Λ p+1 (Ω) → Λ p (Ω)
are exterior differential and divergence operator on forms, with respectively. Recall also that he Hodge Laplace operator is defined by

∆ H = dd * + d * d = (d + d * ) 2 .
In the turbulent neighborhood of boundary, we can identify the Ω locally as one side of the turbulent neighborhood denoted by

Y + = [0, 0 ) × X, X = {x ∈ R d-1 : |x | < 1}. We denote by ∂Y + = Y + | y=0 and Y 0 + = Y + | y>0 . For x ∈ Ω, we note x = (y, x )
, where y ∈ [0, 0 ), x ∈ X, and x ∈ ∂Ω if and only if x = (0, x ). In this coordinate system, the Euclidean metric dx 2 can be written as matrices

g = 1 0 0 g(y, x ) , g -1 = 1 0 0 g -1 (y, x ) , with |ξ | 2 g -1 (y,x )) = ξ , g -1 (y, x )ξ C d-1 = g jk ξ j ξ k be the induced metric on T * ∂Ω, parametrized by y. Note that |ξ | 2 g -1 (0,x ) = ξ , g -1 (0, x )ξ C d-1 = g jk ξ j ξ k
is the natural norm on T * ∂Ω, dual of the norm on T ∂Ω, induced by the canonical metric on Ω. Write (x, ξ) = (y, x , η, ξ ) and denote by |ξ| the Euclidean norm on T * R d . For u, v ∈ Λ 1 (Y + ) with support in the local chart of turbulence neighborhood, we define the L 2 norms and inner product on

[0, 0 ) × X via u 2 L 2 (Y + ) := 0 0 X u|u det(g)dx dy, (u|v) L 2 (Y + ) := (u|v) Y + := 0 0 X u|v det(g)dx dy, u(y, •) 2 L 2 (∂Y + ) := X u(y, •)|u(y, •) det(g)dx , (u|v) L 2 (∂Y + ) (y) := X u(y, •)|v(y, •) det(g)dx ,
where for u = u 0 dy

+ u j dx j , v = v 0 dy + v j dx j , u|v = u 0 v 0 + u j v k g jk .
In certain situations we also use global notation for L 2 inner product:

(u|v) Ω := Ω u • vdx, (f |g) ∂Ω := ∂Ω f • gdσ(x)
We will identify the unknown vector fields u, v, etc. and their dual 1-form. Formulation of differential form will simplify some calculations. In the turbulence neighborhood, we write a vector field

X = X + X ⊥ ∂ ∂y , X = d-1 j=1 X ,j ∂ ∂x j
and we write X = (X , X ⊥ ). The normal component obeys the following convention: (0, a) = -aν.

As in [START_REF] Chaves-Silva | Spectral inequality and optimal cost of controllability for the stokes system[END_REF], we will write down system (1.1.2) in the turbulent neighborhood. For u = (u , u ⊥ ), equation (1.1.2) can be written as:

         (-h 2 ∆ -1)u + h∇ x q = f , (-h 2 ∆ g -1)u ⊥ + h∂ y q = f ⊥ , h div u + h √ det g ∂ y ( detgu ⊥ ) = 0 (1.2.2)
where

h 2 ∆ = h 2 ∂ 2 y -Λ 2 (y, x , hD x ) + hM (y, x , hD x ) + hM 1 (y, x )h∂ y , h 2 ∆ g = h 2 ∂ 2 y -Λ 2 (y, x , hD x ) + hM ⊥ (y, x , hD x ) + hN 1 (y, x )h∂ y , h div u = h √ det g N -1 j=1 ∂ x j ( det gu ,j ). h 2 Λ 2 (y, x , hD x ) has the symbol λ 2 = |ξ | 2 α(y,•)
, and M ,⊥ are both first-order matrix-valued semi-classical differential operators.

Geometric Preliminaries

Denote by b T Ω the vector bundle whose sections are the vector fields X(p) on Ω with X(p) ∈ T p ∂Ω if p ∈ ∂Ω. Moreover, denote by b T * Ω the Melrose's compressed cotangent bundle which is the dual bundle of b T Ω. Let j : T * Ω → b T * Ω be the canonical map. In our geodesic coordinate system near ∂Ω, b T Ω is generated by the vector fields

∂ ∂x 1 , • • •, ∂ ∂x d-1
, y ∂ ∂y and thus j is defined by

j(y, x ; η, ξ ) = (y, x ; v = yη, ξ ).
The principal symbol of operator

P h = -(h 2 ∆ + 1) is p(y, x , η, ξ ) = η 2 + |ξ | 2 g -1 (y,x ) -1.
By Car(P ) we denote the characteristic variety of p:

Car(P ) := {(x, ξ) ∈ T * R d | Ω : p(x, ξ) = 0}, Z := j(Car(P )).

By writing in another way

p = η 2 -r(y, x , ξ ), r(y, x , ξ ) = 1 -|ξ | 2 g -1 (y,x ) ,
we have the decomposition

T * ∂Ω = E ∪ H ∪ G,
according to the value of r 0 := r| y=0 where

E = {r 0 < 0}, H = {r 0 > 0}, G = {r 0 = 0}.
The sets E, H, G are called elliptic, hyperbolic and glancing, with respectively.

For a symplectic manifold S with local coordinate (z, ζ), a Hamiltonian vector field associated with a real function f is given by

H f = ∂f ∂ζ ∂ ∂z - ∂f ∂z ∂ ∂ζ .
Now for (x, ξ) ∈ Ω far away from the boundary, the Hamiltonian vector field associated to the characteristic function p is given by

H p = 2ξ ∂ ∂x .
We call the trajectory of the flow

φ s : (x, ξ) → (x + sξ, ξ)
bicharacteristic or simply ray, provided that the point x + sξ is still in the interior.

To classify different situations as a ray approaching the boundary, we need more accurate decomposition of the glancing set G. Let r 1 = ∂ y r| y=0 and define

G k+3 = {(x , ξ ) : r 0 (x , ξ ) = 0, H j r 0 (r 1 ) = 0, ∀j ≤ k; H k+1 r 0 (r 1 ) = 0}, k ≥ 0 G 2,± := {(x , ξ ) : r 0 (x , ξ ) = 0, ±r 1 (x , ξ ) > 0}, G 2 := G 2,+ ∪ G 2,-.
No infinite order of contact means that we can decompose G into

G = ∞ j=2 G j .
Given a ray γ(s) with π(γ(0)) ∈ Ω and π(γ(s 0 )) ∈ ∂Ω be the first point who attaches the boundary. If γ(s 0 ) ∈ H, then η ± (γ(s 0 )) = ± r 0 (γ(s 0 )) be the two different roots of η 2 = r 0 at this point. Notice that the ray starting with direction η -will leave Ω, while the ray with direction η + will enter the interior of Ω. This motivates the following definition of broken bicharacteristic:

Definition 1.1 ([28])
A broken bicharacteristic arc of p is a map:

s ∈ I \ B → γ(s) ∈ T * Ω \ {0},
where I is an interval on R and B is a discrete subset, such that 1. If J is an interval contained in I \ B, then s ∈ J → γ(s) is a bicharacteristic of p over Ω.

2.

If s ∈ B, then the limits γ(s + ) and γ(s -) exist and belongs to T *

x Ω \ {0} for some x ∈ ∂Ω, and the projections in T *

x ∂Ω \ {0} are the same hyperbolic point.

When a ray γ(s) arrives at some point ρ 0 ∈ G, there are several situations. If ρ 0 ∈ G 2,+ , then the ray passes transversally over ρ 0 and enters T * Ω immediately. If ρ 0 ∈ G 2,-or ρ 0 ∈ G k for some k ≥ 3, then we can continue it inside T * ∂Ω as long as it can not leave the boundary along the trajectory of the Hamiltonian flow of H -r 0 . We now give the precise definition.

Definition 1.2 ([28])

A generalized bicharacteristic ray of p is a map:

s ∈ I \ B → γ(s) ∈ (T * Ω \ T * ∂Ω) ∪ G
where I is an interval on R and B is a discrete set of I such that p • γ = 0 and the following:

1. γ(s) is differentiable and dγ ds = H p (γ(s)) if γ(s) ∈ T * Ω \ T * ∂Ω or γ(s) ∈ G 2,+ . 2. Every s ∈ B is isolated, γ(s) ∈ T * Ω \ T * ∂Ω if s = t
and |s -t| is small enough, the limits γ(s ± ) exist and are different points in the same fibre of T * ∂Ω.

γ(s) is differentiable and dγ

ds = H -r 0 (γ(s)) if γ(s) ∈ G \ G 2,+ . Remark 1.1
The definition above does not depend on the choice of local coordinate, and in the geodesic coordinate system, the map

s → (y(s), η 2 (s), x (s), ξ (s))
is always continuous and s → (x (s), ξ (s))

is always differentiable and satisfies the ordinary differential equations

dx dt = - ∂r ∂ξ , dξ dt = ∂r ∂x ,
the map s → y(s) is left and right differentiable with derivative 2η(s ± ) for any s ∈ B (hyperbolic point).

Moreover, there is also the continuous dependence with the initial data, namely the map

(s, ρ) → (y(s, ρ), η 2 (s, ρ), x (s, ρ), ξ (s, ρ))
is continuous. We denote the flow map by γ(s, ρ).

Remark 1.2

Under the map j : T * Ω → b T * Ω, one could regard γ(s) as a continuous flow on the compressed cotangent bundle b T * Ω, and it is called the Melrose-Sjöstrand flow. We will also call each trajectory generalized bicharacteristic or simply ray in the sequel.

It is well-known that if there is no infinite contact in G, a generalized bicharacteristic is uniquely determined by any one of its points. In other words, the Melrose-Sjöstrand flow is globally well-defined. See [START_REF] Hörmander | The analysis of linear partial differential operators. iii, volume 274 of grundlehren der mathematischen wissenschaften (fundamental principles of mathematical sciences[END_REF] for more discussion.

definition of defect measure

We follow closely as in [START_REF] Burq | Semi-classical estimates for the resolvent in nontrapping geometries[END_REF] and the one can find in [START_REF] Gérard | Ergodic properties of eigenfunctions for the dirichlet problem[END_REF] for a little different but comprehensive introduction.

We denote by S m the usual symbol class. Define the partial symbol class S m ∂ and the class of boundary h-pseudo-differential operators A m h as follows

S m ∂ := {a(y, x , ξ ) : sup α,β,y∈[0, 0 ] |∂ α x ∂ β ξ a(y, x , ξ )| ≤ C m,α,β (1 + |ξ |) m-β }. A m h =: Op comp h (S m ) + Op h (S m ξ ) := A m h,i + A m h,,∂ .
Consider functions of the form a = a i +a ∂ with a i ∈ C ∞ c (Ω×R d ) which can be viewed as a symbol in S 0 , and

a ∂ ∈ C ∞ c (Y + × R d-1
) can be viewed as a symbol in S 0 ξ . We quantize a as follows:Take

ϕ i ∈ C ∞ c (Ω) equal to 1 near the x-projection of supp(a i ) and ϕ ∂ ∈ C ∞ c (R d ) equal to 1 near the x-projection of supp(a ∂ ). Define Op ϕ i ,ϕ ∂ h (a)f (y, x ) = 1 (2πh) d R 2d e i(x-z)ξ h a i (x, ξ)ϕ i (z)f (z)dzdξ + 1 (2πh) d-1 R 2(d-1) e i(x -z )ξ h a ∂ (y, x , ξ )ϕ ∂ (y, z )f (y, z )dz dξ .
According to the symbolic calculus, the operator Op

ϕ i ,ϕ ∂ h (a) does not depend on the choice of functions ϕ i , ϕ ∂ , modulo operators of norms O L 2 loc →L 2 comp (h ∞ )
, and we will use the notion Op h (a) in the sequel. Notice that the acting of tangential operator Op h (a ∂ ) can be viewed as pseudo-differential operator on the manifold ∂Ω, parametrized by the parameter y ∈ [0, 0 ). The bounded family of operators A m h,∂ is defined uniquely up to a family of operators with norms uniformly dominated by Ch, as h → 0. See [START_REF] Gérard | Ergodic properties of eigenfunctions for the dirichlet problem[END_REF] for more details. Moreover, for any family (A h ), such that

A h -Op h (a ∂ ) L 2 →L 2 = O(h),
the principal symbol σ(A) is determined uniquely as a function on T * ∂Ω, smoothly depending on y, i.e. σ(A) ∈ C ∞ ([0, 0 ) × T * ∂Ω).

When we deal with vector-valued functions, we could require the symbol a to be matrixvalued. Now for any sequence of vector-valued function w k , uniformly bounded in L 2 (Ω), there exists a subsequence (still use w k for simplicity), and a nonnegative definite Hermitian matrix-valued Radon measure µ i on T * Ω such that

lim k→0 (Op h k (a i )w k |w k ) L 2 = µ i , a := T * Ω tr (adµ i ).
For a proof, see for example [START_REF] Burq | Semi-classical estimates for the resolvent in nontrapping geometries[END_REF], and the micro-local version was appeared in [START_REF] Gérard | Microlocal defect measures[END_REF].

From now on the symbols and operators will be scalar-valued unless otherwise specified. Suppose u k be a sequence of solutions to

-h 2 k ∆u k -u k + h k ∇q k = f k , (u k , f k ) ∈ (H 2 (Ω) ∩ V ) × H, h k divu k = 0, in Ω (1.2.3)
under the assumptions below:

u k L 2 (Ω) = O(1), f k ∈ H and f k L 2 (Ω) = o(h k ), h∇q k L 2 (Ω) = O(1), Ω q k dx = 0. (1.2.4)
The following result shows that the interior measure µ i is supported on the Car(P ).

Proposition 1.1 Let a i ∈ C ∞ c (Ω × R d
) be equal to 0 near Car(P ), then we have lim

k→∞ (Op h k (a i )u k |u k ) L 2 = 0.
Proof. Note that the symbol b(x, ξ) = a i (x,ξ) |ξ| 2 -1 ∈ S 0 is well-defined from the assumption on a i . From symbolic calculus, we have

Op h k (a i ) = B h k (-h 2 k ∆ -1) + O L 2 →L 2 (h k ). Therefore (B h k (-h 2 k ∆ -1)u k |u k ) L 2 = (B h k f k |u k ) L 2 -(B h k h k ∇q k |u k ) L 2 = o(1) + ([h k ∇, B h k ]q k |u k ) L 2 -(h k ∇B h k q k |u k ) L 2 = o(1), as k → ∞,
where in the last line we have used the symbolic calculus, integrating by part, and Lemma 1.3. Now we denote by Z = j(Car(P )). Proposition 1.1 indicates that the interior defect measure µ i is supported on Z. To define the defect measure near the boundary, we have

to check that if a ∂ ∈ C ∞ c (U × R d-1
) vanishing near Z (i.e. a ∂ is supported in the elliptic region for all y small) then

lim k→∞ (Op h k (a ∂ )u k |u k ) L 2 = 0.
Indeed, this can be ensured by the analysis of boundary value problem in the elliptic region, which will be given later. Now for any family of operator A h ∈ A 0 h , let a = σ(A h ) be the principal symbol of A h and we define κ(a) ∈ C 0 (Z) via κ(a)(ρ) := a(j -1 (ρ)). Note that Z is a locally compact metric space and the set {κ(a) : a = σ(A h ), A h ∈ A 0 h } is a locally dense subset of C 0 (Z). We then have the following proposition, which guarantees the existence of a Radon measure on Z: Proposition 1.2 There exists a subsequence of u k , h k and a nonnegative definite Hermitian matrix-valued Radon measure µ, such that

lim k→∞ (A h k u k |u k ) L 2 = µ, κ(a) , a = σ(A h ), ∀A h ∈ A 0 h .
The proof of this result can be found in [START_REF] Burq | Semi-classical estimates for the resolvent in nontrapping geometries[END_REF], see also [START_REF] Burq | Mesures de défaut de compacité, application au systeme de lamé[END_REF] and [START_REF] Gérard | Microlocal defect measures[END_REF] for its micro-local counterpart. Notice that if we write a = a i + a ∂ , then

(A k u k |u k ) → T * Ω tr (a i (ρ)dµ i (ρ)) + Z tr (a ∂ (ρ)dµ(ρ)).
The following result shows that information about frequencies higher than the scale h -1 k does not lost, and the measure µ contains the relevant information of the sequence (u k ).

Proposition 1.3

The sequence of solution (u k ) is h k -oscillating in the following sense:

lim R→∞ lim sup k→∞ |ξ|≥Rh -1 k | ψu k (ξ)| 2 dξ = 0, ∀ψ ∈ C ∞ c (Ω), lim R→∞ lim sup k→∞ 0 0 dy |ξ |≥Rh -1 k | ψu k (y, ξ )| 2 dξ = 0, ∀ψ ∈ C ∞ c (Ω),
where in the second formula, the Fourier transform is only taken for the x direction.

The proof will be given in appendix.

1.3 A priori information about the system

1.3.

Information about the trace

We consider the semi-classical Stokes system

-h 2 ∆u -u + h∇q = f, (u, f ) ∈ (H 2 (Ω) ∩ V ) × H h divu = 0, in Ω (1.3.1) Assume that u L 2 (Ω) = O(1), f L 2 (Ω) = o(h).
Taking inner product with u and integrating by part, we have h∇u L 2 (Ω) = O(1). Since q ∈ L 2 (Ω)/R, we may assume that Ω qdx = 0.

From the regularity theory of steady Stokes system, (see [START_REF] Temam | Navier-Stokes equations: theory and numerical analysis[END_REF], page 33), Poincaré inequality, we have

h 2 ∇ 2 u L 2 (Ω) = O(1), q L 2 (Ω) = O(h -1 ), h∇q L 2 (Ω) = O(1).
The following is a direct consequence of trace theorem for q 0 = q| ∂Ω .

Lemma 1.1 q 0 H 1/2 (∂Ω) = O(h -1 ).
There is hidden regularity for the normal derivative.

Lemma 1.2 h∂ ν u| ∂Ω = (h∂ ν u , 0) and h∂ ν u| ∂Ω L 2 (∂Ω) = O(1).
The proof of this lemma will be given in appendix A in which we use the fact that Ω is star-shaped. We will recover some information for low frequencies from the following lemma:

Lemma 1.3 Suppose u 0 in L 2 (Ω).
Then after extracting to subsequences, we have h∇q 0 weakly in L 2 (Ω) and hq → 0 strongly in H 1/2 (Ω).

Proof. We may assume that h∇q r weakly in L 2 (Ω) and Rellich theorem implies that hq → P strongly in L 2 (Ω), and thus ∇P = r with the property Ω P = 0. Moreover it is easy to see that ∆P = 0 in Ω. Since the sequence (h 2 ∇ 2 u) is bounded in L 2 , then up to a subsequence, h 2 ∇ 2 u W weakly in L 2 . From Rellich theorem, the sequence (h 2 u) converges strongly in L 2 and the strong limit must be 0 due to u 0 weakly in L 2 . Thus W = 0 and this implies that ∇P = 0. Finally, we must have P = 0 since it has zero mean value. The last assertion follows from Rellich theorem.

Semi-classical parametrix of the pressure term

In system (2.4.1), the family of pressures q satisfy the boundary value problem of Laplace equation -h 2 ∆q = 0, in Ω, q| ∂Ω = q 0 with unknown boundary data q 0 . We denote by PI(q 0 ) the Poisson integral of the corresponding harmonic function with trace q 0 . Let N be the Dirichlet-Neumann operator satisfying

N q 0 = ∂ ν PI(q 0 )| ∂Ω .
Next we study the behaviour of the sequence of pressure q in the regime of frequency scale h -1 . We always fix the notation

λ(y, x , ξ ) = |ξ | g -1 (y,x ) ∼ |ξ |.
Let Y = (-0 , 0 ) y × X x and Y + = [0, 0 ) y × X x . We first have the L 2 bound of q, microlocally away from ξ = 0.

Lemma 1.4 Let (f h ) 0<h<1 be a h-dependent distributions such that f h L 2 (R n ) = O(h -N ). Assume that for any χ ∈ C ∞ c (R 2n ), vanishing near ξ = 0, we have χ(x, hD x )f h H 1 2 (R n ) = O(h -1 ). Then χ(x, hD x )f h L 2 (R n ) = O(h -1 2 ). Proof. Assume that {|ξ| ≤ 2δ 0 } ∩ supp(χ) = ∅. Take Φ ∈ C ∞ c (R n ) such that Φ(ξ) = 1, |ξ| ≤ δ 0 , Φ(ξ) = 0, |ξ| > 2δ 0 .
We write

χ(x, hD x )f = Φ(hD x )χ(x, hD x )f + (1 -Φ(hD x ))χ(x, hD x )f.
From the support property we have

Φ(hD x )χ(x, hD x )f = O H ∞ (h ∞ ). Thus (1-Φ(hD x ))χ(x, hD x )f = O H 1 2 (h -1 ). Let b(ξ) = |ξ| 1/2 (1 -Φ(ξ)), and we have b(hD x )χ(x, hD x )f = O L 2 (h -1 2 ). Since b(ξ) = 0 on supp(χ), we have χ(x, hD x )f L 2 (R n ) ≤ C b(hD x )χ(x, hD x )f L 2 (R n ) + Ch χ(x, hD x )f L 2 (R n ) ≤ Ch -1 2 . Lemma 1.5 Given δ 0 > 0 and ϕ, ϕ ∈ C ∞ c (Y + ). For any χ δ 0 ∈ C ∞ c (Y + × R d-1 ) such that χ δ 0 | ≡ 0 if λ(y, x , ξ ) ≤ 2δ 0 , we have ϕOp h (χ δ 0 )(ϕq) L 2 (R d + ) + h 1/2 ( ϕOp h (χ δ 0 )(ϕq)) | y=0 L 2 (R d-1 ) ≤ C δ 0 ,ϕ, ϕ . Proof. Write D j = 1 i ∂ ∂x j , we have hD j (ϕq) L 2 (R d + ) = O(1). Note that ξ j |ξ | 2 χ δ 0 (y, x , ξ ) ∈ S 0 ∂ ,
and if we let

χ j = ξ j |ξ | 2 χ δ 0 , then ϕχ δ 0 (y, x , hD x )(ϕq) = d-1 j=1 ϕχ j (y, x , hD x )hD j (ϕq) + O L 2 (R d + ) (1) = O L 2 (R d + ) (1),
where the implicit bound in big O depends on δ 0 , ϕ, ϕ. For the boundary term, we observe that ϕOp h (χ

δ 0 )(ϕq)| y=0 = O H 1/2 (R d-1 ) (h -1
) from trace theorem. Thus from Lemma ??,

ϕOp h (χ δ 0 )(ϕq)| y=0 = O L 2 (R d-1 ) (h -1/2 ).
We express semi-classical Laplace operator h 2 ∆ g in the geodesic coordinates of turbulence neighborhood Y by

P 0 = h 2 ∂ 2 y + i,j g ij ∂ i ∂ j + h j M j (y, x )h∂ j + hH(y, x )h∂ y
where ∂ j = ∂ x j . We make the ansatz q(y, x ) := 1 (2πh) d-1 a(y, h, x , ξ )e ix ξ h θ(ξ )dξ , then we calculate

P 0 ( q)(y, x , ξ ) = 1 (2πh) d-1 h 2 ∂ 2 y a + g jk (h 2 ∂ j ∂ k a -g jk ξ j ξ k a) e ix ξ h θ(ξ )dξ + 1 (2πh) d-1 ihg jk ξ k ∂ j a e ix ξ h θ(ξ )dξ + 1 (2πh) d-1 (h 2 M j ∂ j a + ihM j ξ j a) + h 2 H∂ y a e ix ξ h θ(ξ )dξ .
We next look for the formal semi-classical expansion a(y, h, x , ξ ) j≥0 h j a j (y, h, x , ξ ) with a j ∈ S -j ∂ and h k ∂ k y a j ∈ S -j+k ∂ . We obtain

P 0 q 1 (2πh) d-1 ((h 2 ∂ 2 y a 0 -g ij ξ i ξ j a 0 ) +h(ig jk ξ k ∂ j a 0 + iM j ξ j a 0 + h 2 H∂ y a 0 ) +h(h 2 ∂ 2 y a 1 -g jk ξ j ξ k a 1 ) +h 2 (g jk ∂ j ∂ k a 0 + M j ∂ j a 0 ) +h 2 (ig jk ξ k ∂ j a 1 + iM j ξ j a 1 + h 2 H∂ y a 1 ) +h 2 (h 2 ∂ 2 y a 2 -g jk ξ j ξ k a 2 ) + • ••)e ix ξ h θ(ξ )dξ . Pick ϕ 1,0 = ϕ 1 | ∂Ω , ϕ 1 ∈ C ∞ c (Y ).
For q 0 = ϕ 1,0 q 0 , we put

θ(ξ ) = F h ( q 0 (ξ )) = (2πh) -(d-1) R d-1 q 0 (x )e -ix ξ /h dx , a 0 (0, •) ≡ 1, a j (0, •) ≡ 0, ∀j ≥ 1,
and we define the functions a j inductively as follows: firstly we define a 0 a 0 (y, x , ξ ) = e -yλ(y,x ,ξ ) h , λ(y, x , ξ ) =:

g ij ξ i ξ j ∼ |ξ |,
and the quantity

(h 2 ∂ 2 y -λ 2 )a 0 = h h 2 λ 2 y 2 λ 2 h 2 (∂ y λ) 2 + 2yλ h ∂ y λ -2∂ y λ e -yλ h
can be viewed as of order h. Next we set a j , j ≥ 1 implicitly by solving a sequence of linear ODEs:

h 2 ∂ 2 y a 1 -λ 2 a 1 = -h -1 (h 2 ∂ 2 y -λ 2 )a 0 -(ig jk ξ k ∂ j a 0 + iM j ξ j a 0 + h 2 H∂ y a 0 ). h 2 ∂ 2 y a n -g ij ξ i ξ j a n = -(g ij ∂ i ∂ j a n-2 + M j ∂ j a n-2 ) -(ig jk ξ k ∂ j a n-1 + iM j ξ j a n-1 + h 2 H∂ y a n-1 ), n ≥ 2.
Unfortunately, the functions a j constructed above are not symbols, since they have singularities when ξ = 0. This indicates that we can only obtain information of q(h) from such parametrix away from ξ = 0. We modify the construction above by setting

A 0 (y, x , ξ ) = e -yλ h ψ δ 0 (λ)ϕ 2 (y, x ), (y, x , ξ ) ∈ R d + × R d-1 , with ψ δ 0 = ψ(δ -1 0 •), ψ ∈ C ∞ (R + ) satisfying ψ(s) ≡ 1 when s ≥ 1 and ψ(s) = 0 when 0 < s ≤ 1 2 .
We next modify other A j in the same manner. Indeed, the ODEs which define A j are linear ODEs in y variable. Thus for j ≥ 1, A j (y, x ξ ) ≡ 0 when λ(y, x , ξ ) ≤ δ 0 2 . We define the particular class of symbols in S j ∂ .

Definition 1.3

E j ∂ := a ∈ S j ∂ : |(h∂ y ) l ∂ α x ,ξ a(y, x , ξ )| ≤ C l,α e - C l,α y h . Lemma 1.6
The symbols constructed above can be chosen to satisfy A j ∈ E -j ∂ for all j ∈ N.

The proof will be given in appendix.

In summary, we have A j≥0 h j A j , and a tangential symbol B δ 0 (y, x , ξ ) compactly supported in λ(y, x , ξ ) ≤ δ 0 2 , such that

ϕP 0 A(y, x , hD x )(ϕ 1,0 q 0 ) =ϕB δ 0 (y, x , hD x )(ϕ 1,0 q 0 ) + O H ∞ (h ∞ ), ϕ 0 A(0, x , hD x )(ϕ 1,0 q 0 ) =ϕ 0 Op h (ψ δ (λ))(ϕ 1,0 q 0 ) + O H ∞ (h ∞ ).
The following proposition states that the parametrix constructed above is an approximation of the pressure q in the relevant semi-classical scale.

Proposition 1.4

There exists A ∈ S 0 ∂ with principal symbol

A 0 (y, x , ξ ) = exp - yλ(y, x , ξ ) h ψ δ 0 (λ(y, x , ξ ))ϕ 1 (y, x ),
which satisfies asymptotic expansion A ∼ j≥0 h j A j , A j ∈ E -j ∂ . Moreover, for any

ϕ, ϕ 1 ∈ C ∞ c (Y + ), ϕ 1 | supp(ϕ) ≡ 1, we have ϕOp h (χ δ 0 A j )(ϕ 1,0 q 0 ) = O L 2 (R d + )
(1) for all j, and

ϕOp h (χ δ 0 )(ϕ 1 q) = ϕOp h (χ δ 0 A)(ϕ 1,0 q 0 ) + O L 2 (R d + ) (h 3/4 ), ϕOp h (χ δ 0 )h∂ y (ϕ 1 q) = ϕOp h (χ δ 0 λA)(ϕ 1,0 q 0 ) + O L 2 (R d + ) (h 3/4 ), ϕOp h (χ δ 0 )h∂ y (ϕ 1 q) = ϕOp h (χ δ 0 λA)(ϕ 1,0 q 0 ) + O H 2 3 (R d + ) (h 1/4 ), where ϕ 0 = ϕ| ∂Ω , ϕ 1,0 = ϕ 1 | ∂Ω , χ δ 0 ,0 = χ δ 0 | y=0 .
We postpone the proof of this proposition in appendix. A direct consequence is that the singularity of the pressure q must concentrate in a very thin strip near the boundary.

Lemma 1.7 With the same χ δ 0 ∈ C ∞ c (Y + × R d-1 ) and ϕ 1 , ϕ ∈ C ∞ c (Y + ) for any 0 < y 0 < 0 , we have 0 y 0 ϕOp h (χ δ 0 )(ϕ 1 q) 2 L 2 (R d-1 ) dy ≤ C δ 0 (e -cy 0 h + h),
where the constant C δ 0 only depends on δ 0 and is independent of y 0 and h.

Proof. The second term appearing on the right hand side comes from all the possible remainder terms. It suffices to estimate the term

0 y 0 ϕOp h (χ δ 0 A 0 )(ϕ 1 q 0 ) 2 L 2 (R d-1 ) dy. Since ϕ 1,0 q 0 = O L 2 (R d-1 ) (h -1/2
), micro-locally, we have for each fixed y > 0 that

ϕOp h (χ δ 0 A 0 )(ϕ 1,0 q 0 ) L 2 (R d-1 ) ≤Ch -1/2 |β|≤Cd h |β| 2 sup y>0,(x ,ξ ) |∂ β x ,ξ (χ δ 0 A 0 )| + O(h ∞ ) ≤Ch -1/2 e -cy h 1 + 1≤m,n≤Cd h m/2 y h n + O(h ∞ ).
Squaring and Integrating the right hand side in y variable yields the desired conclusion.

Main Steps of the Proof

The proof of Theorem 1.2 can be divided into several steps according to different geometric situations. We want to show that for any given point

ρ 0 ∈ b T * Ω, if ρ 0 / ∈ supp µ, then γ(s, ρ 0 ) / ∈ supp µ for any s > 0. The first step is to show that if ρ 0 ∈ T * Ω, ρ 0 / ∈ supp µ, then γ(s, ρ 0 ) / ∈ supp µ for all s > 0 provided that π(γ(•, ρ 0 )| [0,s] ) ∩ ∂Ω = ∅.
This can be summarized by the following proposition, in which we have stronger conclusion that the measure is also invariant under the flow. Proof. Let A = Op h (a) and P = -h 2 ∆-1. We apply equation and Lemma ??

to calculate i h ([P, A]u|u) Ω = i h (Au|P u) Ω - i h (AP u|u) Ω = i h (Au|f -h∇q) Ω - i h (A(f -h∇q)|u) Ω = - i h (Au|h∇q) Ω + i h (Ah∇q|u) Ω + o(1) = - i h ([A, hdiv ]u|q) Ω + i h ([A, h∇]q|u) Ω + o(1) = i(Op h (∇a) • u|q) Ω -i(Op h (∇a)q|u) Ω + o(1) = i(u|Op h (∇a)q) Ω -i(Op h (∇a)q|u) Ω + o(1). (1.4.1)
where we have used integrating by part freely, thanks to the fact that A has compact support in x ∈ Ω. Now we claim that for any χ ∈ C ∞ c (T * Ω), we have

(u|Op h (∇χ)q) Ω = o(1).
We know that q = O L 2 (Ω) (1) micro-locally away from ξ = 0 since h∇q = O L 2 (1). On the other hand, h 2 ∆(Op h (∇χ)q) = O L 2 (Ω) (h) and this implies that Op h (∇χ)q = o L 2 (1) since the symbol of h 2 ∆Op h (∇χ) vanishes away near ξ = 0 as well as x near the boundary. In view of the definition of µ, this completes the proof.

For the second step, we need prove that if ρ 0 ∈ E, then µ = 0 in a neighborhood of ρ 0 .

Proposition 1.6 µ1 E = 0. If we let ν be the semi-classical defect measure of the sequence

(h k ∂ ν u k | ∂Ω , h k ), then ν1 E = 0.
The third step consists of proving that after reflection near a hyperbolic point, the measure µ is still zero. Proposition 1.7 Suppose ρ 0 / ∈ suppµ and there exists s 0 > 0 such that γ(s 0 , ρ 0 ) ∈ H and π(γ(s, ρ 0 )) ∈ Ω for all 0 ≤ s < s 0 . Then there exists δ > 0 such that

π(γ(•, ρ 0 )| [s 0 ,s 0 +δ] ) ∩ suppµ = ∅.
Next step is to prove the propagation near a diffractive point.

Proposition 1.8 Suppose ρ 0 / ∈ suppµ and there exists s 0 > 0 such that γ(s 0 , ρ 0 ) ∈ G 2,+ and π(γ(s, ρ 0 )) ∈ Ω for all 0 ≤ s < s 0 . Then γ(s 0 , ρ 0 ) / ∈ suppµ.

To deal with higher order contact, we will use induction. First let us introduce Definition 1.4 (k-propagation property) For k ≥ 2, we say that k-propagation property holds, if along generalized ray γ(s, ρ 0 ), the following statement is true: For some

σ 0 > 0, if γ(•, ρ 0 )| [0,σ 0 ) ∩ supp(µ) = ∅ (or γ(•, ρ 0 )| (-σ 0 ,0] ∩ supp(µ) = ∅) and γ(σ 0 , ρ 0 ) ∈ 2≤j≤k G j (or γ(-σ 0 , ρ 0 ) ∈ 2≤j≤k G j ), then γ(σ 0 , ρ 0 ) / ∈ supp(µ) (or γ(-σ 0 , ρ 0 ) / ∈ supp(µ) ).
The last step for the proof of Theorem 1.2 can be reduced to Proposition 1.9 k-propagation property holds for all k ≥ 2.

Near E

This section is devoted to the proof of Proposition 1.6. We set Q(y, x , ξ

) := √ λ 2 -1 and define Q h = ϕOp h (Qχ 0 )ϕ 1 with χ 0 ∈ C ∞ c (R d-1 ξ ) with support near E in which 1 + δ < λ < C.
With a bit abuse of notation, we refer q 0 , q to be ϕOp h (χ 0 )ϕ 1 q 0 , ϕOp h (χ 0 )ϕ 1 q and u to be ϕOp h (χ 0 )ϕ 1 u. In this manner, we can combine the parametrix in last section to write the system (1.1.2) as

(-h 2 ∂ 2 y + Q 2 h )u ,j + g jk h∂ x k (Op h (A 0 )q 0 ) = R ,j = O L 2 (R d + ) (h), (-h 2 ∂ 2 y + Q 2 h )u ⊥ + h∂ y (Op h (A 0 )q 0 ) = R ⊥ = O L 2 (R d + ) (h).
(1.5.1)

Note that the symbol A 0 (y, x , ξ ) is defined in last section which equals to e -yλ h since λ > 1.

Take ψ ∈ C ∞ (R + ), with ψ| [0, 0 ] ≡ 1, ψ [2 0 ,∞) ≡ 0. Denoting the extended distributions of u by w = (w , w ⊥ ) = (u , u ⊥ )ψ(y)1 y≥0 , we have from standard elliptic parametrix construction (see Appendix A) that modulo that O H ∞ (R d + ) (h ∞ ), w ,j = E(y, x , hD y , hD x )(-ψ(y)g jk h∂ x k (Op h (A 0 )q 0 ) + 2hv j ⊗ δ y=0 + ψ(y)R ,j ), w ⊥ = E(y, x , hD y , hD x )(-hψ(y)∂ y (Op h (A 0 )q 0 ) + ψ(y)R ⊥ ). (1.5.2) where v = h∂ y u | y=0 = O L 2 x
(1). Recall that the principal symbol of E is given by

E 0 := χ 0 (ξ )ϕ(y, x ) η 2 + λ(y, x , ξ ) 2 -1 ,
Now we need a lemma which deals with the trace of error terms:

Lemma 1.8 Assume that R = ϕOp h (χ 0 )ϕ 1 R + O H ∞ (h ∞ ), then if ψ(y)R L 2 (R d + ) = O(h), we have E(y, x , hD y , hD x )(ψ(y)R)| y=0 L 2 (R d-1 x ) = O(h 1/3 ).
Proof. From the parametrix construction above, we know that

|∂ α y,x ,η,ξ E(y, x ; η, ξ )| ≤ C α η 2 + Q(y, x , ξ ) 2 .
Therefore, the symbols ηE(y, x ; η, ξ ) and λ(y, x , ξ )E(y, x ; η, ξ ) are uniformly bounded in S 0 . Thus E(y, x ; hD y , hD

x )(ψR) = O L 2 (R d + ) (h) = O H 1 (R d + ) (1)
, and from interpolation, we have E(y, x ; hD y , hD x )(ψR

) = O H 2/3 (R d + ) (h 1/3
). The conclusion then follows from trace theorem that

H s (R d + ) → H s-1/2 (R d-1 ) is bounded for s > 1/2.
Proof of Proposition 1.6. Denote by F h (q 0 ) = θ the semi-classical Fourier transform of q 0 , we calculate

E(y, x , hD y , hD x )(ψ(y)h∂ y Op h (A 0 )q 0 ) = 1 (2πh) d e i(y-z)η h dzdη e ix ξ h θ(ξ )ϕ(y, x )χ 0 (ξ ) η 2 + Q 2 (y, x , ξ ) ψ(z)h∂ z (e -zλ(z,x ,ξ ) h )dξ + R 1 = - h (2πh) d ξ θ(ξ )e i(yη+x ξ ) h B 1 (η, x , ξ )ϕ(y, x )χ 0 (ξ ) η 2 + Q 2 (y, x , ξ ) dηdξ - h 2 (2πh) d e ix ξ h θ(ξ )dξ e iyη h B 0 (η, x , ξ )ϕ(y, x )χ 0 (ξ ) η 2 + Q(y, x , ξ ) 2 dη + R 1 , (1.5.3
)

with reminder term R 1 = O L 2 (R d ) (h), where λ 0 = λ| y=0 , B 1 (η, x , ξ ) = ∞ 0 ψ(z)e - iη+λ(z,x ,ξ ) h z λ(z, x , ξ ) ξ 1 h dz, and
B 0 (η, x , ξ ) = ∞ 0 ψ(hz)z(∂ z λ)(hz, x , ξ )e -(iη+λ(hz,x ,ξ ))z dz.
We notice that

K 0 (y, x , ξ ) := e iyη h B 0 (η, x , ξ )ϕ(y, x )χ 0 (ξ ) η 2 + Q(y, x , ξ ) 2 dη
is a bounded symbol in S 0 ξ . Thus the second term on the right hand side of (??) is equal to

R 2 = O C 0 (Ry;L 2 (R d-1 x
)) (h) and we may concentrate on the first term. Write

B 1 (η, x , ξ ) = ∞ 0 ψ(hz)e -(iη+λ(hz,x ,ξ ))z λ(hz, x , ξ ) ξ dz.
Taylor expansion gives e -λ(hz,x ,ξ )z λ(hz, x , ξ )ψ(hz) =e -λ 0 (x ,ξ )z λ 0 (x , ξ ) +

1 0 d dt e -λ(htz,x ,ξ )z λ(htz, x , ξ )ψ(htz) dt =e -λ 0 (x ,ξ )z λ 0 (x , ξ ) + h 1 0 P t (z, x , ξ )e -λ(htz,x ,ξ )z dt with P t (z, x , ξ ) = -z 2 (λ∂ y λ)(htz, x , ξ )ψ(htz)+z(∂ y λ)(htz, x , ξ )ψ(htz)+zλ(htz, x , ξ )ψ (htz). Thus we have B 1 (η, x , ξ ) = λ 0 (x , ξ ) (iη + λ 0 (x , ξ )) ξ + h B 1 (η, x , ξ ),
where

B 1 (η, x , ξ ) = 1 0 ∞ 0 e -(iη+λ(htz,x ,ξ ))z 1 ξ P t (z, x , ξ )dzdt.
Note that near a point in E, |∂ α x ∂ β ξ P t (z, x , ξ )| ≤ C αβ z 2 , independent of t, h, hence the symbol

K 1 (y, x , ξ ) = e iyη h ϕ(y, x )χ 0 (ξ ) η 2 + Q(y, x , ξ ) 2 B 1 (η, x , ξ )dη ∈ S 0 ξ .
Therefore, the symbol in the principal term of E(y, x , hD y , hD x )(ψ(y)h∂ y Op h (A 0 )q 0 ) equals to

K 1 (y, x , ξ ) =λ 0 (x , ξ ) e iyη h ϕ(y, x )χ 0 (ξ ) (η 2 + Q(y, x , ξ ) 2 )(iη + λ 0 (x , ξ )) dη + h K 1 (y, x , ξ ) =2πλ 0 ϕ(y, x )χ 0 (ξ ) e -yQ h 2(λ 0 -Q)Q - e -yλ 0 h λ 2 0 -Q 2 + h K 1 (y, x , ξ ) =2πλ 0 ϕ(y, x )χ 0 (ξ ) e -yQ h -e -yλ 0 h 2(λ 0 -Q)Q + e -yλ 0 h 2Q(λ 0 + Q) + h K 1 (y, x , ξ ).
Note that

E 1 (y, x , ξ ) = 2πλ 0 ϕ(y, x )χ 0 (ξ ) e -yQ h -e -yλ 0 h 2(λ 0 -Q)Q + e -yλ 0 h 2Q(λ 0 + Q) > 0 near E, we have E(y, x , hD y , hD x )(ψ(y)h∂ y Op h (A 0 )q 0 ) = E 1 (y, x , hD x )q 0 + R 1 + R 2 , with R 2 = O C(Ry;L 2 (R d-1 x )) (h). We claim that R 1 = O H 1 (R d ) (1)
and thus by interpolation, R 1 H 2/3 (R d ) = O(h 1/3 ). Indeed, the reminder terms R 1 comes from symbols of the form hS -1 (in both η and ξ variables), and the symbolic calculus yields

∂ y R 1 = O L 2 (R d ) (1), and ∂ x R 1 = O L 2 (R d ) (1).
We next calculate the parallel component

1 (2πh) d ψ(z)e i(y-z)η h dzdη e ix ξ h - zλ(z,x ,ξ ) h θ(ξ )g jk (z, x )ξ k ϕ(y, x )χ 0 (ξ ) η 2 + Q(y, x , ξ ) 2 dξ = 1 (2πh) d e i(x ξ +yη) h ξ k θ(ξ )ϕ(y, x )χ 0 (ξ ) η 2 + Q(y, x , ξ ) 2 dηdξ ∞ 0 ψ(z)e -iη+λ(z,x ,ξ ) h z g jk (z, x )dz = h (2πh) d e i(x ξ +yη) h ξ k θ(ξ )ϕ(y, x )χ 0 (ξ ) (η 2 + Q(y, x , ξ ) 2 ) B 2,jk (η, x , ξ )dηdξ =: E 2 (y, x , hD x )q 0 .
where

B 2,jk (η, x , ξ ) = ∞ 0 ψ(z)e -iη+λ(z,x ,ξ ) h z g jk (z, x ) 1 h dz. Define K 2,jk (y, x , ξ ) = e iyη h B 2,jk (η, x , ξ )ϕ(y, x )χ 0 (ξ ) η 2 + Q(y, x , ξ ) 2 dη,
and from similar argument we can write

K 2,k (y, x , ξ ) = g jk (0, x ) e iyη h ϕ(y, x )χ 0 (ξ ) (η 2 + Q(y, x , ξ ) 2 )(iη + λ 0 (x , ξ )) + h K 2 (y, x , ξ )
and the principal symbol of E 2 (y, x , hD x ) is elliptic if λ 0 (ξ ) > 1 and y small enough.

Finally,

E(y, x , hD y , hD x )(2hv ⊗ δ y=0 ) = 2h (2πh) d F h (v)(ξ )e i(yη+x ξ ) h ϕ(y, x )χ 0 (ξ ) η 2 + Q(y, x , ξ ) 2 dξ dη + O L 2 (R d ) (h) = 2h (2πh) d F h (v)(ξ )e ix ξ h πe -yQ(y,x ,ξ ) h ϕ(y, x )χ 0 (ξ ) Q(y, x , ξ ) dξ + O L 2 (R d ) (h) = : E 3 (y, x , hD x )v + O L 2 (R d ) (h), (1.5.4)
and again, E 3 (y, x , hD x ) is elliptic near λ 0 (ξ ) > 1. Moreover, the reminder terms are indeed of O H 2/3 (R d ) (h 1/3 ) from the same argument as for R 1 . Now the boundary condition (w , w ⊥ )| y=0 = 0 and trace theorem yields

E 1 (0, x , hD x )q 0 = O L 2 (R d-1 x ) (h 1/3 ), E 2 (0, x , hD x )q 0 + E 3 (0, x , hD x )v = O L 2 (R d-1 x ) (h 1/3 ).
Therefore, from the ellipticity of E 1 , E 2 , E 3 , the measure of pressure at the elliptic region vanishes, so does the measure of v, namely σ| E = ν| E = 0. In summary, the proof of Proposition 1.6 is complete.

Near H

We take

ϕ 1 , ϕ ∈ C ∞ c (Y + ) such that ϕ 1 | supp(ϕ) ≡ 1. For any tangential symbol b ∈ C ∞ c (Y + × R d-1
), we define the pseudo-differential operator B h = ϕOp h (b)ϕ 1 , with compact support in Y + . We will change the notation of tangential variables (x , ξ ) to (x, ξ). We always work in local coordinate (y, x) and sometimes abuse the notation u = ϕ 1 u, q = ϕ 1 q as compactly supported functions in Y + . Note that q 0 , the trace of q is not bounded in L 2 in priori. Fortunately, it turns out that q 0 = O L 2 (1), micro-locally near a point in H.

L 2 bound of boundary datums

Take b(y, x, ξ), b 1 (y, x, ξ) ∈ C ∞ c ([0, 0 ) × H), such that b 1 | [0, 0 /2)×supp(b) ≡ 1. Let Q(y, x, ξ) = 1 -λ(y, x, ξ) 2 b 1 (y, x, ξ
). We will first factorize the operator (-h 2 ∆-1) near a hyperbolic point.

Lemma 1.9 For 0 ≤ y < 0 , we have

B h (-h 2 ∆ -1) = -(hD y -Q + h )(hD y -Q - h ) + R = -(hD y -Q - h )(hD y -Q + h ) + R , where R , R ∈ C ∞ ([0, 0 ], h ∞ Ψ -∞ (∂Ω))
, and Q ± h have principal symbol ±Q(y, x, ξ).

Proof. The proof is quite standard, and we follow the construction in [START_REF] Burq | Mesures de défaut de compacité, application au systeme de lamé[END_REF] by translating word by word to the semi-classical setting. In local coordinate, we have

B h (-h 2 ∆ -1) = h 2 D 2 y + R(y, x, hD x ) + hM 1 (y, x )hD y + hM 0 (y, x)hD x with σ(R) = Q 2 . Set q + 1 = √ Q(y, x , ξ ), Q + 1 = Op h (q + 1 ) and Q - 1 = -Q + 1 -hM 1 . Direct calculation gives (hD y -Q + 1 )(hD y -Q - 1 ) = h 2 D 2 y -(Q + 1 ) 2 -hQ + 1 M 1 -(Q + 1 + Q - 1 )hD y - h i ∂ y (Q - 1 ) = h 2 D 2 y -(Q + 1 ) 2 + hM 1 hD y -h(Q + 1 M 1 -i∂ y (Q - 1 )). Thus B h (-h 2 ∆ -1) -(hD y -Q + 1 )(hD y -Q - 1 )
= hT 1 , with some operator T 1 , bounded in L 2 . Now for j ≥ 1, suppose that we have

B h (-h 2 ∆ -1) -(hD y -Q + j )(hD y -Q - j ) = h j T j , then we set Q ± j+1 := Q ± j + h j S ± j+1 with S + j+1 + S - j+1 = 0, σ(S + j+1 ) = σ(T j ) 2σ(Q + j )
and obtain that

B h (-h 2 ∆ -1) -(hD y -Q + j+1 )(hD y -Q - j+1 ) =h j T j + h j (S + j+1 Q - j + Q + j S - j+1 ) -h j (S + j+1 + S - j+1 )hD y - h j+1 i ∂ y (S - j+1 ) + h 2j S + j+1 S - j+1 =:h j+1 T j+1 ,
for some operator T j+1 bounded in L 2 . The proof is then complete by induction.

Define w = ϕ 1 u -h∇(ϕ 1 q), w ± = B h (hD y -Q ± h )w and its boundary values w ± 0 := w ± | y=0 . Note that ϕP h w = ϕf . Proposition 1.10 B h h∂ y w ⊥ L 2 (R d + ) = O(1), and consequently, w ± ⊥ L 2 (R d + ) = O(1).
Proof. From h div u = 0, we have ϕh div w = 0, hence

ϕ(h∂ y w ⊥ + h div w ) = O L 2 (R d + ) (h), where in local coordinates, div w = 1 det(g) d-1 j=1 ∂ x j ( det(g)w ,j ).
Therefore,

B h h∂ y w ⊥ L 2 (R d + ) ≤ O(h) + B h hdiv w L 2 (R d + ) = O(1).
Now we recall a semi-classical version of hyperbolic energy estimate.

Lemma 1.10

Suppose A h = Op h (a) is ellptic (with real-valued symbol a smoothly depending on t) of order 0 on a compact manifold M and w are solutions of the h-dependent equations

(hD t ± A h )w = g, (t, x) ∈ R × M.
Assume that for any compact time interval I and small h,

w L 2 (I×M ) ≤ C(I), g L 2 (I×M ) ≤ C(I)h,
then we have for all small h,

sup t∈I w(t) L 2 (M ) ≤ C(I ), ∀I ⊂ Icompact.
Proof. By symmetry, we only need to treat the case hD t -A h . Take χ(t) ∈ C ∞ c (I ), and we may assume that 0 ∈ I with χ(0) = 1. Multiplying by χ(t) to the equation, we have

(hD t -A h )(χw) = χg + [χ, hD t -A h ]w =: r = O L 2 (R×M ) (h).
We now calculate

h d dt (χw|χw)(t) L 2 (M ) = (ihD t χw|χw) L 2 (M ) + (χw|ihD t χw) L 2 (M ) = i(A h (χw) + r|χw) L 2 (M ) -i(χw|A h (χw) + r) L 2 (M ) = i((A h -A * h )χw|χw) L 2 (M ) + i(r|χw) L 2 (M ) -i(χw|r) L 2 (M )
Integrating the formula above from 0 to sup I , we finally have w(0

) 2 L 2 (M ) = O(1). Lemma 1.11 w ± 0 L 2 (R d-1 ) = O(1).
Proof. From Proposition 1.10, we have

(hD y -Q ∓ h )w ± ⊥ = O L 2 (R d + ) (h). Applying the previous lemma to w ± ⊥ , we have w ± 0,⊥ L 2 (R d-1 x ) = O(1)
. Combining the boundary condition, we have

B h (Q + h -Q - h )(h∂ y q)| y=0 = -B h (Q + h -Q - h )hN q 0 = w + 0,⊥ -w - 0,⊥ = O L 2 (R d-1 ) (1)
.

Remark that in priori, N is a first order pseudo-differential operator, and we only have

B h hN q 0 L 2 (R d-1 ) ≤ B h H -1 →L 2 h N q 0 H -1 (R d-1 ) = O(h -1 ).
From the exact pricipal symbol of

Q ± h , we have B h hN q 0 L 2 (R d-1 ) = O(1)
, and the constant in big O depends on the micro-local cut-off b(y, x , ξ ). As a consequence,

w ± 0,⊥ L 2 (R d-1 ) = O(1).
It remains to study w ± . Notice that their boundary values are

w ± 0, = B h (v -(hD y h∇ q)| y=0 ) -B h Q ± h h∇ q 0 , where v = (h∂ y u)| y=0 = O L 2 (R d-1 ) (1). All terms are obviously bounded in L 2 (R d-1
) except the trace of B h h∇ hD y q. To bound it, we use the support property of b and Proposition

1.4, hence B h h∇ hD y q| y=0 = -B h h∇ hN q 0 = O L 2 (R d-1 x ) (1)
.

Again by hyperbolic estimates, we can establish the following results:

Proposition 1.11 w ± L 2 (R d + ) = O(1). In particular, B h hD y w L 2 (R d + ) + B h hN q 0 L 2 (R d-1 x ) + B h h 2 ∆ 0 q 0 L 2 (R d-1 x ) = O(1),
where ∆ 0 = ∆ ∂Ω , the Laplace-Beltrami operator on ∂Ω.

Proof. It remains to prove

B h h 2 ∆ 0 q 0 L 2 (R d-1 x ) = O(1). Indeed, B h h∂ y w ⊥ =B h h∂ y u ⊥ -h 2 B h ∂ 2 y q =h 2 B h ∂ y u ⊥ + h 2 B h 1 det(g) 1≤j,k≤d-1 ∂ j (g jk ∂ k q 0 ) + ∂ y det(g) det(g) ∂ y q . Thus B h h∂ y w ⊥ | y=0 = B h h 2 ∆ 0 q 0 + O L 2 (R d-1 ) (1), thanks to h∂ y u ⊥ = 0 and B h hN q 0 = O L 2 (R d-1 ) (h -1 ). From w ± 0,⊥ = B h hD y w ⊥ | y=0 + B h Q ± h hN q 0 = O L 2 (R d-1 ) (1), we deduce that B h hD y w ⊥ | y=0 = O L 2 (R d-1 ) (1)
, and these yield

B h h 2 ∆ 0 q 0 L 2 (R d-1 ) = O(1). Corollary 1.1 B h h∇q L 2 (R d + ) = o(1).
Proof. We will go back to the global notation in this calculation. It would be suffices to show that

B h h∇q = ϕOp h (b)ϕ 1 h∇q = o L 2 (Ω) (1)
since there are only change of bounded weight in the integral with respect to the measure det(g)dydx and dydx in local coordinate, and the former admit us to apply integrating by part and the structure of the equation in a simple way. We calculate

(B h h∇q|B h h∇q) L 2 (Ω) =([B h , h∇]q|B h h∇q) L 2 (Ω) + (h∇B h q|B h h∇q) L 2 (Ω) =o(1) -(A h B h q|hdivB h h∇q) L 2 (Ω) +(B h hq 0 |B h (h∂ ν q)| ∂Ω ) L 2 (∂Ω) =o(1) + (B h hq 0 |B h hN q 0 ) L 2 (∂Ω) ,
where we have used the fact that hq = o L 2 (Ω) (1) and ∆q = 0 in the calculation. Now from Lemma 1.3, we know that hq 0 weakly in H 1 (Ω) and

hq 0 → 0 strongly in L 2 (∂Ω). The last term is o(1) since B h hN q 0 = O L 2 (∂Ω) (1).

propagation of singularity

In this subsection, we prove Proposition 1.7.

We factorize

-h 2 ∆ -1 as (hD y -Q ± h )(hD y -Q ∓ h ) + R ± near z 0 ∈ H and choose Q ± h with principal symbols ±Q(y, x, ξ) = √ 1 -λ 2 b 1 (y, x, ξ)
, as in the previous subsection. Take ψ ∈ C ∞ c ([0, 0 ))with ψ ≡ 1 in a neighborhood of y = 0. By an abuse of notation, we introduce

w ± = B ± h (hD y -Q ∓ h )w, with B ± h has principal symbols ψ(y)b ± (y, x, ξ) where b ± are solutions of ∂b ± ∂y ∓ H Q(y,x,ξ) b ± = 0, b ± | y=0 = b 0 , (1.6.1) b 0 is another micro-localization near z 0 with b 1 | supp(b 0 ) = 1 and H Q b = {Q, b}.
Note that the compact support of ψ(y)b ± can be chosen to be arbitrarily close to the semi-bicharacteristic curves γ ± corresponding to the principal symbol p. Moreover, b ± are invariant along γ ± . Under these notations, Proposition 1.7 can be rephrased as follows Proposition 1.12 Let µ be the defect measure of u.

If b + µ1 0<y≤ 0 = 0(b -µ1 0<y≤ 0 = 0), then we have b -µ1 0<y≤ 0 = 0(b + µ1 0<y≤ 0 = 0).

Moreover, we have in fact b

+ µ = b -µ = 0 in this case.
The proof will be divided into several lemmas. First we calculate

(hD y -Q ± h )w ± = [hD y -Q ± h , B ± h ](hD y -Q ∓ h )w + B ± h (hD y -Q ± h )(hD y -Q ∓ h )w,
and

[hD y -Q ± h , B ± h ] = h i Op h (∂ y b ± ∓ H Q b ± )ψ(y) + h i ψ (y)B ± h + R .
The first operator vanishes thanks to the definition of b ± , and the remainder term

R = O L 2 (R d + ) (h 2 ). Therefore we have R (hD y -Q ∓ h )w L 2 (R d + ) = O(h 2 ).
and consequently

(hD y -Q ± h )w ± = h i ψ (y)w ± + g ± , with g ± = o L 2 (R d + ) (h). Lemma 1.12 Let µ ± be the semi-classical defect measure of w ± , b is defined as above. Suppose b ± µ ± 1 0<y≤ 0 = 0, then we must have b ± µ ± ≡ 0 and µ ± 0 = 0, where µ ± 0 is the defect measure of w ± 0 = w ± | y=0 .
Proof. Take y 0 = 0 /2, we first claim that w ± (y 0 ) L 2 x = o(1). Indeed, from the assumption and compactness, the measure µ ± vanishes in a small neighborhood of semi-bicharacteristic curve γ ± . Thus

w ± L 2 ([y 0 , 0 ]×R d-1 ) = o(1)
, provided that we choose supp(b 0 ) small enough in the definition of w ± . Finally, repeat the argument in the proof of Lemma 1.10, we have

-h w ± (y 0 ) 2 L 2 (R d-1 ) = i 0 y 0 ((Q ± h -(Q ± h ) * )χw ± |χw ± ) L 2 (R d-1 ) (y)dy + o(h).
The claim then follows.

Integrating the identity

h d dy (w ± |w ± ) L 2 (R d-1 ) = (i(Q ± h -(Q ± h ) * )w ± |w ± ) L 2 (R d-1 ) + 2h(ψ (y)w ± |w ± ) L 2 (R d-1 ) + 2Im(w ± |g ± ) L 2 (R d-1 )
from y = z < y 0 to y = y 0 , we have

w ± (z) 2 L 2 (R d-1 ) ≤ C y 0 z w ± (y) 2 L 2 (R d-1 ) dy + o(1).
Using

y 0 0 w ± (y) 2 L 2 (R d-1 ) dy = o(1), we obtain that w ± 0 L 2 (R d-1 ) = o(1).

Remark 1.3

Away from the boundary, the defect measure of u equals to the defect measure of w, and it propagates along the bi-characteristic curves γ ± . Since we can essentially decompose w into w + and w -near a hyperbolic point, we call w + (w -)the incoming wave and the out-coming wave. Thus the above proposition asserts that if we have no singularity of w + (w -) along incoming wave(out-coming wave) near the boundary but strictly away from the boundary, then there is no singularity of the boundary data of incoming wave(out-coming wave).

Changing the role of y = y 0 and y = 0 in the proof of Lemma 1.12, we conclude that if

µ ± 0 = 0, then b ± µ ± = 0.
To finish the proof, we need understand how the singularity transfers form boundary data of in-coming wave to the boundary data of out-coming wave.

Lemma 1.13 µ ± 0 = 0 implies that µ ∓ 0 1 ξ =0 = 0. Consequently, µ ∓ 1 ξ =0 = 0.
Proof. By symmetry, we only need deduce µ -

0 1 ξ =0 = 0 from µ + 0 = 0. For δ > 0, we define b 0,δ (x, ξ) = b 0 (x, ξ) 1 -ψ λ(0, x, ξ) δ , with some ψ ∈ C ∞ c (R), ψ| [-2,2] ≡ 1.
We define b ± δ (y, x, ξ) by solving ODE (1.6.1) with initial data b 0,δ .Let B ± δ,h be the associated PdO with principal symbol b ± δ . From compactness and continuity dependence of the initial data, we have such that δ < λ(y, x, ξ

) < c 0 < 1 on supp(b δ (y)) for 0 ≤ y ≤ 0 . Since in Y + × R d-1 , λ(y,x,ξ) |ξ| ∼ 1. Since b ± (y, x, ξ) = b 0 • γ ± (y) -1 (x, ξ), b ± δ (y, x, ξ) = b 0,δ • γ ± (y) -1 (x, ξ), we have that b ± δ b ± is a smooth function with compact support in Y + × R d-1 . Denote by w ± δ = B ± δ,h (hD y -Q ∓ h )w, associated with semi-classical defect measure µ ± δ , we have µ ± δ = µ ± b ± δ b ± 2 . In particular, µ ± δ,0 = µ ± 0 1 -ψ λ 0 δ 2
and supp(µ ± δ ) ⊂ supp(µ ± ). On the boundary, B + δ,h and B - δ,h coincide and will be denoted by B 0 δ,h . Taking the trace of w ± δ , we have

w + δ,0, = -iB 0 δ,h v + ih 2 B 0 δ,h ∂ y (∇q) | y=0 + B 0 δ,h Q + h h∇ q 0 , w + δ,0,⊥ = iB 0 δ,h h 2 ∂ 2 y q| y=0 + B 0 δ,h Q + h h∂ y q| y=0 , where v = h∂ y u| y=0 = O L 2 x (1). Similarly, we have w - δ,0, = -iB 0 δ,h v + ih 2 B 0 δ,h ∂ y (∇q) | y=0 + B 0 δ,h Q - h h∇ q 0 , w - δ,0,⊥ = iB 0 δ,h h 2 ∂ 2 y q| y=0 + B 0 δ,h Q - h h∂ y q| y=0 . Notice that σ(Q + h ) = -σ(Q - h ), and write α = -B 0 δ,h h 2 ∆ 0 q 0 , β = B 0 δ,h Q + h hN q 0 , we have w ± δ,0,⊥ = iα ∓ β + O L 2 x (h).
From assumption w + δ,0,⊥ L 2 = o(1), we have iα -

β 2 L 2 = o(1)
, and this implies that

α 2 + β 2 -2Im(α|β) = o(1). We claim that Im(α|β) = o(1).
Indeed, we first note that from Proposition 1.11 and the ellipticity of N , we have that

q 0 = O L 2 (R d-1 ) (1)
, micro-locally away from ξ = 0. Now from trace theorem and Proposition 1.4, we have

β = A δ,h q 0 + O L 2 (R d-1 x
) (h 1/3 ) for some PdO with real-valued principal symbol a δ , compactly supported and vanishing for λ(y, x, ξ) ≤ δ/4. Similarly,

α = A δ,h q 0 + o L 2 (R d-1 ) (1)
for some PdO with real-valued principal symbol a δ . Thus Im(α|β) L 2 = o(1), since all the principal symbols involved in the inner product are real-valued. Now from 1), one can deduce that the terms on the righthand side of w ± δ,0, involving pressure are also o L 2

α L 2 = o(1), β L 2 = o(
x (1), and ). Therefore µ - δ,0 = 0 and consequently µ - δ = 0 from Lemma 1.12. This implies that µ -

v = o L 2 x (1) follows since w - δ,0, = o L 2 (R d-1 ) (1
0 1 λ>δ = µ -1 λ>δ = 0. Since δ > 0 is arbitrary, we have that b -µ -1 ξ =0 = 0. Moreover, Corollary 1.1 implies that µ1 ξ =0 = 0.
Now we finish the proof of Proposition 1.12 by showing the following lemma.

Lemma 1.14 µ + = 0 implies that µ -= 0.

Proof. We need deal with ξ = 0. Take ψ be a cut-off function which equals 1 in a neighborhood of 0. Pick any > 0, we define the operator

B ,± h = Op h ( ψ(λ(y, x, ξ)/ ))B ± h . Applying divergence equation for w ± B ,± h hdiv w ± + B ,± h h∂ y w ± ⊥ = O L 2 (R d + ) (h), we have B ,± h h∂ y w ± ⊥ L 2 (R d + ) ≤ B ,± h hdiv w ± L 2 (R d + ) + R (h) with R (h) → 0, as h → 0 for each fixed > 0.
By estimating the operator norm from its symbol, we have

B ,± h h∂ y w ± ⊥ L 2 (R d + ) ≤ C + R (h), and lim sup h→0 + B ,± h h∂ y w ± ⊥ L 2 (R d + ) ≤ C .

Using the equation hD

y w ± ⊥ = Q ± h w ± ⊥ + O L 2 (R d + ) (h), we have lim sup h→0 + B ,± h Q ± h w ± ⊥ L 2 (R d + ) ≤ C
Finally let → 0, we have µ ± ⊥ 1 ξ=0 = 0. Therefore µ - ⊥ = 0. As a consequence of Lemma 1.12(more precisely, using the proof of Lemma 1.12), µ ± 0,⊥ 1 ξ=0 = 0. Now let µ α , µ β be the defect measures of α = -B 0 h h 2 ∆ 0 q 0 , β = B 0 h Q h hN q 0 , and let µ iα±β be the defect measure of iα ± β. Denote also by µ αβ the limit corresponding to the quadratic form (A h α|β). Similarly for µ βα . Note that µ αβ = µ βα . From

µ iα+β , 1 ξ=0 = µ α , 1 ξ=0 + µ β , 1 ξ=0 -2 Im µ αβ , 1 ξ=0 = 0 µ iα+β , 1 ξ=0 = µ α , 1 ξ=0 + µ β , 1 ξ=0 + 2 Im µ αβ , 1 ξ=0 = 0,
we have µ α 1 ξ=0 = µ β 1 ξ=0 = 0. Next we consider parallel components. The key claim is that the measure corresponding to B 0 h Q ± h h∇ q 0 vanishes on the set {ξ = 0}. Indeed, from Lemma 1.3 and trace theorem, hq 0 → 0 strongly in L 2 (∂Ω). From the ellipticity of N , there exists a classical pseudo-differential operator E of order -1 such that EN = I + R, where R is a non semi-classical smoothing operator.

We want to show that

lim →0 lim sup h→0 B 0 h B ,0 h Q ± h h∇ q 0 L 2 (R d-1 ) = 0.
From symbolic calculus and the strong convergence of

hq 0 in L 2 (R d-1 x ), it suffices to prove lim →0 lim sup h→0 h∇ B 0 h B ,0 h Q ± h q 0 L 2 (R d-1 ) = 0. (1.6.2)
We write

h∇ B 0 h B ,0 h Q ± h q 0 =h∇ B 0 h B ,0 h Q ± h EN q 0 -h∇ B 0 h B ,0 h Q ± h Rq 0 =∇ EB 0 h B ,0 h Q ± h hN q 0 + h∇ [B 0 h B ,0 h Q ± h , E]N q 0 -h∇ B 0 h B ,0 h Q ± h Rq 0 .
(1.6.3)

Here we are taking the commutator between a semi-classical operator and a classical pseudo-differential operator, hence the semi-classical symbolic calculus is not applicable. Instead, it is not difficult to check that for any

a ∈ C ∞ c (T * ∂Ω), E ∈ S -1 x,ξ , [a(x, hD x ), E(x, D x )] = hOp(S -1 ) + Op(S -2 ),
where the implicit constants only depend on the semi-norms of the symbols a(x, ξ) and

E(x, ξ). Notice that h∇ B 0 h , B ,0 h , Q ± h are uniformly bounded operators in L 2 x with respect to h, thus ∇ B 0 h B ,0 h Q ± h R, ∇ B 0 h Op(S -2 )N , h∇ B 0 h Op(S -1 )N are uniformly bounded op- erators in L 2
x with respect to h. Thus from the strong convergence of hq 0 , the last two terms on the right hand side of (1.6.3) are killed when we let h → 0 first. Thus (1.6.2) follows from the vanishing of the measure of ±β = B 0 h Q ± h hN q 0 on the set {ξ = 0}. Combining the assumption that µ + 0, 1 ξ=0 = 0, we deduce that µ - 0, 1 ξ=0 = 0. The proof is now complete.

Near G 2,+

In this section, we will follow the strategy of V.Ivrii (see [START_REF] Ivrii | Microlocal analysis and precise spectral asymptotics[END_REF] or [START_REF] Hörmander | The analysis of linear partial differential operators. iii, volume 274 of grundlehren der mathematischen wissenschaften (fundamental principles of mathematical sciences[END_REF]) to prove Proposition 1.8. Denote by G = det(g) and

P h = h 2 ∆ H -1, we have Lemma 1.15
In local coordinate Y + , we have

P h = -h 2 g √ G ∂ ∂y √ Gg -1 ∂ ∂y + R h = h 2 D 2 y + Op h (r) + O L 2 →L 2 (h).
Moreover, R h is a matrix-valued second order differential operator in x with scalar principal symbol r(y, x, ξ) = 1 -λ(y, x, ξ) 2 , which is self-ajoint with respect to the

(•|•) L 2 (Y + ) .
The proof is direct calculation and will be given in the appendix.

In a fix local coordinate in Y + , we identify u = ϕ 1 u, q = ϕ 1 q and all the operators B by ϕBϕ 1 .

Proposition 1.13

For any tangential operator B with scalar principal symbol b(y, x, ξ) which vanishing near ξ = 0, we have lim sup

h→0 BhD y u L 2 (Y + ) ≤ sup ρ∈supp (b) |r(ρ)| 1/2 |b(ρ)|.
Proof. We calculate

(BhD y u|BhD y u) Y + =([B, hD y ]u|BhD y u) Y + + (hD y Bu|BhD y u) Y + =O(h) + (Bu|Bh 2 D 2 y u) L 2 (Y + ) =O(h) -(Bu|BRu) L 2 (Y + ) + (Bu|BP h u) Y + =O(h) -(Bu|BRu) Y + -(Bu|Bhdq) Y + .
Integrating by part and using symbolic calculus, we have

(Bu|Bhdq) Y + =(Bu|hdBq) Y + + (Bu|[B, hd]q) Y + = -([hd * , B]u|Bq) Y + + (Bu|[B, hd]q) Y + =O(h),
thanks to the fact that B has scalar-valued principal symbol.

The proof of Proposition 1.8 is based on the following integrating by part result.

Proposition 1.14 Given real scalar-valued tangential symbols a 0 , a 1 , there exist tangential operators A 0 , A 1 (constructed in the local coordinate) with real, scalar-valued principal symbol a 0 , a 1 and A = A 1 hD y +A 0 , such that for any 1-form w with compact support in Y + , we have

2 h Im(P h w|Aw) Y + = (A 1 hD y w|hD y w) ∂Y + + Re 2 j=0 (C j (hD y ) j w|w) Y + + O(h),
where the tangential operators C j have scalar-valued principal symbol c j (y, x, ξ) and

2 j=0 c j (y, x, ξ)η j = {p, a}.
Proof. We first calculate

I = 1 ih - h 2 √ G gh∂ y ( √ Gg -1 h∂ y w) Aw Y + - 1 ih Aw - h 2 √ G gh∂ y ( √ Gg -1 h∂ y w) Y + =(hD y w|A 1 hD y w) ∂Y + + (A 1 hD y w|hD y w) ∂Y + + 1 ih (hD y w|hD y Aw) Y + - 1 ih (hD y Aw|hD y w) Y + =(hD y w|A 1 hD y w) ∂Y + + (A 1 hD y w|hD y w) ∂Y + + 1 ih (hD y w|[hD y , A]w) Y + - 1 ih ([hD y , A]w|hD y w) Y + + 1 ih (hD y w|AhD y w) Y + - 1 ih (AhD y w|hD y w) Y + ,
and the last two terms on the right hand side equal to

1 ih (A * hD y w|hD y w) -(A * 1 hD y w|hD y w) ∂Y + - 1 ih (AhD y w|hD y w) Y + .
We want to construct operators A 0 , A 1 such that

A * 1 = A 1 + O(h 2 ) and A * = A + O(h 2 ). Assume that a 1 a (0) 1 + h i a (1) 1
with real-valued a (j)

1 (not necessarily scalar-valued). From

Y + A 1 u|v √ Gdydx = Y + g -1 A 1 u, v C d-1 √ Gdydx, the symbol of A * 1 is equal to the symbol of g √ G Op h ( a 1 * ) √ Gg -1 , which can be expressed by b 1 (y, x, ξ) k≥0 h i k b (k) 1 (y, x, ξ), with b (k) 1 (y, x, ξ) = 1 j=0 |α|+|β|+j=k (-1) j ∂ β ξ g √ G ∂ α ξ ∂ α x a (j) 1 • ∂ β x ( √ Gg -1 ), k ≥ 1.
We have that b

(0) 1 = a (0) 1 , b (1)
1 = -a

1

+ |α|+|β|=1 ∂ β ξ g √ G ∂ α ξ ∂ α x a (0) 1 • ∂ β x ( √ Gg -1 )
We set

a (0) 1 = a 1 , a (1) 1
= 1 2 |α|+|β|=1 ∂ β ξ g √ G ∂ α ξ ∂ α x a (0) 1 • ∂ β x ( √ Gg -1 ), thus A * 1 = A 1 + O(h 2 ). Note that a (1)
1 is matrix-valued while a (0)

1 is real and scalar-valued. The construction of A 0 is similar. We observe that (hD y

) * = hD y + h g √ G D y ( √ Gg -1 ) and set a 0 = a (0) 0 + h i a (1)
0 . A * 0 has symbol which can be expanded as

b 0 k≥0 h i k b (k) 0 with b (0) 0 = a (0) 0 and b (k) 0 (y, x, ξ) = 1 j=0 |α|+|β|+j=k (-1) j ∂ β ξ g √ G ∂ α ξ ∂ α x a (j) 0 • ∂ β x ( √ Gg -1 ), k ≥ 1. (1.7.1) Note that (hD y ) * A * 1 -A * 1 hD y =[(hD y ) * , A * 1 ] + A * 1 (hD y ) * -A * 1 hD y = h i (∂ y A * 1 ) + h i g √ G ∂ y ( √ Gg -1 ), A * 1 + h i A * 1 g √ G ∂ y ( √ Gg -1 ),
and its symbol can be expanded as

k≥0 h i k κ k (y, x, ξ)
with κ 0 = 0 and

κ 1 = ∂ y b 1 + b 1 g √ G ∂ y ( √ Gg -1 ) , κ k = |α|=k-1 1 i |α|+1 {∂ α ξ , ∂ α x } g √ G ∂ y ( √ Gg -1 ), b 1 h i ∂ y b 1 + b 1 g √ G ∂ y ( √ Gg -1 ) + h i |α|≥1 h |α| i |α| {∂ α ξ , ∂ α x } g √ G ∂ y ( √ Gg -1 ), b 1 , k ≥ 2, where {∂ α ξ , ∂ α x }(f 1 , f 2 ) = ∂ α ξ f 1 ∂ α x f 2 -∂ α ξ f 2 ∂ α x f 1 .
We set b (0) 0 = a 0 and a

0 such that a

(1)

0 = b (1)
0 + κ 1 (it has a solution thanks to (1.7.1)). Finally, we construct A j by ϕ 1 Op h ( a j )ϕ 1 in local coordinate and one verify easily that

A * 1 = A 1 + O L 2 →L 2 (h 2 ), A * = A + O L 2 →L 2 (h 2 ).
Therefore

I = (hD y w|A 1 hD y w) ∂Y + + 1 ih (hD y w|[hD y , A]w) Y + - 1 ih ([hD y , A]w|hD y w) Y + + O(h).
We next calculate

1 ih (Rw|Aw) Y + - 1 ih (Aw|Rw) Y + = 1 ih ((A * R -R * A)w|w) Y + = 1 ih ([A, R]w|w) Y + + O(h) since R is self-ajoint and A * -A = O L 2 →L 2 (h 2 ). Moreover, the principal symbol of 1 ih [A, R] is {r, a}. This completes the proof.
Now assume that we are working near a diffractive point ρ ∈ G 2,+ in Y + where ∂r ∂y

≥ c 0 > 0

The following lemma is a semi-classical version of Lemma 24.4.5 in [START_REF] Hörmander | The analysis of linear partial differential operators. iii, volume 274 of grundlehren der mathematischen wissenschaften (fundamental principles of mathematical sciences[END_REF]. The proof is a little more complicated, due to the different equation that we are considering.

Lemma 1.16

Let B j = ϕB j ϕ 1 , with real, scalar-valued tangential principal symbol b j , j = 0, 1, 2, compactly supported and

2 j=0 b j (y, x, ξ)η j = -ψ(y, x, η, ξ) 2 when η 2 = r(y, x, ξ), with some smooth function ψ ∈ C ∞ c (R d × (R d \ {0})). Further assume that dr = 0, ∂r ∂y > 0, on {y = r = 0} ∩ 2 j=1
supp(b j ).

Then one can chose compactly supported, tangential operators Ψ j , j = 0, 1 with real, scalarvalued principal symbols ψ j , j = 0, 1 which satisfy

ψ 0 (y, x, ξ) = ψ(y, x, 0, ξ), ψ 1 (y, x, ξ) = ∂ η (y, x, 0, ξ) when η = r(y, x, ξ) = 0,
so that for any solution u of P h u = f -h∇q, hdivu = 0 with u| y=0 = 0 in trace sense, we have

Re 2 j=0 (B j (hD y ) j v|v) Y + + Ψ 0 v + Ψ 1 hD y v 2 L 2 (Y + ) + (ΘP h v|v) Y + =o (1)
(1.7.2)

as h → 0, where v = ϕOp h (χ)ϕ 1 u and χ ∈ C ∞ c (Y + × R d-1
) has support near ρ ∈ G 2,+ . Θ is a tangential operator which depends on ψ j , b j , whose principal symbols are scalar-valued.

The proof is based on the following elementary lemma, for which the proof can be found as Lemma 24.4.3 in [START_REF] Hörmander | The analysis of linear partial differential operators. iii, volume 274 of grundlehren der mathematischen wissenschaften (fundamental principles of mathematical sciences[END_REF],

Lemma 1.17 Let X be an open subset of R n + = {x ∈ R n : x 1 ≥ 0}, and let r ∈ C ∞ (X)
. Assume that r is real-valued, that dr = 0 when r = 0 and that ∂r ∂x 1 > 0 when r = x 1 = ∂r ∂x j = 0 for j = 1. Let

F (t, x) = 2 j=0 f j (x)t j be a quadratic polynomial in t with coefficients in C ∞ (X) such that F (t, x) = -ψ(t, x) 2 when t 2 = r(x),
where ψ ∈ C ∞ (R×X). Then one can find ψ 0 , ψ 1 , θ ∈ C ∞ (X) with ψ 0 (x) = ψ(0, x), ψ 1 (x) = ∂ψ ∂t (0, x) when r(x) = 0, and

F (t, x) + (ψ 0 (x) + tψ 1 (x)) 2 ≤ θ(x)(t 2 -r(x)); t ∈ R, x ∈ X.
Proof of Lemma 1.16. Choose C ∞ functions ψ 0 (y, x, ξ) and ψ 1 (y, x, ξ) as in the previous Lemma with ψ j (y, x, ξ) = ∂ j η ψ| y=0 , j = 0, 1 when η = r(y, x, ξ) = 0 and so that

2 j=0 b j η j + (ψ 0 + ηψ 1 ) 2 ≤ θ(y, x, ξ)(η 2 -r).
Since ψ 0 , ψ 1 and each b j are compactly supported in variables (y, x, ξ), we may assume that θ is smooth and with compact support. Define Θ = ϕOp h (θ)ϕ 1 , Ψ j = ϕOp h (ψ j )ϕ 1 , j = 0, 1 and consider the quantity

Re 2 j=0 (B j (hD y ) j v|v) Y + + ((Ψ 0 + Ψ 1 hD y ) 2 v|v) Y + -(ΘhD y v|hD y v) Y + + (ΘR h v|v) Y + .
The expression above can be written in the form

2 j=0 C j (hD y ) j v|v Y + ,
where the tangential operators C j have real, scalar-valued principle symbol. Moreover,

2 j=0 c j (y, x, ξ)η j ≤ 0.
However, since the symbol is not bounded in η and we can not apply sharp Gårding inequality directly. To resolve this issue, we extend each

c j to c j ∈ C m c (R × R 2d-2
) who agrees with c j on y ≥ 0 up to order m, any given order, of derivatives. This is possible since any order of y derivatives of all the symbols has continuous limit as y → 0. We still use the notation c j in what follows. Let v = v1 y≥0 and we use the boundary condition v| y=0 = 0 and calculate

2 j=0 C j (hD y ) j v v Y + = 2 j=0 C j (hD y ) j v v Y + = ψ hD y A 2 j=0 C j (hD y ) j v v Y + + 1 -ψ hD y A 2 j=0 C j (hD y ) j v v Y + =:I + II,
for any big number A > 0. Now we apply sharp Gåding inequality to the first term to get

I ≤ C A h,
with some constant C A depending on A. The second term is essentially in the elliptic region and we define its symbol

Ξ(y, x, η, ξ) := 1 -ψ η A 2 j=0 c j (y, x, ξ)η j η 2 -r(y, x, ξ) ∈ S 0 (R 2d ),
hence we can bound

|II| ≤O(h) + C Ξ(y, x, hD y , hD x )χ(y, x, hD x )P h u 1 -ψ 2hD y A v Y + =O(h) + C Ξ(y, x, hD y , hD x )χ(y, x, hD x )(2hw ⊗ δ y=0 ) 1 -ψ 2hD y A v Y + +C Ξ(y, x, hD y , hD x )χ(y, x, hD x )(1 y≥0 h∇q) 1 -ψ 2hD y A v Y +
, with w = hD y u| y=0 . Note that to obtain the expression above, one can not use symbolic calculus to deal with commutator between tangential symbol and usual symbol. However, since P h is a differential operator, we can compute its commutator with χ(y, x, hD x ) directly.

Now from Proposition 1.22, the limsup of the third term on the right hand side when h → 0 can be bounded by (A) with lim A→∞ (A) = 0. Here we can use the flat metric to estimate the L 2 norm. The second term on the right hand side can be bounded by

Ch (1 -h 2 ∆ y,x ) -s 2 (w ⊗ δ y=0 ) L 2 (R d ) (1 -h 2 ∆ y,x ) s 2 v L 2 (R d ) ≤ Ch 1-s ,
for any s ∈ 1 2 , 1 . Here we have used the fact that δ y=0 ∈ H -s (R y ) for any s > 1 2 and

h s v is bounded in H s (R d ) since v| y=0 = 0 and h∇ y,x v is bounded in L 2 (R d ).
Therefore, for any A > 0, we have proved that

lim sup h→0 |II| ≤ (A),
and this completes the proof.

Adapting to the notations in this section, Proposition 1.8 can be rephrased as follows Proposition 1.15 Suppose ρ ∈ G 2,+ , and

ρ 0 ∈ T * Ω approaching to ρ such that ∂ y r(ρ 0 ) ≥ 1 2 ∂ y r(ρ) ≥ c 0 . Let γ -= [ρ 0 ,
ρ] be a segment of the generalized ray issued from ρ 0 to ρ (the trajectory under the canonical projection is tangent to the boundary at ρ). Suppose ρ 0 / ∈ suppµ. Then we have ρ / ∈ suppµ.

Proof. Take a small neighborhood Γ 0 of ρ 0 such that Γ 0 ∩suppµ = ∅. Take a small neighborhood W 0 ⊂ Ω × R d-1 such that ∂r ∂y (y, x, ξ) ≥ c 0 /4 > 0. Shrinking W 0 if necessary, we assume that each point (y, x, ξ) ∈ W 0 with r(y, x, ξ) ≥ 0 can be connected by a (possibly broken) ray issued from Γ 0 with at most one reflection or tangency at ∂Ω. It suffices to prove the following statement:

For any χ ∈ C ∞ c (Ω × R d-1
) with supp χ ⊂ W 0 , small enough, we have

ϕOp h (χ)ϕ 1 u = o L 2 (1), h → 0.
As in [START_REF] Hörmander | The analysis of linear partial differential operators. iii, volume 274 of grundlehren der mathematischen wissenschaften (fundamental principles of mathematical sciences[END_REF], we construct test functions which satisfy some properties as follows:

Lemma 1.18 There exists a(y, x, η, ξ) = a 0 (y, x, ξ) + a 1 (y, x, ξ)η, a j ∈ C ∞ c (W 0 ) with the following properties: 1. a 1 (0, x, ξ) = -t(x, ξ) 2 , for some t ∈ C ∞ c (T * ∂Y + ), 2.
For some large M ≥ 0, when p = η 2 -r(y, x, ξ) = 0, we have

{p, a} + aM = -ψ(y, x, η, ξ) 2 + ω(y, x, ξ)(η -r 1/2 (y, x, ξ)), a = s 2 , where s ∈ C ∞ (Y + × (R d \ {0})),ψ ∈ C ∞ c (Y + × R d \ {0}) and ω ∈ C ∞ c (W 0 ). Moreover, r| supp ω > 0.
The construction is exactly the same as in [START_REF] Hörmander | The analysis of linear partial differential operators. iii, volume 274 of grundlehren der mathematischen wissenschaften (fundamental principles of mathematical sciences[END_REF] and will be given in the appendix D for the sake of completeness. Now we take χ ∈ C ∞ c (W 0 ) with χ ≡ 1, in a neighborhood of supp a 1 ∪ supp a 2 . Let v = ϕOp h (χ)ϕ 1 u, and we calculate

(P h v|Av) Y + =(ϕOp h (χ)ϕ 1 P h u|Av) Y + + ([P h , ϕOp h (χ)ϕ 1 ]u|Av) Y + =(ϕOp h (χ)ϕ 1 f |Av) Y + -(ϕOp h (χ)ϕ 1 hdq|Av) Y + +([P h , ϕOp h (χ)ϕ 1 ]u|Av) Y + .
Here we have used the differential form formulation to calculate the inner product. Notice that {p, χ} = 0 on supp a j and

f = o L 2 (h), hD y u ⊥ | y=0 = 0, thus 2 h Im(P h v|Av) Ω = o(1) - 2 h Im([ϕOp h (χ)ϕ 1 , hd]q|AϕOp h (χ)ϕ 1 u) Y + + 2 h Im(Op h (χ)q|hd * (AϕOp h (χ)ϕ 1 u)) Y + .
(1.7.3)

From Proposition 1.14,

2 j=0 (C j (hD y ) j v|v) Y + = -(A 1 hD y v|hD y v) ∂Y + - 2 h Im([ϕOp h (χ)ϕ 1 , hd]q|Av) Y + + 2 h Im(ϕOp h (χ)ϕ 1 q|hd * (AϕOp h (χ)ϕ 1 u)) Y + + o(1). (1.7.4)
Since the principal symbol of A is scalar-valued, by using d * u = 0, we could write

2 h Im(ϕOp h (χ)ϕ 1 q|hd * (AϕOp h (χ)ϕ 1 u)) Y + = (ϕOp h (χ)ϕq|Υu) Y + + O(h)
where

Υ = Υ 0 + Υ 1 hD y , Similarly, - 2 h Im([ϕOp h (χ)ϕ 1 , hd]q|Av) Y + = (Υ 2 q|Av) Y + + O(h),
where Υ j are matrix-valued tangential pseudo-differential operators with principal symbol supported in supp(χ) Using Lemma 1.16 for the function

2 j=0 c j η j + aM -ω(η -r 1/2 ) = -ψ 2 ,
we have

Re 2 j=0 C j (hD y ) j u|u Y + -Re(Φ(hD y -Q + )v|ϕ(y, x, hD x )v) Y + + Re(M v|Av) Y + + (ΘP h v|v) Y + + Ψ 0 v + Ψ 1 hD y v 2 L 2 (Y + ) ≤o(1) + Ch v 2 L 2 (Y + ) , (1.7.5)
where the compact supported tangential operator Φ has scalar-valued principal symbol φ ∈ C ∞ c (W 0 ) and r| supp φ > 0, φ = 1 in a neighborhood of supp ω. Q + is the operator constructed in the hyperbolic region with principal symbol r 1/2 . This is possible since in the proof of Lemma 1.18, we indeed have r ≥ δ 2 |ξ| 2 on the support of ω. Note that the principal symbol of A is positive on η 2 -r = 0, we can apply Lemma 1.16 again to the term (M v|Av) Ω and bound it from below by

o(1) -|(Θ 1 P h v|v) Y + |.

Thus we have

-(A 1 hD y v|hD y v) ∂Y + + Ψ 0 v + Ψ 1 hD y v 2 L 2 (Y + ) ≤ o(1) + Ch v 2 L 2 (Y + ) + C|(ΘP v |v) Y + | + C|(Θ 1 P h v|v) Y + | + Re(ϕOp h (φ)ϕ 1 (hD y -Q + )v|ϕOp h (ω)ϕ 1 v) Y + + |(Υ 2 q|Av) Y + | + |(ϕOp h (χ)ϕ 1 q|Υu) Y + |. (1.7.6)
The terms on the left hind side are essentially positive from semi-classical sharp Gårding inequality, hence we only need to control the terms on the right hind side. The term 1) follows from the equation and symbolic calculus since the principal symbols of Θ and Θ 1 are scalar-valued. We claim that

|(ΘP h v|v) Ω | + |(Θ 1 P h v|v) Ω | = o(
Re(ϕOp h (φ)ϕ 1 (hD y -Q + )v|ϕOp h (ω)ϕ 1 v) Y + = o(1), h → 0. (1.7.7)
Indeed, micro-locally on supp φ, r δ 2 > 0, hence in the region where λ 2 (y, x, ξ) < 1, we could construct Q + , Q -micro-locally such that

P h = (hD y -Q -)(hD y -Q + ) + O(h ∞ )
as we have done in the hyperbolic case. From symbolic calculus and Corollary 1.1, we have

(hD y -Q -)(hD y -Q + )u = O L 2 y,x (h) + h∇q = o L 2 y,x (1)
, micro-locally on supp φ. Therefore the measure µ concentrates on {η = -√ r} ∪ {η = √ r}, on the support of φ. For any point ρ 1 ∈ supp φ∩supp µ, with η(ρ 1 ) = -r(ρ 1 ) < 0, the backward generalized ray issued from ρ 1 must enter Γ 0 without meeting any point in G 2,+ , since along the backward flow, η is decreasing. Consequently, away from the boundary,

u = o L 2 (Y + ) (1) and hence (hD y -Q + )u = o L 2 (Y + ) (1), micro-localized near η = - √ r. From the fact that hD y -Q -is micro-locally elliptic near η = √ r.
Near the boundary and some point ρ 1 ∈ H ∩ supp(φ), any point can be connected backwardly to Γ 0 by at most transversal reflection. Thus (1.7.7) holds true.

It remains to control the last two terms involving pressure. We just treat one of them, and the other can be treated in the same way.

Pick ϕ 0 ∈ C ∞ c ((-2, 2)) which equals 1 on (-1, 1). Define χ (y, x, ξ) = χ(y, x, ξ)ϕ 0 r(y, x, ξ) -1 .
We fix any > 0, small enough, and write

(Υ 2 q|Av) Y + =(Υ 2 q|AϕOp h (χ )ϕ 1 u) Y + +(Υ 2 q|AϕOp h (χ -χ )ϕ 1 u) Y + =:I h, + II h, . (1.7.8)
We first deal with I h, . Notice that from Proposition 1.13, we have

lim sup h→0 hD y ϕOp h (χ )ϕ 1 u L 2 (Y + ) ≤ C 1/2 .
We then apply Cauchy Schwartz to estimate

y 0 0 ϕOp h (χ )ϕ 1 u 2 L 2 (∂Y + , √ Gdx) dy ≤ Ch -2 y 0 0 y 0 hD y ϕOp h (χ )ϕ 1 u(s, x)ds 2 dxdy ≤ Cy 2 0 h 2 hD y ϕOp h (χ ϕ 1 u) 2 L 2
x,y .

Choose θ ∈ (0, 1/2), and y 0 = h -θ , we estimate

|I h, | ≤ y 0 0 + 0 y 0 |(Υ 2 q|AϕOp h (χ )ϕ 1 u) L 2 (∂Y + , √ Gdx) |dy ≤ C 1 2θ ( hD y Op h (χ )u 2 L 2 x,y + O(h)) + Ce -c θ ,
where we have used Lemma 1.7. Note that that lemma is applicable even when the microlocal cut-off χ δ 0 there is matrix-valued.

In summary we have lim sup

h→0 |I h, | ≤ C( 1-2θ + e -c θ ).
We now turn to the estimates of II h, . This can be done from geometric argument.

Let S := {(y, x, ξ) : r(y, x, ξ) ≥ , y ≤ 4 /c 0 } ∩ W 0 .
We claim that for any ray γ with γ(0) ∈ Γ 0 and Γ(s

0 ) ∈ S , γ| [0,s 0 ] ∩ G 2,+ = ∅.
Indeed, by contradiction, assume that for some γ and

s 1 ∈ [0, s 0 ], we have ρ 1 γ(s 1 ) ∈ G 2,+ . After time s 1 , along γ we have ẏ = 2η, η = ∂ y r ≥ c 0 /4, with y(s 1 ) = η(s 1 ) = 0, η(s 0 ) ≥ √ . This implies that s 0 -s 1 ≥ 4 √ /c 0 and y(s 0 ) ≥ c 0 T 2 /4 ≥ 4 δ 0 c 0 . The claim follows. Now we write II = (ϕ 0 (c 0 y/ ) Υ 2 q|AϕOp h (χ -χ )ϕ 1 u) Y + + ((1 -ϕ 0 (c 0 y/ )) Υ 2 q|AϕOp h (χ -χ )ϕ 1 u) Y + .
From the discussion above, the first term on the right hand side above tends to 0 as h → 0 for any fixed > 0. while the second term is controlled from above by 0 δ 0 4C |C(y, x, hD x )q| 2 dxdy for some zero order pseudo-differential operator with principal symbol c(y, x, ξ), and supp c ∩ {ξ = 0} = ∅. Using Lemma 1.7, we have lim sup h→0 |II | = 0 holds for any > 0. Notice that the left hand side of (1.7.5) is independent of , we have

lim sup h→0 ((-A 1 hD y v|hD y v) ∂Y + + Ψ 0 v + Ψ 1 hD y v 2 L 2 (Y + ) ) = 0.
From the construction of a 0 , a 1 and the corresponding expression of ψ 0 , ψ 1 , we can choose another different a 0 , a 1 , such that the function

ψ 0 + ψ 1 η is independent of ψ 0 + ψ 1 η on supp χ(see appendix D). It follows then v L 2 (Y + ) + hD y v L 2 (Y + ) = o(1), h → 0,
and this completes the proof.

1.8 Near G 2,-and G k for k ≥ 3

This section is devoted to the proof of Proposition 1.9. Before proving it, we need some preparation. In what follows, we take tangential operator

A = ϕOp h (a)ϕ 1 , A * = A + O L 2 (∂Y + ) (h 2 ).
Proposition 1.16

1 h (([P, A]u|u) Y + = 1 h (Au|P u) Y + - 1 h (AP u|u) Y + + O(h).
Proof. The proof goes in exactly the same way and much simpler than the diffractive case, and we omit it here.

Let r 0 = r| y=0 and r 1 = ∂ y r| y=0 . Direct calculation gives

H p a = 2η ∂a ∂y + ∂r ∂y ∂a ∂η + H -r a.
Pick ρ 0 ∈ G 2,-⊂ T * ∂Ω {0} and a small neighborhood U ⊂ T * ∂Ω {0} of ρ 0 . Let L ⊂ U be a co-dimension 1 hypersurface containing ρ 0 in T * ∂Ω and transversal to the vector field H -r 0 . For small positive numbers δ, τ > 0, define

L ± (δ, τ ; ρ 0 ) := {exp(tH -r 0 )(ρ) ∈ U : ρ ∈ L, dist (ρ, ρ 0 ) ≤ δ 2 , 0 ≤ ±t ≤ τ }.
When there is no risk of confusion, we write it simply as L ± (δ, τ ). Define also

F ± (δ, τ ) := {(y, x, ξ) : 0 ≤ y ≤ δ 2 , (x, ξ) ∈ L ± (δ, τ )}, F (δ, τ ) = F + (δ, τ ) ∪ F -(δ, τ ).
Let C 1 > 0 sufficiently large and δ 0 > 0, τ 0 > 0 sufficiently small so that δ < δ 0 , τ < τ 0

|r(y, x, ξ)| ≤ 1 2 C 2 1 δ 2 (1.8.1) in F (δ, τ ) for all 0 < δ ≤ δ 0 , 0 < τ ≤ τ 0 .
With the same constant C 1 , we further define the sets

V ± (δ, τ ) :={(y, x, η, ξ) : 0 ≤ y ≤ δ 2 /2, (x, ξ) ∈ L ± (δ, τ )} ∪{(y, x, η, ξ) : δ 2 /2 ≤ y ≤ δ 2 , (x, ξ) ∈ L ± (δ, τ ), |η| ≤ C 1 δ}, W ± (δ, τ ) :={(y, x, η, ξ) : 0 ≤ y ≤ δ 2 /2, (x, ξ) ∈ L ± (δ, τ )} ∪{(y, x, η, ξ) : δ 2 /2 ≤ y ≤ δ 2 , (x, ξ) ∈ L ± (δ, τ ), |η| ≤ 2C 1 δ}, V (δ, τ ) := V + (δ, τ ) ∪ V -(δ, τ ), W (δ, τ ) = W + (δ, τ ) ∪ W -(δ, τ ).
We need test functions constructed in [START_REF] Richard | Singularities of boundary value problems[END_REF]:

Lemma 1.19 ([48]) Let I = [0, 0 ) y . There exist σ > 0, δ 0 > 0, τ 0 > 0 small enough with δ σ 1 and families of smooth functions a δ ∈ C ∞ c (I × U ), g δ , h δ ∈ C ∞ (Y + × R η × R d-1 ξ \ {0}
) for any 0 < δ ≤ δ 0 , with the following properties:

1. a δ ≥ 0, suppa δ ⊂ F + (δ, σδ) ∪ F -(δ, δ 2 ) . 2. a δ (0, exp(tH -r 0 (ρ 0 ))) = 0, ∀0 ≤ t < δσ. 3. a δ > 0 on suppa δ if 0 < δ < δ ≤ δ 0 and a δ is independent of y for 0 ≤ y < δ 2 /2. 4. g δ + h δ = -H p a δ .
5. in W (δ, τ ), g δ ≥ 0 and g δ > 0 when a δ = 0.

For any m > 1 and any multiple index

α ∈ N d , |g -1 m δ ∂ α g δ | = O δ (1)
, locally uniformly on W (δ, τ 0 ), where the implicit constant inside O δ (1) depends on α, m and δ.

7. supph δ ⊂ I × L -(δ, δ 2 ) × R η , and suppg δ ∪ supph δ ⊂ suppa δ , g δ , h δ are independent of η for 0 ≤ y ≤ δ 2 /2.
For the convenience of the reader, we will recall the proof in the appendix D.

According to the lemma, we have ∂(g

1/2 δ ) = 2g -1/2 δ ∂g δ = O(1), this implies that g 1/2 δ ∈ C ∞ (W (δ, τ )). Set b δ := g 1/2 δ ∈ C ∞ (W (δ, τ ))
. Note b δ may not be smooth and with compact support. We need split it into two parts as follows: Let

φ 1 ∈ C ∞ (R) such that φ 1 ≡ 1 if 0 ≤ y ≤ δ 2 4 and φ 1 ≡ 0 if y > 3δ 2 8 . Let φ 2 ∈ C ∞ (Ω × R d \ {0}
) with compact support in x, ξ, η variables, such that φ 2 ≥ 0 and φ 2 ≡ 0 whenever y ≤ δ 2 4 or |η| > 2C 1 δ. Indeed, we can choose κ ∈ C ∞ c (R), non-negative, smooth and with compact support, such that κ(z) ≡ 0 when |z| > 2C 1 δ and κ(z) ≡ 1 when |z| ≤ 3 2

C 1 δ. Now let φ 2 (y, x, η, ξ) 2 = (1 -φ 1 (y) 2 )κ(δ -1 η)χ δ (y, x, ξ) with χ δ | supp(a δ ) ≡ 1, supp(χ δ ) ⊂ F + (δ, σδ) ∪ F -(δ, σδ). We observe that W ± (δ, τ ) ∩ supp (1 -φ 2 1 -φ 2 2 ) ⊂ (y, x, η, ξ) : δ 2 4 ≤ y ≤ δ 2 , |η| > 3 2 C 1 δ, .
We finally put b δ,j :=

φ j b δ , j = 1, 2. Note that b δ,1 ∈ C ∞ c (F (δ, τ )) is a tangential symbol ( since for y ≥ δ 2 2 , supp(g δ ) ⊂ supp(a δ ) is compact ) while b δ,2 ∈ C ∞ c (W (δ, τ )
) is a usual interior symbol with compact support in T * Ω.

Gliding case

The propagation of support of µ near a gliding point in G 2,-can be stated as follows:

Proposition 1.17 Suppose ρ 0 ∈ G 2,-and L + (δ 0 , τ 0 ) ∪ L -(δ 0 , τ 0 ) ⊂ G 2,-for some δ 0 , τ 0 > 0. Then for any σ > 0 with σδ 0 < τ 0 , such that if {(y, x, η, ξ) : 0 ≤ y ≤ δ 2 , (x, ξ) ∈ L -(δ, δ 2 ; ρ 0 )} ∩ supp(µ) = ∅ for some 0 < δ ≤ δ 0 , then exp(tH -r 0 )(ρ 0 ) / ∈ supp(µ) for any t ∈ [0, σδ).
We need several lemmas.

Lemma 1.20 Suppose a ∈ C ∞ c (R 2d ), b ∈ C ∞ ([0, 1], C ∞ c (R 2(d-1)
)) with the following support property:

a(y, x, η, ξ) ≡ 0 if y ≤ c 0 < 1 or |η| > C 0 |ξ|.
Then the usual symbolic calculus for a(y, x, hD y , hD x )b(y, x, hD x ) still valid. In particular,

a(y, x, hD y , hD x )b(y, x, hD x ) = c(y, x, hD y , hD x ) + O L 2 →L 2 (h), with c(y, x, η, ξ) = a(y, x, η, ξ)b(y, x, ξ)
We postpone the proof in the appendix F.

Lemma 1.21

Given any ρ 1 ∈ G, there exists

δ 1 > 0, τ 1 > 0, σ 1 > 0 with δ 1 σ 1 and σ 1 δ 1 < τ 1 such that if ρ ∈ T * ∂Ω and dist(ρ, ρ 1 ) ≤ δ 2 for some 0 < δ ≤ δ 1 , then dist(γ(s, ρ), γ(s, ρ 1 )) ≤ Cδ 2 for |s| ≤ σ 1 δ. In particular, γ(s, ρ) ∈ W (δ, τ 1 ) for all |s| ≤ σ 1 δ.
Proof. Write γ(s, ρ) and exp(sH -r 0 )(ρ) in coordinate as γ 1 (s) = (y(s), η(s), x(s), ξ(s)) and γ 2 (s) = (ỹ(s), η(s), x(s), ξ(s)).

From ẏ = 2η, η = O(1), we have y(s) ≤ Cs 2 and the same estimate holds for ỹ(s). Let

d(s) = |x(s) -x(s)| 2 + |ξ(s) -ξ(s)| 2 ,
and then ḋ(s) ≤ C d(s). This implies d(s) ≤ C 1 δ 2 for all |s| ≤ σ 1 δ. By the same argument, we have dist(exp(sH -r 0 )(ρ), exp(sH -r 0 )(ρ 1 )) ≤ Cδ 2 . The conclusion then follows from triangle inequality.

We will see the crucial role of ρ 0 ∈ G 2,-in the following lemma: Lemma 1.22 Assume that δ 1 , τ 1 are parameters given in the previous lemma. Suppose that -C 0 ≤ ∂ y r(ρ) ≤ -c 0 < 0 for all ρ ∈ W (δ 1 , τ 1 ). Define S = W (δ 1 , τ 1 ) ∩ {r ≥ , y ≤ } for sufficiently small > 0. Then along any ray γ(s, ρ 1 ) in W (δ 1 , τ 1 ) with ρ 1 ∈ S , if y(γ(-t, ρ 1 )) = 0 for some 0 ≤ t ≤ τ 1 , we have r(y(γ(t, ρ 1 )) ≥ c 1 , where c 1 depends only on W (δ 1 , τ 1 ).

Proof. Assume ρ 1 = (y 1 , x 1 , η 1 , ξ 1 ) ∈ S and γ(s, ρ 1 ) = (y(s), x(s); η(s), ξ(s)). Let s 3 = inf{0 ≤ s ≤ τ 1 : y(-s) = 0}. For s ∈ [-s 3 , 0], ẏ = 2η, -C 0 ≤ η = ∂ y r ≤ -c 0 .
There are two possibilities. If

η 1 ≥ √ , then η(-s + 3 ) ≥ η 1 > 0 since η < 0. Otherwise, η 1 ≤ - √
, and we denote by

s 2 = inf{s ∈ [0, s 1 ] : η(-s) = 0}. From η 1 = 0 -s 2 ηds ≥ -C 0 s 2 ,
we have

s 2 ≥ |η 1 | C 0 . Moreover, y 1 -y(-s 2 ) = 2η 1 s 2 - 0 -s 2 ds 0 s ÿds ≤ 2η 1 s 2 + C 0 s 2 2 ≤ - |η 1 | 2 C 0 . Now from y(-s 2 ) = y(-s 2 ) -y(-s 3 ) = - -s 2 -s 3 ds -s 2 s ÿds ≤ C 0 |s 3 -s 2 | 2 , we have |s 3 -s 2 | 2 ≥ y(-s 2 ) C 0 ≥ (y(-s 2 ) -y 1 ) C 0 ≥ |η 1 | 2 C 2 0
and finally

η(-s + 3 ) = - -s 2 -s 3 ηds ≥ c 0 |s 3 -s 2 | ≥ c 0 √ C 0 .
Proof of Proposition 1.17. For any δ , we define the operator

N δ = 1 ih [P, A δ ] with principal symbol n δ = -H p a δ = g δ + h δ . Define operators B δ ,j := Op h (b δ ,j ), j = 1, 2, N δ,3 = Op h ((1 -φ 2 1 -φ 2 2 )n δ ). Write h δ ,j = φ 2 j h δ , H δ ,j = Op h (h δ ,j ), j = 1, 2.
The proposition will follow if we can show that for any δ < δ,

lim h→0 2 j=1 B δ ,j u 2 L 2 (Y + ) = 0 (1.8.2)
We remark that h δ ,1 , b δ ,1 are both tangential symbols while h δ ,2 , b δ ,2 are interior symbols vanishing near the boundary. Observe also that N δ ,3 is interior pseudo-differential operator with symbol vanishing near the boundary as well as on p -1 (0), thanks to the fact that in

W (δ , τ ), |r(y, x, ξ)| ≤ 1 2 C 2 1 δ 2 . Thus N δ ,3 u = o L 2 (Y + ) (1)
as h → 0 for δ > 0 small enough. Moreover, from the assumption on the support of µ near the original point ρ 0 we have

H δ ,j u = o L 2 y,x (1)
. Now set

M δ ,j = φ 2 j N δ ,j -B * δ ,j B δ ,j -H δ ,j , j = 1, 2.
From symbolic calculus, we have

M δ ,1 = O L 2 →L 2 (h) is a tangential operator.
Note that definition of M δ ,2 , we will encounter the composition of tangential operator with interior operator Op h (φ 2 2 ). Since φ 2 has support far away form y = 0 and η = 0, the symbolic calculus still valid thanks to Lemma 1.20. Therefore M δ ,2 = O L 2 →L 2 (h) is an interior operator. Finally, we have obtained

N δ = N δ ,3 + 2 j=1 (B * δ ,j B δ ,j + H δ ,j ) + O L 2 (Y + )→L 2 (Y + ) (h).
Combining all the analysis above and the Proposition 1.16, we have

2 j=1 B δ ,j u 2 L 2 (Y + ) ≤ o(1) + 2 h Im([A δ , hd]q|u) Y + + 1 h Im(q|hd * (A δ u)) Y + =o(1) + |(q|Υ 1 u) Y + | + |(Υ 2 q|u) Y + | (1.8.3)
where Υ 1 , Υ 2 are compactly supported matrix-valued tangential operators with principal symbols vanishing outside supp(a δ ).

To finish the proof, we need show that the right hand side of (

1.8.3) is o(1) as h → 0. Pick χ ∈ C ∞ c (R) such that χ(s) ≡ 1 if 0 ≤ s ≤ 1 2 and χ(s) ≡ 0 is s ≥ 1. Let χ (y, x, ξ) = χ( -1 r(y, x, ξ)). Denote by I h, = (Υ 1 ϕOp h (χ )ϕ 1 u|q) Y + , II h, = (Υ 1 (1 -ϕOp h (χ )ϕ 1 )u|q) Y + .
The treatment of I h, is exactly the same as in the diffractive case, so we have

lim →0 lim sup h→0 I h, = 0.
For II h, , we may assume that the the integral of y variable is only from [0, ], since for y ≥ we can use the rapid decreasing of q as in the treatment of I h, . According to Lemma 1.21 and Lemma 1.22, the measure of Υ 1 (1 -ϕOp h (χ )ϕ 1 )u vanishes, since all the backward generalized rays starting from each point in S will enter the small neighborhood of ρ 0 ∈ G 2,-by at most reflection at boundary. From the propagation theorem in the hyperbolic case(Proposition 1.7), the proof of Proposition 1.17 is complete.

Remark 1.4

We remark that as a consequence of Proposition 1.17, the measure of q(or h∇q) also vanishes along exp(tH -r 0 ) for t ∈ [0, σδ).

high order contact

In this subsection we will use a new coordinate system in a neighborhood

W k of ρ k ∈ G k in [0, 0 ] × T * ∂Ω: (y, η, x, ξ) → (y, η, z, ζ), z = (z 1 , z ), ζ = (ζ 1 , ζ ) with p = η 2 -r, r = ζ 1 + yr 1 (z, ζ) + O(y 2 ), ζ 1 = r 0 , where r 0 = r| y=0 , r 1 = ∂ y r| y=0 . This is possible since d x,ξ r 0 = 0, if ξ = 0. Along the generalized bicharacteristic flow γ(s), (z, ζ) satisfies ż = -∂ ζ r(y(s), z(s), ζ(s)), ζ = ∂ z r(y(s), z(s), ζ(s)).
This implies that in W k , -ż1 ∼ 1 > 0, as y → 0, and thus s → z 1 (s) is strictly decreasing. Moreover, ζ1 ∼ y∂ z 1 r 1 , as y → 0.

Suppose now k ≥ 3, we have locally that

G k := {(z, ζ) : ζ 1 = 0, ∂ l z 1 r 1 (z, ζ) = 0, ∀l ≤ k -3, ∂ k-2 z 1 r 1 (z, ζ) = 0}. Define Σ k := {(z, ζ) : ∂ k-3 z 1 r 1 (z, ζ) = 0, ∂ k-2 z 1 r 1 (z, ζ) = 0}.
From implicit function theorem, Σ k is locally a hypersurface and we can write it as

Σ k = {(z, ζ) : z 1 = Θ k (z , ζ)}.
G k can be viewed locally as a closed subset of Σ k . Since the map s → z 1 (s) is bijective, we may assume that along each ray,

z 1 (0) = Θ k (z (0), ζ(0)), and z 1 (s) < Θ k (z (s), ζ(s)), s > 0, z 1 (s) > Θ k (z (s), ζ(s)), s < 0.
We see that all the generalized rays are transversal to the codimension 2 manifold(locally) Σ k . Moreover, a ray passes Σ k if and only if y(0) = 0 and ζ 1 (0) = 0. Now we define the set near ρ k ,

Σ ± k := {(y, η, z, ζ) ∈ Car(P ) ∩ W k : z 1 ∓ Θ k (z , ζ) > 0}.
Note that the gliding rays exp(sH -r 0 ) intersect transversally to Σ k and H -r 0 = -∂ z 1 inside T * ∂Ω. Thus we can re-parametrize the gliding flows by z 1 . Moreover, Σ ± k ∩ G j = ∅, ∀j ≥ k, provided that we choose W k small enough. In other word, z 1 gives a foliation of T * ∂Ω near

Σ k for small |z 1 -Θ k (z , ζ)|.
The following proposition is a long time extension of Proposition 1.17, adapted to the notations introduced above.

Proposition 1.18 Suppose ρ 0 ∈ G 2,-near ρ k ∈ Σ k with coordinate (z, ζ), z 1 > Θ k (z , ζ).
Then there exists δ 0 > 0, sufficiently small such that if

{(y, x, η, ξ) : 0 ≤ y ≤ δ 2 , (x, ξ) ∈ L -(δ, δ 2 ; ρ 0 )} ∩ supp(µ) = ∅ for 0 < δ ≤ δ 0 , then exp(sH -r 0 )(ρ 0 ) / ∈ supp(µ) for any s < z 1 -Θ k (z , ζ).
In other words, each generalized ray, issued from gliding set outside supp(µ) does not carry any singularity until it touches some point in G k for k ≥ 3.

Proof. The proof is purely topological. For each

ρ 0 = (z, ζ) / ∈ supp(µ) and z 1 > 0, let s 1 := sup{s : s ≤ z 1 -Θ k (z , ζ), exp(s H -r 0 ) / ∈ supp (µ), ∀s ∈ [0, s)}. The existence of s 1 is guaranteed by Proposition 1.17. It remains to show that s 1 = z 1 -Θ k (z , ζ). By contra- diction, suppose s 1 < z 1 -Θ(z , ζ), then the point ρ 1 = (z 1 -s 1 , z , ζ) is in G 2,
-. We can apply Proposition 1.17 again to obtain that for some small σ 1 > 0, exp(θσH -r 0 )(ρ 1 ) / ∈supp (µ) for any θ ∈ [0, 1]. This is a contradiction of the choice of s 1 .

As a consequence, we have Corollary 1.2 Suppose ρ 0 ∈ G 2,-and ρ 0 / ∈ supp(µ). Let γ(s) be the generalized ray passing ρ 0 with γ(0) = ρ 0 . Then γ(s) / ∈ supp(µ) for any s ∈ [-s 0 , s 0 ], provided that γ| [-s 0 ,s 0 ] ⊂ G 2,-.

Combining the analysis near a diffractive point and a gliding point, we have already established the k-propagation property for k = 2. We will argue by induction to prove k-propagation property for all k ≥ 3. To this end, we need an intermediate step. Let us first introduce a notation

Γ(ρ 0 ; δ) := {(y, x; z, ζ) : 0 ≤ y ≤ δ 2 , (z, ζ) ∈ L -(δ, δ 2 ; ρ 0 )}
and a definition Definition 1.5 (k-pre-propagation property) For k ≥ 2, we say that k-pre-propagation property holds, if the following statement is true:

For any ρ k ∈ G k , there exists a neighborhood V k of ρ k in T * ∂Ω, and δ k > 0, σ k > 0, δ k σ k , depending on V k , such that for any ρ 0 ∈ G 2,-∪ 3≤j≤k G j ∩ V k , if Γ(ρ 0 ; δ) ∩ supp(µ) = ∅ for some 0 < δ < δ k , then exp(sH -r 0 )(ρ 0 ) / ∈ supp(µ) for all s ∈ [0, σ k δ).
The key step is the following inductive proposition.

Proposition 1.19

Suppose k ≥ 3 and (k -1)propagation property holds true, then k-pre-propagation property also holds true.

We do some preparation before proving this proposition. Select a neighborhood

W k of ρ k ∈ G k in T * ∂Ω (and contained in W k ) with compact closure such that in W k such that |∂ k-2 z 1 r 1 (ρ)| ≥ c 0 > 0 for all ρ ∈ W k .
By abusing the notation, we will refer G k to be G k ∩ W k in the sequel. According to the asymptotic behaviour of the flow exp(sH -r 0 ) as s → 0, we have for any given

(z 1 = Θ k (z 0 , ζ 0 ), z 0 , ζ 0 ) ∈ G k , r 1 • exp(sH -r 0 )(z 0 , ζ 0 ) = b k (z 0 , ζ 0 )s k-2 + O(s k-1 ),
where b k = 0 can be viewed as a function of points in G k . From compactness, we can choose σ > 0, θ > 0 depending only on W k such that for all ρ ∈ G k ,

|b k (ρ)| ≥ θ > 0, |r 1 • exp(s H -r 0 )(ρ)| ≥ 1 2 |b k s k-2 |, ∀s ∈ [-σ, 0) ∪ (0, σ].
Now we define a smaller neighborhood V k of ρ k such that for any ρ 0 ∈ V k , and , near S δ, .

δ k > 0, σ k > 0, exp(sH -r 0 )(L ± (δ k , δ 2 k ; ρ 0 )) ⊂ W k for all |s| ≤ σ k δ k . Moreover, |r 1 • exp(sH -r 0 )(ρ 0 )| ≤ δ k . We also put W k = [0, δ 2 k ] × W k , V k = [0, δ 2 k ] × V k . Choosing a cut-off ãδ ∈ C ∞ c with ãδ ≡ 1 near ρ k ,
We divide the proof of Proposition 1.19 into several lemmas.

Lemma 1.23

Given any generalized ray γ(s)

= (y(s), η(s), z(s), ζ(s)) with γ(s 0 ) ∈ Γ(ρ 0 ; δ) ∩ G 2,-and γ(s 1 ) ∈ S δ, . Assume that γ| [s 0 ,s 1 ] ⊂ Car(P ) ∩ W k , then γ(s) / ∈ G k for all s ∈ [s 0 , s 1 ].
Proof. Take Γ + (ρ 0 ; δ) := Γ(ρ 0 ; δ) ∩ Σ + k and identify points in Σ ± k as their projection to (y, x, ξ). Let F k (may be empty) be the union of generalized rays issued from Γ + (ρ 0 ; δ) which meet G k . Note that along both real trajectories γ(s) and exp(sH -r 0 ), s → z 1 is strictly decreasing, it suffices to show that F k ∩ S δ, ⊂ Σ + k since generalized rays intersect with Σ k transversally,.

We argue by contradiction. Assume that some ray in F k satisfies γ(s 0 ) ∈ Γ + (ρ 0 ; δ), γ(0) ∈ G k , and γ(s 1 ) ∈ S δ, for s 0 < 0 < s 1 . Write exp(sH -r 0 )(γ(0)) = (z(s), ζ(s)), and

r 1 • exp(sH -r 0 )(z (0), ζ(0)) = r 1 (z(s), ζ(s)) = b k s k-2 + O(s k-1 ), s → 0,
More precisely, we have

|b k (z (0), ζ(0))| ≥ θ > 0, |r 1 (z(s), ζ(s))| ≥ 1 2 |b k s k-2 |, ∀s ∈ [-σ, 0) ∪ (0, σ].
After shrinking support of a δ if necessary, we may assume that s 1 < σ. According to the parity of k and the sign of b k , there are several situations.

If b k < 0, then no longer k is γ(s) ∈ G 2,-for all s ∈ (0, σ). This is impossible since r • γ(s 1 ) ≥ . Otherwise b k > 0, in this case we have r 1 (z(s), ζ(s)) ≥ b k s k-2 /2,
for all s ∈ (0, σ), and

(∂ z 1 r 1 )(z(s), ζ(s)) =(∂ z 1 r 1 ) • exp(sH -r 0 (z (0), ζ(0)) = -∂ s (r 1 • exp(sH -r 0 (z (0), ζ(0))) = -(k -2)b k s k-3 + O(s k-2 ) ≤ 0, ∀s ∈ [0, σ), (1.8.4) thanks to ∂ ζ r| y=0 = ∂ z r| y=0 = 0.
Taking the difference with real trajectory γ(s) = (y(s), η(s); z(s), ζ(s)), we have

∂ z 1 r 1 (y(s), z(s), ζ(s)) -∂ z 1 r 1 (z(s), ζ(s)) = (∂ z 1 r 1 (0, z(s), ζ(s)) -∂ z 1 r 1 (z(s), ζ(s))) + (∂ z 1 r 1 (z(s), ζ(s)) -∂ z 1 r 1 (z(s), ζ(s))) +(∂ z 1 r 1 (y(s), z(s), ζ(s)) -∂ z 1 r 1 (0, z(s), ζ(s)))
Using the fact that (z(0), ζ(0)) = (z(0), ζ(0)) and y(s) = O(s 2 ), we have

∂ z 1 r 1 (y(s), z(s), ζ(s)) -∂ z 1 r 1 (z(s), ζ(s)) = O(s).
This together with (1.8.4) imply that

ζ1 ≤ y∂ z 1 r 1 (y(s), z(s), ζ(s)) + C 0 y 2 ≤ C 0 (y 2 + ys), ẏ = 2η, η 2 = ζ 1 + yr 1 (z, ζ) + O(y 2 ), (ζ 1 (0), y(0)) = (0, 0), (1.8.5)
where the constant C 0 and the implicit constant inside the big O only depend on supp a δ .

Applying the formula

H k p y(0) = 2(H -r 0 ) k-2 r 1 = 2b k (k -2)! > 0 and
Taylor expansion, we have

y(s) = 2b k k(k -1) s k + O(s k+1 ) ≥ b k k(k -1) s k , s ∈ (0, σ), ẏ(s) = 2b k k -1 s k-1 + O(s k ) > b k k -1 s k-1 > 0, s ∈ (0, σ).
(1.8.6)

Injecting in (1.8.5), we have ζ1 (s) ≤ C 0 ( 2 + s) for all s > 0 small as long as y(s) ≤ and γ(s) / ∈ S δ, . For these s,

ζ 1 (s) ≤ C 0 ( 2 s + s 2 /2).
Setting s 2 = inf{0 ≤ s ≤ s 1 : γ(s) ∈ S δ, }, we know that along the flow, 2 √ = 2η(s 2 ) = ẏ(s 2 ), and this implies that 1) . In summary, we have

s 2 ∼ 1 2(k-1) since y(s) > if s > 1 2(k-
≤ r • φ s 2 ≤ 2C 0 1+ 1 2(k-1) + δ k + C 1 2 .
However, this contradicts to

r = ζ 1 + yr 1 + O(y 2 ), provided that δ k < 1, δ k < 1.

Lemma 1.24

The conclusion of Proposition 1.

19 holds if ρ 0 ∈ G 2,-
Proof. Adapting the notations and argument in the proof of Proposition 1.17, we have

2 j=1 B δ,j u 2 L 2 (Y + ) ≤ o(1) + 2 h Im([A δ , hd]q|u) Y + + 1 h Im(q|hd * (A δ u)) Y + . (1.8.7)
The goal is to show that the last two terms on the right hand side tend to 0 as h → 0.

We denote by γ(s) the gliding ray exp(sH -r 0 ) such that γ(s 0 ) = ρ 0 for some s 0 < 0. Suppose γ(0) = ρ ∈ G k for some k ≥ 3 and γ(s) ∈ G 2,-for s ∈ (s 0 , 0). In view of Corollary ??, we may assume that ρ 0 is close enough to ρ, and

|s 0 | is small. Pick χ ∈ C ∞ c (R) such that χ(s) ≡ 1 if 0 ≤ s ≤ 1
2 and χ(s) ≡ 0 if s ≥ 1. For any > 0, let χ (y, x, ξ) = χ( -1 r(y, x, ξ)). Let

I h, = 2 h (hd * (ϕOp h (χ )ϕ 1 u)|q) Y + , II h, = 2 h (hd * (1 -ϕOp h (χ )ϕ 1 )u|q) Y + .
The treatment of I h, is exactly the same as in the diffractive case, we have

lim →0 lim sup h→0 I h, = 0.
For II h, , we only concern about the integral from [0, ] in y variable.

From Lemma 1.23, any ray entering S δ, can at most pass G j for j < k. Applying (k -1) propagation property, we deduce that for any cut-off ϕ with supp(ϕ

) ⊂ S δ, , supp (ϕ )∩ supp(µ) = ∅. Therefore lim h→0 II h, = 0
for any > 0. This completes the proof.

Lemma 1.25

The conclusion of Proposition 1.19 holds if ρ 0 ∈ G j for some 3 ≤ j ≤ k.

Proof. Taking a micro-local cut-off ψ(y, x, ξ) with support near ρ 0 , we have

ϕOp h (ψ)ϕ 1 u L 2 (Y + ) = o(1)
from the assumption that ρ 0 / ∈ supp (µ). Note that along the flow of H -r 0 and on supp (1-ψ)∩V k we have |r 1 (0, x, ξ)| ≥ c(ψ, δ) > 0. Hence from Corollary 1.2, if exp(tH -r 0 )(ρ 0 ) ∈ G 2,-for all t ∈ (0, σδ), and then exp(tH -r 0 )(ρ 0 ) / ∈ supp(µ). Otherwise exp(tH -r 0 )(ρ 0 ) ∈ G 2,+ for all t ∈ (0, σδ), we claim that we still have exp(tH -r 0 )(ρ 0 ) / ∈supp(µ) from geometric consideration.

Indeed, by considering the backward generalized ray, we conclude that for any s 0 ∈ (0, σ k δ), there exists ρ ∈ W k , so that γ(s 0 , ρ) = exp(s 0 H -r 0 )(ρ 0 ) where γ(s, ρ) is the generalized ray issued from ρ. From this fact we must have γ(s, ρ) / ∈ G k for s ∈ (0, s 0 ), since any ray intersecting with G k will enter T * Ω or G 2,-immediately, provided that the neighborhood W k is chosen to be small enough.

Therefore, to conclude, we only need to show that

ρ ∈ {(y, z, ζ) : 0 ≤ y ≤ δ 2 , |(z, ζ) -ρ 0 | ≤ δ 2 }.
We will prove this by comparing two flows exp(sH -r 0 )(ρ 0 ) = (z(s), ζ(s)) and γ(s, ρ) = (y(s), η(s), z(s), ζ(s)). Taking the difference the two dynamics, we have

d ds (z 1 (s) -z1 (s)) = -∂ ζ 1 r(y(s), z(s), ζ(s)) + ∂ ζ 1 r(0, z(s), ζ(s)) = O(y(s)),
y(s) ≤ C(s -s 0 ) 2 for all s ∈ [0, s 0 ]. Hence y(0) ≤ Cσ 2 k δ 2 < δ 2 , provided that σ 2 k < 1/C. Moreover, |(z(0), ζ(0)) -ρ 0 | ≤ Cs 3 0 ≤ Cσ 3 k δ 3 ≤ δ 2 .
Proposition 1.20 Suppose that (k-1)-propagation property holds. Then k-pre-propagation property implies k-propagation property.

Proof. Up to re-parameter the flow, we may assume that ρ 0 ∈ G k and γ(s) is the generalized ray such that γ(0) = ρ 0 . We also denote γ(s) by γ(s, ρ 0 ) in view of flow map. Suppose γ(s 0 ) / ∈supp µ for some s 0 < 0 and φ| [s 0 ,0) ∩ supp (µ) = ∅. Our goal is to show that ρ 0 / ∈supp µ. Let σ k-1 > 0 be the required length in the definition of (k -1)-propagation property.

Let δ k > 0, σ k > 0 and V k , neighborhood of ρ 0 ∈ G k in T * ∂Ω and V k , neighborhood of ρ 0 in [0, 0 ] × T * ∂Ω, as in the definition of k-pre-propagation property which satisfy the conditions in the paragraph in front of Lemma 1.23. Note in particular that we have V k ∩ G j = ∅ for all j > k. Without loss of generality, we may assume that |s 0 | < min{σ k-1 , σ k } and γ(s 0 ) ∈ V k , since otherwise we can choose s 0 < 0, |s 0 | small enough and replace γ(s 0 ) by γ(s 0 ).

Let Γ 0 ⊂ V k be a neighborhood of γ(s 0 ) so that Γ 0 ∩ supp(µ) = ∅. For δ 1 > 0 small with δ 1 σ k , we set ρ 1 = exp -σ k δ 1 2 H -r 0 (ρ 0 ) and define U δ 1 := {ρ = (y, η, z, ζ) ∈ Car(P ) : 0 ≤ y ≤ δ 2 1 , |(z, ζ) -ρ 1 | ≤ δ 2 1 }.
From continuous dependence of the generalized bi-characteristic flow, we have

U δ 1 ⊂ γ(s 0 , Γ 0 ), provided that δ 1 small enough .
Now we claim that for possibly smaller δ 1 > 0, we have

γ(s 1 , U δ 1 ) ∩ j≥k G j = ∅, ∀s 1 ∈ (s 0 , 0).
Indeed, it suffices to prove that γ(s 1 , U δ 1 ) ∩ G k = ∅ since there are no point of G j in V k for j > k. First from transversal intersection between the flow exp(sH -r 0 ) and Σ k , we deduce that at ρ 1 , z 1 > Θ k (z , ζ). By choosing δ 1 smaller, there is some constant 1 > 0, such that for all ρ ∈ U δ 1 , z 1 > Θ k (z , ζ) + 1 holds. In particular, U δ 1 ⊂ Σ + k . We calculate

d ds Θ k (z (s), ζ(s)) = ∂Θ k ∂z dz ds + ∂Θ k ∂ζ dζ ds = - ∂Θ k ∂z ∂r ∂ζ + ∂Θ k ∂ζ ∂r ∂z . Note that in V k , we can write r = ζ 1 + yr 1 (z, ζ) + O(y 2 ), hence d ds Θ k (z (s), ζ(s)) = O(y(s)).
Next we argue by contradiction, assume that for some s 1 ∈ (s 0 , 0) and ρ ∈ U Combining with ż1 ∼ -1, we have

Θ k (z (s 1 ), ζ(s 1 )) ≤Θ k (z (0), ζ(0)) + C 0 s 1 |s -s 1 |ds <z 1 (0) + Cs 2 1 =z 1 (s 1 ) + 0 s 1 dz 1 ds ds + Cs 2 1 ≤z 1 (s 1 ) -C 1 |s 1 | + Cs 2 1 ≤z 1 (s 1 ),
provided that |s 0 | is chosen to be small enough. This implies that γ(s 1 , ρ) ∈ Σ + k , which is a contradiction.

From (k -1)-propagation property, we know that U δ 1 ∩ supp(µ) = ∅. Therefore, applying k-pre-propagation property with respect to ρ 1 and U δ 1 , we deduce that ρ 0 / ∈ supp(µ), and this completes the proof.

1.A Proof of Lemma 1.2

Proof of Lemma 1.2. The first assertion follows from hdivu = 0 and Dirichlet boundary condition, while we apply a multiplier method to prove the second. From the geometric assumption on Ω, we can find a vector field L ∈ C 1 (Ω) such that L| ∂Ω = ν(see [START_REF] Manuel Milla | Introduction to exact control theory[END_REF], page 36). In global coordinate system, we write L = L j (x)∂ x j . By using the equation, we have

Ω Lu • f dx = Ω Lu • (-h 2 ∆u -u + h∇q)dx, - Ω Lu • udx = - Ω L j (x)∂ x j u i u i dx = - Ω ∂ x j L j (x)u i u i dx + Ω div L(x)|u| 2 dx = Ω L j (x)u i (x)∂ x j u i dx + Ω div L(x)|u| 2 dx = Ω Lu • udx + Ω div L(x)|u| 2 dx,
and thus

Ω Lu • udx = - 1 2 Ω div L(x)|u| 2 dx = O(1). Next we calculate h Ω Lu • ∇qdx = -h Ω u i ∂ x j (L j ∂ x i q) dx = -h Ω u • L(∇q)dx -h Ω (div L(x))u • ∇qdx = -h Ω u • [L, ∇]qdx -h Ω div L(x)u • ∇qdx = O(1), -h 2 Ω Lu i ∆u i dx = -h 2 ∂Ω ∂ ν u i 2 dσ + h 2 Ω ∇L(∇u i , ∇u i )dx + h 2 Ω L j (x)∂ 2 x j x k u i ∂ x k u i = -h 2 ∂Ω ∂ ν u i 2 dσ + h 2 Ω ∇L(x)(∇u i , ∇u i )dx + h 2 Ω ∂ x j L j ∂ x k u i ∂ x k u i dx -h 2 Ω div L(x)∇u i • ∇u i (x)dx, h 2 Ω ∂ x j L j ∂ x k u i ∂ x k u i dx = h 2 ∂Ω L • ν ∂ ν u i 2 dσ -h 2 Ω L j (x)∂ x k u i ∂ 2 x j x k u i dx, -h 2 Ω Lu i ∆u i dx = - h 2 2 ∂Ω ∂ ν u i 2 dσ+ Ω ∇L(x)(h∇u i , h∇u i )dx- h 2 2 Ω div L(x)|∇u i | 2 dx.
Observing that Ω Lu • f dx = o(1), we have

∂Ω |h∂ ν u| 2 dσ = O(1).

1.B Standard elliptic theory

The differential operator is given by

P h = Op h (η 2 + λ(y, x , ξ ) 2 -1 + hm(y, x , η, ξ )),
where the principaly symbol p = η 2 + λ 2 -1 is scalar while m is matrix valued. When micro-locally near the region p > 0, we want to construct the parametrix of the inverse of P . Denote by U the turbulent neighborhood (two sided ) of ∂Ω.

Take ϕ ∈ C ∞ c (U ), χ 0 ∈ C ∞ (R d-1
) and the support of ϕ is contained in a coordinate patch near the boundary. We put

E 0 := Op h χ 0 (ξ )ϕ(y, x ) η 2 + λ(y, x , ξ ) 2 -1 ,
and we define matrix valued pdo E l , l ≥ 1 inductively via

E 1 × p = - |α|=1 1 i ∂ α ξ ,η E 0 × ∂ α x ,y p -E 0 × m, E n × p = - |α|+k=n,k =n 1 i |α| ∂ α ξ ,η E k × ∂ α x ,y p - |α|+k=n-1 1 i |α| ∂ α ξ ,η E k × ∂ α x ,y m. (1.B.1)
For any N ∈ N, we set

E N = N j=0 h j E j ,
and then

E N • P = χ 0 (hD x )ϕ(y, x )Id + R N , R N L 2 →L 2 = O(h N +1 ).

Proposition 1.21

The sequence of solution (u k ) is h k -oscillating in the following sense:

lim R→∞ lim sup k→∞ |ξ|≥Rh -1 k | ϕu k (ξ)| 2 dξ = 0, ∀ψ ∈ C ∞ c (Ω), lim R→∞ lim sup k→∞ 0 0 dy |ξ |≥Rh -1 k | ϕu k (y, ξ )| 2 dξ = 0, ∀ψ ∈ C ∞ c (Ω),
where in the second formula, the support of ϕ is contained in some local coordinate patch near the boundary, and the Fourier transform is only taken for the x direction.

Proof. We drop the subindex k in the proof. For the first formula, one can use the equation of u to obtain

(-h 2 ∆ -1)(ϕu) = g = O L 2 (1),
and

|ξ|≥Rh -1 | ϕu(ξ)| 2 dξ ≤ h|ξ|≥R | g(ξ)| 2 |h 2 |ξ| 2 -1| 2 dξ ≤ C (R 2 -1) 2 .
For the second formula, it will be sufficient to show that [START_REF] Alinhac | Opérateurs pseudo-différentiels et théorème de Nash-Moser[END_REF]. We apply the parametric construction above with χ 0 (ξ ) = 1-χ ξ R . Let e N (y, x , η, ξ ) be the symbol of the operator E N , which is meromorphic in η with poles η ± 0 = ±i λ 2 (y, x , ξ ) 2 -1. Moreover,

lim R→∞ lim sup k→∞ 1 -χ hD x R (ϕu) L 2 = 0 for some χ ∈ C ∞ c (-1, 1). We write w = u1 y≥0 , g = g1 y≥0 , v = h∂ y u| y=0 = O L 2 (y=0)
|∂ α e N (y, x , η, ξ )| ≤ C N,α R . (1.B.2)
We take ϕ be a slight enlargement of ϕ such that ϕϕ = ϕ. Then

E N • ϕP w = 1 -χ hD x R (ϕw) + R N w.
From the jump formula, P w = g + 2hv(x ) ⊗ δ y=0 . We have

1 -χ hD x R (ϕw) = E N ( ϕ g + ϕ2hv ⊗ δ y=0 ) + O L 2 y,x (h N +1 ).
From symbolic calculus,

E N ( ϕ g) L 2 y,x ≤ |α|≤Cd sup (y,x ,η,ξ )∈R 2d
|∂ α e N (y, x , η, ξ )| + Ch, and it vanishes after the limit procedure, thanks to (1.B.2). We next express

E N ( ϕ2hv ⊗ δ y=0 ) = π (2πh) d-1 R 2(d-1) e i(x -z )•ξ h ω N (y, x , ξ ) ϕ(0, z )v(z )dz dξ , with ω N (y, x , ξ ) = ∞ -∞
e N (y, x , η, ξ )e iyη h dη.

To calculate ω N for y > 0, we deform the contour of integral in η in the half plane Im η > 0.

From Residue formula, we have ω N (y, x , ξ ) = 2πiRes(e N (y, x , η, ξ ); iη + 0 )e iyη + 0 h .

The principal symbol of ω N is given by

π exp - yQ(y, x , ξ ) h ϕ(y, x ) 1 -χ ξ R 2Q(y, x , ξ ) , Q(y, x , ξ ) = λ(y, x , ξ ) 2 -1.
Therefore

lim sup h→0 E N ( ϕ2hv ⊗ δ y=0 ) L 2 y,x ≤C N,d ∞ 0 |α|≤Cd sup (x ,ξ ) |∂ α x ,ξ ω N (y, x , ξ )| v L 2 x dy ≤ C R ,
where we have used the point-wise estimate

|∂ α x ,ξ ω N (y, x , ξ )| ≤ C α e -y √ R 2 -1 2Rh √ R 2 -1 1 + y h |α| . Given χ(y, x , ξ ) ∈ C ∞ c ([0, 0 ) × R 2d-2
), the following proposition can be deduced in the same manner.

Proposition 1.22 Let w k = χ(y, x , hD x )(ϕu k ), w k = 1 y≥0 u k . Then for χ 1 ∈ C ∞ c (R), 0 ≤ χ 1 ≤ 1, χ 1 = 1 near 0, we have lim R→∞ lim sup k→∞ 1 -χ 1 h k D y R w k L 2 (R d ) = 0.
Proof. We have

P w = g + 2hv(x ) ⊗ δ y=0 with g L 2 y,x = O(1), v L 2 x = O(1)
. Note that the functions g, v here may not coincide with the functions in the proof of Proposition 1.3. We define

E N = N j=0 h j E j + R N , R N L 2 →L 2 = O(h N +1 ), with E 0 = Op h ϕ(y, x ) 1 -χ 1 η R η 2 + λ(y, x , ξ ) 2 -1
Id and E l , l ≥ 1 as in (1.B.1). This implies that

1 -χ 1 hD y R (ϕw) = E N ( ϕg + 2h ϕv ⊗ δ y=0 ) -R N w.
Consequently, we have

lim R→∞ lim sup h→0 E N ( ϕg) L 2 y,x = 0. E N (2h ϕv ⊗ δ y=0 )(y, x ) = π (2πh) d-1 R 2(d-1) e i(x -z )ξ h a(y, x , ξ ) ϕ(0, z )v(z )dz dξ , with a(y, x , ξ ) = ∞ -∞ e iyη h e N (y, x , η, ξ )dη.
Observe that

sup (x ,ξ ) |∂ α x ,ξ e N (y, x , η, ξ )| ≤ C α 1 -χ 1 η R (1 + η 2 )(1 + y 2 ) ,
and this implies that

lim R→∞ lim sup h→0 E N (2h ϕv ⊗ δ y=0 ) L 2 y,x = 0.
1.C. Proof of technical results in section 3

1.C Proof of technical results in section 3

Proof of Lemma 1.6. The proof can be reduced to the point-wise estimate of the solution F (y) of the ODE:

-h 2 d 2 F dy 2 + λ(y) 2 F (y) = G(y), F (0) = F (0) = 0, with 0 < c 1 ≤ λ(y) 2 ≤ c 2 , G ∈ C ∞ ([0, ∞))
, and |G(y)| ≤ Ce -cy h for all y ≥ 0. By rescaling z = y h , it reduces to prove the exponential decay of the solution W of the ODE:

- d 2 W dz 2 + V (z)W (z) = g(z), W (0) = W (0) = 0, with 0 < c 1 ≤ V (z) ≤ c 2 , g ∈ C ∞ ([0, ∞)) and |g(z)| ≤ Ce -cz for all z ≥ 0.
For this, we first notice that W is smooth and in H s (R + ) for all s ≥ 0. To prove the exponential decay, we pick θ (z) = e 2δ 0 z 1+ z with δ 0 > 0 to be chosen later. One observe easily that 0 < θ (z) ≤ 2δ 0 θ (z) for z ≥ 0. We multiply by θ W to the both sides of the equation and integrate it, then

∞ 0 (θ W ) W + θ V W 2 dz = ∞ 0 θ W Gdz. Notice that θ W W ≥ -2δ 0 θ |W ||W | ≥ -δ 0 θ (|W | 2 + |W | 2 ), by choosing δ 0 < min c 1 4 , 1 4 , c 2
,
we have that

∞ 0 θ |W | 2 + |W | 2 dz ≤ 2 Ge 2δ 0 z L 2 (R + ) W L 2 (R + ) ,
thanks to Cauchy-Schwartz and the fact that θ ≤ e 2δ 0 z , uniformly in . From dominating convergence theorem, we have W e δ 0 z ∈ L 2 (R + ) and W e δ 0 z ∈ L 2 (R + ). Finally, from theory of elliptic regularity, we have that

W e δ 0 z ∈ L ∞ (R + ),W e δ 0 z ∈ L ∞ (R + ). Proof of Proposition 1.4. We choose ϕ 2 ∈ C ∞ c (Y + ) such that ϕ 1 | supp(ϕ 2 ) = 1, ϕ 2 | supp(ϕ) = 1.
We first claim that

ϕ 2 Op h (χ δ 0 A j )(ϕ 1,0 q 0 ) = O L 2 (R d + ) (1). (1.C.1)
Indeed, we can write

ϕ 2 Op h (A j χ δ 0 )(ϕ 1,0 q 0 ) = ϕ 2 Op h (A j )ϕ 1 ϕOp h (χ δ 0 )(ϕ 1,0 q 0 ) + hϕ 2 Op h (B j χ δ 0 )(ϕ 1,0 q 0 )
with B j ∈ E -j ∂ and χ δ 0 has similar support property as χ δ 0 . Thus from symbolic calculus, we have for each fixed y > 0, By taking supp(χ δ 0 ) small such that ϕ 2 χ δ 0 = χ δ 0 , we have that

ϕ 2 Op h (A j (y)χ δ 0 (y))(ϕ 1,0 q 0 ) 2 L 2 (R d-1 ) ≤ C j e -c j y h h -1 1 + y h
ϕ 2 (Op h (χ δ 0 )(ϕ 1 q) -Op h (χ δ 0 A)(ϕ 1,0 q 0 )) = w + O L 2 (R d + ) (h) with w = ϕ 2 (Op h (χ δ 0 )(ϕ 1 q) -Op h (χ δ 0 )ϕ 1 ϕ 2 Op h (A)(ϕ 1,0 q 0 )) Lemma 1.5 implies that w = O L 2 (R d + ) (1) and hD y w = O L 2 (R d + ) (1)
. The trace of w satisfies

w| y=0 = ϕ 2,0 Op h (χ δ 0 (1 -ψ δ 0 (λ 0 )))(ϕ 1,0 q 0 ) = O H ∞ (R d-1 ) (h ∞ ).
w satisfies the equation (we use ϕ 2 = ϕ 1 ϕ 2 here)

P 0 w = ϕ 1 [P 0 , ϕ 2 Op h (χ δ 0 )] (ϕ 1 q -ϕ 2 Op h (A)(ϕ 1,0 q 0 )) + O H ∞ (R d ) (h ∞ ). (1.C.2) Notice that ϕ 1 q -ϕ 2 Op h (A)(ϕ 1,0 q 0 ) = O L 2 (R d + ) (1), micro-locally for λ ≥ δ 0 2 , the right hand side of (1.C.2) is equal to O L 2 (R d + ) (h) as well as O H 1 (R d + ) (1)
. Multiply by w = ϕ 1 w to the both sides of (1.C.2) and integrate it, we have

∞ 0 R d-1
w(y, x )P 0 wdx dy = O(h).

The left hand side can be integrated by part as

∞ 0 R d-1 wP 0 wdx dy = - ∞ 0 R d-1
|h∂ y w| 2 dx dy -

R d-1 h 2 (w∂ y w)| y=0 dx - ∞ 0 R d-1 1≤j,k≤d-1 g jk h∂ j wh∂ k wdx dy + O(h).
This implies that

hD y,x w L 2 (R d + ) + w L 2 (R d + ) = O(h 1/2
). Using this smallness and redo the integrating by part argument, we can improve each bound in the procedure above and obtain that

hD y,x w L 2 (R d + ) + w L 2 (R d + ) = O(h 3/4
). To conclude, we observe that

ϕh∂ y w =ϕh∂ y Op h (χ δ 0 )(ϕ 1 q) -ϕh∂ y Op h (χ δ 0 )ϕ 2 Op h (A)(ϕ 1,0 q 0 ) =ϕOp h (χ δ 0 )h∂ y (ϕ 1 q) -ϕOp h (χ δ 0 )ϕ 2 Op h (h∂ y A)(ϕ 1,0 q 0 ) +O L 2 (R d + ) (h) =ϕOp h (χ δ 0 )h∂ y (ϕ 1 q) -ϕOp h (χ δ 0 λA)(ϕ 1,0 q 0 ) +O L 2 (R d + ) (h), (1.C.3)
where we have used symbolic calculus and Lemma 1.5 several times. Plugging into (1.C.2), we have that

P 0 w = O L 2 (R d + ) (h), w| y=0 = O H ∞ (R d-1 ) (h ∞ ).
We decompose w = w 1 + w 2 with P 0 w 1 = P 0 w, w 1 | y=0 = 0 and P 0 w 2 = 0, w 2 | y=0 = w| y=0 . From elliptic regularity of boundary value problem, we have 4 ). Observe that the error terms on the right hand side of (1.C.3) can be also bounded by 4 ) by interpolation. This completes the proof.

h 2 w 1 = O H 2 (R d + ) (h) and h 2 w 2 = O H 2 (R d + ) (h ∞ ) and thus h∂ y w = O H 1 (R d + ) (1). From interpolation, we have h∂ y w = O H 2 3 (R d + ) (h 1
O H 1 (R d + ) (1) and thus O H 2 3 (R d + ) (h 1

1.D Construction of test functions

We first give the detailed construction of a = a 0 + a 1 η used in the first step of the proof of Proposition1.15, which closely follows from [START_REF] Hörmander | The analysis of linear partial differential operators. iii, volume 274 of grundlehren der mathematischen wissenschaften (fundamental principles of mathematical sciences[END_REF].

Proof of Lemma 1.18 . : Given χ 1 ∈ C ∞ c (-2, 2) with χ 1 | (-1,1) = 1 and χ 2 ∈ C ∞ c (-3, 3) such that χ 2 | (-2,2) = 1.
Consider χ 0 (t) = e -1/t 1 t>0 with smoothness. We work in the local coordinate (y, x, ξ), and assume (0,

x 0 , ξ 0 ) ∈ G 2,+ with |ξ 0 | ∼ 1. Set φ = φ 0 + φ 1 η with φ 1 (y, x, ξ) = 1 |ξ| , φ 0 (y, x, ξ) = y 2 + |x -x 0 | 2 + |ξ -ξ 0 | 2 .
We calculate

H p φ = η (2∂ y φ 0 -{r, φ 1 }) + φ 1 ∂ y r -{r, φ 0 } ≥ 2c > 0
provided that |η| ≤ c 0 for some c 0 > 0 and W 0 is choosing small enough such that ∂r ∂y ≥ 4c in it. The positivity then follows from the direct calculation:

{r, φ 0 } =2∂ ξ r • (x -x 0 ) -2∂ x r • (ξ -ξ 0 ), ∂ y φ 0 =2y, {r, φ 1 } = ∂ x r • ξ |ξ| 3 .
We next take

f (y, x, η, ξ) := χ 2 φ 0 δ 2 χ 0 1 - φ δ .
The desired functions a 0 , a 1 are chosen to be the remainder when f is divided by p = η 2 -r(y, x, ξ) thanks to the Malgrange preparation theorem:

f (y, x, η, ξ) = (η 2 -r(y, x, ξ))g(y, x, η, ξ) + a 1 (y, x, ξ)η + a 0 (y, x, ξ).

On the support of f , we observe that

φ 0 (y, x, ξ) = |(y, x, ξ) -(0, x 0 , ξ 0 )| 2 ≤ 3δ, η |ξ| + φ 0 ≤ δ,
which implies η ≤ δ|ξ|. Moreover, on supp f ∩supp∂χ 2 (δ -1 φ 0 ), we have φ 0 ≥ 2δ, φ 0 +φ 1 η ≤ δ, and these imply η ≤ -δ|ξ|, hence r(y, x, ξ) = η 2 ≥ δ 2 |ξ| 2 , when p = η 2 -r = 0.

Direct calculation yields

H p f + f M |ξ| + ψ 2 = χ 0 1 - φ δ H p χ 2 φ 0 δ 2 -1 -χ 1 η δ|ξ| 2 N, with N = χ 2 φ 0 δ 2 χ 0 1 - φ δ H p φ δ -χ 0 1 - φ δ M |ξ| ∈ C ∞ , ψ = χ 1 η δ|ξ| N 1/2 .
Here N ≥ 0 on supp(ψ) if we choose δ > 0 small enough. Observe that when η = r 1/2 ≥ 0,

we have χ 0 (1 -δ -1 φ)H p χ 2 (δ -1 φ 0 ) 2 = 0, 1 -χ 1 η δ|ξ| 2 N = 0. We then define a function ϕ(y, x, ξ) = - χ 0 1 -φ δ H p χ 2 φ 0 δ 2 -1 -χ 1 η δ|ξ| 2 N 2r 1/2 | η=-r 1/2 1 r(y,x,ξ)>0
and then

H p f + f M |ξ| + ψ 2 = ϕ(η -r 1/2 ), when p = η 2 -r = 0.
Therefore, on p = 0, we have

H p a + aM |ξ| + ψ 2 = ϕ(η -r 1/2 ).
It is left to check the smoothness of functions ϕ, ψ and ρ. Indeed, on the support of ψ,|φ 1 η| ≤ 2δ, φ 0 ≤ 3δ, and then1 -φ δ ≤ 3. Notice that χ 0 (t) χ 0 (t) = t 2 , we have

N 1/2 = χ 2 φ 0 δ χ 0 1 - φ δ H p φ δ G M |ξ|δ H p φ , 1 - φ δ ∈ C ∞ , since the function G(a, t) = √ 1 -at 2 ∈ C ∞ for t ≤ 3, |a| 1. This implies that ψ ∈ C ∞ provided that δ is chosen small enough.
For ϕ, the smoothness comes from the fact that on the support of

χ 0 1 - φ δ H p χ 2 φ 0 δ 2 | η 2 =r -1 -χ 1 η δ|ξ| 2 N | η 2 =r , we have r ≥ δ 2 |ξ| 2 . Moreover, ϕ has compact support.
Finally, from the definition of a, we have

a 1 (y, x, ξ) = f (y, x, η, ξ) -f (y, x, -η, ξ) 2η | η= √ r(y,x,ξ) ∈ C ∞ c , a 0 (y, x, ξ) = f (y, x, η, ξ) + f (y, x, -η, ξ) 2 | η=± √ r(y,x,ξ) ∈ C ∞ c .
we deduce that ∂f ∂η = -

φ 1 δ χ 2 φ 0 δ 2 χ 0 1 -φ δ < 0, hence f (y, x, η, ξ) -f (y, x, -η, ξ) 2η = 1 2 1 -1 ∂f ∂η (y, x, sη, ξ)ds < 0. Define t(y, x, ξ) = - 1 2 1 -1 ∂f ∂η (y, x, sη, ξ)ds η= √ r(y,x,ξ) 1/2
, one can show that t is a smooth function with compact support.

The last observation is that a = f > 0 on p = 0,

hence s = f 1/2 | p=0 ∈ C ∞ c .
We give some more calculations: Let ψ 0 = ψ| y=0 , ψ 1 = ∂ψ ∂η | y=0 , when η = r = 0. Thus at (x 0 , ξ 0 ),

t(x 0 , ξ 0 ) = χ 0 (1)φ 1 (x 0 , ξ 0 ) δ , ψ 0 (x 0 , ξ 0 ) 2 =χ 0 (1) ∂r ∂y (0, x 0 , ξ 0 ) φ 1 (0, x 0 , ξ 0 ) δ -χ 0 (1)M |ξ| > 0, 2ψ 0 ψ 1 (x 0 , ξ 0 ) = -χ 0 (1) ∂r ∂y (x 0 , ξ 0 ) φ 1 (x 0 , ξ 0 ) 2 δ 2 -χ 0 (1){r, φ 1 }(0, x 0 , ξ 0 ) 1 δ + χ 0 (1)M φ 1 (0, x 0 , ξ 0 ) δ > 0,
for δ small enough since χ 0 (1) < 0.

Observe that near (x 0 , ξ 0 ), we have ψ 1 ψ 0 ∼ -

φ 1 (x 0 ,ξ 0 )χ 0 (1) 2χ 0 (1)δ
, provided that δ is small enough.

Now if we make a different choice of δ > 0, the difference between two ratios ψ 1 ψ 0 and

ψ 1 ψ 0
is non-zero. This implies that we can choose a further cut-off χ near (0,

x 0 , ξ 0 ) such that ϕOp h (χ)ϕ 1 u L 2 (Y + ) = o(1) and ϕOp h (χ)hD y ϕ 1 u L 2 (Y + ) = o(1) from ϕOp h (ψ 0 + ψ 1 η)ϕ 1 u L 2 (Y + ) = o(1) and ϕOp h ( ψ 0 + ψ 1 η)ϕ 1 u L 2 (Y + ) = o(1).
Next we recall the proof of Lemma 1.19, which is essentially given in [START_REF] Richard | Singularities of boundary value problems[END_REF].

Proof of Lemma 1.19. : From the transversal property, we can choose a new coordinate (s, t) in U such that ρ 0 = (0, 0) and H -r 0 = ∂ t in this coordinate.

Step 1. Consider the function χ(u) = e 1 u-3/4 1 u<3/4 . It is easy to check that χ is smooth and non-increasing with the property:

∂ N χ(u) = O((-χ ) 1/m ), ∀N ∈ N, m > 1, locally uniformly.
Step 2. Next we choose β ∈ C ∞ (R) such that β ≥ 0 vanishing on (-∞, -1) and be strictly increasing on (-1, - 1 2 ) and be equal to 1 on (-1 2 , ∞). We modify β such that

∂ N β = O(β 1/m ), ∀N ∈ N, m > 1, locally uniformly.
Step 3. Choose f ∈ C ∞ (R) so that f vanishes on (-∞, 1/2) and is strictly increasing and convex on (1/2, ∞) with f (1) > 1.

Now we set

a δ = β 3t 4δ 2 χ t σδ + |s| 2 δ 4 + f y δ 2 ,
and

g δ = -β 3t 4δ 2 H p (χ(u)) with u = 3t 4σδ + |s| 2 δ 4 + f y δ 2 . Finally we define h δ = -H p a δ -g δ . Note that supp (a δ ) = (y, s, t) : -δ 2 ≤ t, t σδ + |s| 2 δ 4 + f y δ 2 ≤ 3 4 ,
hence it is clear that (1)(2)(3)(4) in Lemma 1.19 are satisfied.

Since r = r 0 + O(y), when H -r acts on functions independent of η we have

H p = ∂ t + O(y)∂ s + O(y)∂ t + O(δ)∂ y , since |η| = O(δ)
. Therefore, we have

-g δ = β t δ 2 1 δσ + O(δ 2 ) |s| δ 4 + O(δ 2 ) 1 δσ + O(δ) 1 δ 2 χ (u) + O(1)χ(u) = β t δ 2 3 4δσ + O δ σ + O 1 δ + O(1)))χ (u) ∼ β t δ 2 1 δσ χ (u).
provided that δ, σ 1. In the calculation above, we have used the fact that χ(u) = O(χ (u)) on the support of g δ . Thus (5) in Lemma 1.19 follows.

(6) follows from the construction of χ.

To check [START_REF] Bourgain | Control for schrödinger operators on 2-tori: rough potentials[END_REF], we observe that supp g δ ∪supp h δ ⊂supp a δ . Moreover, from the construction, g δ , h δ is independent of η whenever 0 ≤ y < δ 2 2 . Finally, to check the support of h δ , we write

h δ = -H p β t δ 2 χ(u). Since β is independent of y, η, we have H p β t δ 2 = H -r β t δ 2
, which is supported on

I × L -(δ, δ 2 ) × R η , thanks to supp β ⊂ [-1, -1 2 ].
1.E Proof of Lemma 1.15

Lemma 1.26

In local coordinate Y + , we have

P h = -h 2 g √ G ∂ ∂y √ Gg -1 ∂ ∂y + R h = h 2 D 2 y + Op h (r) + O L 2 →L 2 (h).
Moreover, R h is a matrix-valued second order differential operator in x with scalar principal symbol r(y, x, ξ) = 1 -λ(y, x, ξ) 2 , which is self-ajoint with respect to the

(•|•) L 2 (Y + ) . Proof. Denote by y = x 0 , ∂ 0 = ∂ y , ∂ j = ∂ x j , j = 1, 2, • • • d -1. Let u ∈ Λ 1 (Y + ) and w ∈ Λ 2 (Y + ) written in the form u = u 0 dx 0 + u j dx j , w = w 0j dx 0 ∧ dx j + w jk dx j ∧ dx k .
We have from direct calculation that

du =(∂ 0 u j -∂ j u 0 )dx 0 ∧ dx j + ∂ k u j dx j ∧ dx k , d * u = - 1 √ G ∂ 0 (u 0 √ G) - 1 √ G ∂ j (g jk u k √ G), d * w = 1 √ G ∂ k (w 0j g jk √ G)dx 0 - 1 √ G g jl ∂ 0 (w 0k g kl √ G) + ∂ m ( √ Gw pk (g pl g km -g pm g kl ) dx j From direct calculation, h 2 ∆ H u = h 2 (dd * + d * d)u = v 0 dx 0 + v j dx j + R h u with v 0 = -h 2 ∂ 2 0 u 0 -h 2 ∂ 0 ( √ G) √ G ∂ 0 u 0 , v j = -h 2 ∂ 2 0 u l - h 2 √ G g jl ∂ 0 (g kl √ G)∂ 0 u k
and the R h u consists only the tangential derivatives ∂ j . Hence in the matrix form,

v = L h u := -h 2 g √ G ∂ ∂y √ Gg -1 ∂u ∂y .

Moreover, one easily verified that L

* h = L h , thus R * h = R h .

1.F Proof of Lemma 1.20

Proof of Lemma 1.20. For our need, it suffices to prove the last assertion. We first let A h = a(y, x, hD y , hD x ) and B h = b(y, x, hD x ), then Denote by c(y, x, η, ξ) = ϕ(y, y, x, η, ξ), it is obvious that c is an interior symbol, since it can be viewed as a tangential symbol for fixed η, and we have

A h B h u(y, x) = 1 (2πh) d e i(x
(1 + |ξ|) m (1 + |ξ| + |η|) m
for all m ∈ R on the support of c, thanks to the supporting property of a. Now we note C h = c(y, x, hD y , hD x ), and write

A h B h u = C h u + R h u, where R h u(y, x) = 1 0 dt 1 (2πh) d e i(x-x )ξ+i(y-y )η h (y -y )∂ y c t (y , y, x, η, ξ)u(y , x )dy dx dξdη = ih 1 0 dt 1 (2πh) d e i(x-x )ξ+i(y-y )η h ∂ η ∂ y c t (y , y, x, η, ξ)u(y , x )dy dx dξdη =: ih 1 0 C t u(y, x)dt.
with c t (y , y, x, η, ξ) = ϕ(ty + (1 -t)y, y, x, η, ξ). Notice that

∂ η ∂ y c t (y , y, x, η, ξ) = 1 (2πh) d-1 e i(x-z)(ξ -ξ) h ∂ η a(y, x, η, ξ )(∂ y b)(ty + (1 -t)y, z, ξ)dξ dz.
We need to be careful here since ∂ y b only exists for y > 0 and at the point y = 0, the right derivative (∂ m y ) + b(0) := lim y→0 + ∂ m b(y) exists for any order m. Since we are dealing with Dirichlet boundary condition, we always apply a tangential operator B(y, x, hD x ) to functions u(y, x) with u| y=0 = 0 in the trace sense. We could thus extend u(y , x ) by u(y , x )1 y ≥0 in y in the expression of the form 1 (2πh) d e i(x-x )ξ +i(y-y )η h ϕ(y , y, x, η, ξ)u(y , x )dy dx dξdη.

Therefore, we have sup y,y ≥0,0<t<1,z,ξ

|∂ α z ∂ β ξ b(ty + (1 -t)y, z, ξ)| ≤ C m,α,β , ∀m ∈ N, α, β ∈ N d-1 .
Now it is reduced to prove the uniform L 2 boundness of the operator

T h u = R d K h (y , x , y, x)u(y , x )dy dx , with kernel K h (y , x , y, x) = 1 (2πh) d R d e i(x-x )ξ+i(y-y )η h H t (y , y, x, η, ξ)dηdξ
where

H t (y , y, x, η, ξ) = 1 y ,y≥0 1 (2πh) d-1 e izζ h a 1 (y, x, η, ξ + ζ)b 1 (ty + (1 -t)y, x -z, ζ)dzdζ.
From Schur test lemma, we need to show

sup (y,x)∈R d + R d + |K h (y , x , y, x)|dy dx ≤ C 1 < ∞, sup (y ,x )∈R d + R d + |K h (y , x , y, x)|dydx ≤ C 2 < ∞, with C 1 , C 2 independent of h and t.
We first define

k h (y, x, w, v) := 1 (2π) d R d
e ivξ+iwη H t (y -hw, y, x, η, ξ)dηdξ, hence,

T h u(y, x) = 1 h d R d + k h y, x, y -y h , x -x h u(y , x )dy dx .
Notice that H t (y , y, x, η, ξ) are tangential symbol, parametrized by (y , y, η). Moreover, it is compactly supported in (y, x, η, ξ) variables, uniformly in the first variable y . Thus, Therefore, we obtain

∂ m η ∂ α ξ H t (y -hw,
R d + |K h (y , x , y, x)| dy dx = 1 h d R d + k h y, x, y -y h , x -x h dy dx = R d |k h (y, x, w, v)|dwdv ≤ C 1 ,
and

R d + |K h (y, x, y , x )|dydx = 1 h d R d + k h y, x, y -y h , x -x h dydx ≤ R d sup (y,x) |k h (y, x, w, v)|dwdv ≤ C 2 .
Chapter 2

Stabilization of Hyperbolic-Stokes System

This chapter is based on a joint-work with Felipe Chaves-Silva.

Introduction and Main Results

Let Ω ⊂ R d (d ≥ 2) be a smooth, bounded star-shpaed domain whose boundary ∂Ω is smooth, ω be a small subset of Ω and let T > 0.

In this paper, we are interested in the stabilization problem for the following hyperbolic Stokes system:

     ∂ 2 t u -∆u + ∇p + a(x)∂ t u = 0 in R × Ω, div u = 0 in R × Ω, u = 0 on R × ∂Ω, (u(0, x), ∂ t u(0, x)) = (u 0 , v 0 ) ∈ V × H, (2.1.1)
where V and H are the usual spaces in the context of fluid mechanics, i.e.,

V = {u ∈ H 1 0 (Ω) d : div u = 0} and H = {u ∈ L 2 (Ω) d : div u = 0, u • ν| ∂Ω = 0},
and ν(x) is the outward normal to Ω at the point x ∈ ∂Ω. In (2.1.1), the damping term a ∈ L ∞ (Ω) and satisfies a(x) ≥ 0, for all x ∈ Ω.

If u = u(t, x) is a (sufficiently smooth) solution of the system, we define its energy as

E[u](t) = 1 2 Ω (|∂ t u(t, x)| 2 + |∇u(t, x| 2 )dx, ∀t ∈ R,
and when there is no damping, namely a ≡ 0, the energy is conserved, while in general we only have that E[u](t) is non-increasing:

dE[u] dt = - Ω a(x)|∂ t u(t, x)| 2 dx ≤ 0.
As for other hyperbolic systems, the stabilization problem for (??) concerns about the decay rate in time of the energy E[u](t) under appropriate assumptions on the damping term.

It is well-known that stabilization problems are closely related to observability and exact controllability problems in abstract settings. In fact, if we consider the undamped system

     ∂ 2 t u -∆u + ∇p = 0 in R × Ω, div u = 0 in R × Ω, u = 0 on R × ∂Ω, (u(0, x), ∂ t u(0, x)) = (u 0 , v 0 ) ∈ V × H, (2.1.2)
we say that (2.1.2) is observable at time T with observation in ω if there exists C > 0 such that

||u 0 || 2 V + ||v 0 || 2 H ≤ C T 0 ω |∂ t u(t, x)| 2 dxdt, (2.1.3)
for every (u 0 , v 0 ) ∈ V × H.

When (2.1.3) holds, one can show that for any (u 0 , v 0 ) ∈ V × H there exists f ∈ L 2 ((0, T ) × ω) d such that the solution of

     ∂ 2 t u -∆u + ∇p = f 1 ω in R × Ω, div u = 0 in R × Ω, u = 0 on R × ∂Ω, (u(0, x), ∂ t u(0, x)) = (u 0 , v 0 ) ∈ V × H, (2.1.4) satisfies u(T, x) = 0, ∂ t u(T, x) = 0,
that is to say, system (2.1.4) is exact controllable at T with control localized in ω. Nevertheless, it is important to mention that a complete characterization of the sets ω for which (2.1.3) is true remains open. A partial answer to this question was given by the first author in [START_REF] Chaves-Silva | A hyperbolic system and the cost of the null controllability for the stokes system[END_REF].

The motivation for studying the stabilization of system (2.1.1) is two folded. First, system (2.1.2) is the hyperbolic counterpart of Stokes system, which is the linearized version of the well-known Navier-Stokes equation in fluid mechanics. In fact, if we know that system (2.1.2) is exact controllable at some time T > 0, with control applied to some control region ω, then the so-called Control Transmutation Method can be applied to obtain the null controllability at any time and the optimal cost of controllability (in time) for the Stokes system (for more details, see [START_REF] Chaves-Silva | A hyperbolic system and the cost of the null controllability for the stokes system[END_REF]). On the other hand, system (2.1.2) comes from simple models of dynamical elasticity for incompressible materials. More precisely, it can be derived as a limit model of Lamé system in linear elastic theory when one parameter tends to infinity ( [START_REF] Lions | On some hyperbolic equations with a pressure term, partial differential equations and related subjects: Proceedings of the conference dedicated do l. nirenberg[END_REF]). For the sake of completeness, in the Appendix we give a derivation of system (2.1.2) from Lamé system. It is important to remark that the stabilization problem for the Lamé system has been already studied in [START_REF] Burq | Mesures de défaut de compacité, application au systeme de lamé[END_REF].

To state our main results, let us introduce several concepts. Some terminologies and notation will be clear in the next section.

Definition 2.1

We say that the support of a non-negative function a ∈ C(Ω) satisfies the geometric control condition (GCC in short) if there exists T > 0, such that each generalized bicharacteristic ray γ(t) with speed 1 issued from a point ρ ∈ b T * Ω enters the set {x ∈ Ω : a(x) > 0} in a time t < T .

We recall that an open set Ω has no infinite order of contact, if in the decomposition

T * ∂Ω = E ∪ H ∪ G, we have G = ∞ j=2 G j .
Here, the sets E, H, G are called elliptic zone, hyperbolic zone and glancing zone, respectively, and G j are the sets of points with j-th order of contact. The precise definition of this sets will be given in the next section.

Our first main result is as follows.

Theorem 2.1

Suppose Ω ⊂ R d is a smooth, bounded star-shaped domain with no infinite order of contact and a ∈ C(Ω) is a non-negative function whose support satisfies the geometric control condition. Then, there exist positive constants C 0 and α such that for any (u 0 , v 0 ) ∈ V ×H, the corresponding solution u(t) to (2.1.1) has the exponential decay:

E[u](t) ≤ C 0 E[u](0)e -αt , ∀t ≥ 0. (2.1.5)
In what follows, we say that the stabilization of (2.1.1) holds if (2.1.5) holds true.

Remark 2.1

As a byproduct of the proof of Theorem 2.1, we obtain the null (exact) controllability at some time T of system (2.1.4). Namely, there exists T > 0 and a control f ∈ L 2 ([0, T ] × ω) such that the corresponding solution u to (2.1.4) satisfies (u(T ), ∂ t u(T )) = (0, 0). However, we do not know the control time T explicitly, since we prove the observability inequality (2.1.3) by reducing it to a quasi-mode estimate.

Let us mention that if a is supported in a neighborhood of boundary ∂Ω, the same result is true by adapting the strategy in [START_REF] Chaves-Silva | A hyperbolic system and the cost of the null controllability for the stokes system[END_REF], where the author has proved the exact controllability of the system (2.1.4) with ω be a neighborhood of ∂Ω. Our result is a great generalization of the result obtained in [START_REF] Chaves-Silva | A hyperbolic system and the cost of the null controllability for the stokes system[END_REF].

The pioneering work of J.Rauch and M.Taylor [START_REF] Rauch | Exponential decay of solutions to hyperbolic equations in bounded domains[END_REF] related the exponential decay of damped wave equation to geometric control condition (GCC) of damped region on compact Riemannian manifold without boundary.

Until the celebrated work of C. Bardos, G. Lebeau, and J. Rauch [?], the presence of the boundary has been understood and the exactly controllability for wave equation as well as the exponential stabilization are obtained under (GCC). The proof mainly relies on the propagation of singularity under Melrose-Sjöstrand flow. Later on, the tool of micro-local defect measure, introduced by P.Gérard and L.Tartar independently, has been used to simplify the proof of these results and adapt to many other problems, see for example [START_REF] Burq | Mesures de défaut de compacité, application au systeme de lamé[END_REF] for Lamé systems and [START_REF] Dehman | Controllability of two coupled wave equations on a compact manifold[END_REF] for a coupled wave system. The key ingredient of the measurebased proof is the propagation formula, which can be viewed as a transport equation for defect measure. As a consequence, the propagation of singularity can be derived as a special case of measure invariance under bicharacteristic flow. For the present system (2.1.1), the presence of the pressure term ∇p introduces nontrivial difficulties if we want to adapt the strategy in [START_REF] Burq | Mesures de défaut de compacité, application au systeme de lamé[END_REF] directly, due to the rough regularity of time-dependent harmonic function p(t, x). However, follow the semi-classical reduction in [START_REF] Burq | Stabilisation of wave equations on the torus with rough dampings[END_REF], it turns out that the exponentially stabilization of (2.1.1) can be reduced to the following semi-classical version observability estimate: Proposition 2.1 Assume that a ∈ C 0 (Ω) and Ω adx > 0. Suppose the following statement holds true:

∃h 0 > 0, C > 0 such that ∀0 < h < h 0 , ∀(u, q, f ) ∈ H 2 (Ω) ∩ V × H 1 (Ω) × H solves the equation -h 2 ∆u -u + h∇q = f, (2.1.6) implies u L 2 (Ω) ≤ C a 1/2 u L 2 (Ω) + 1 h f L 2 (Ω) . (2.1.7)
Then we have the stabilization of (2.1.1).

Note that the system (2.1.7) is just a quasi-mode equation of stationary Stokes system, and in particular, if f = 0, the solution u(h) is a eigenfunction of Stokes operator corresponding to eigenvalues h -2 .

The proof of (2.1.7) is based on the propagation of semi-classical measure µ treated in the previous chapter. We give a brief recall here. The sequence of pressure q are harmonic, and their impact on the solution only occurs at the boundary. It has been shown that the measure is propagated along bi-characteristic rays which is invariant under the flow. When a ray touches the boundary, more careful analysis between the wave-like propagation phenomenon and the impact of the pressure yield the propagation of the support of the measure µ along generalized bi-characteristic ray defined in [START_REF] Richard | Singularities of boundary value problems[END_REF].

We organize this paper as follows. In section 2, we recall the propagation result given in the previous chapter. In section 3, we follow the strategy in [START_REF] Burq | Stabilisation of wave equations on the torus with rough dampings[END_REF] to reduce the stabilization to semi-classical observability (2.1.7). In section 4, we prove the semi-classical observability by adapting the propagation result. Finally in Appendix, we give the derivation from Lamé system to system (2.1.1).

Review of Semi classical propagation of singularity

Now let us recall the several results proved in the last chapter.

-

h 2 ∆u -u + h∇q = f, (u, f ) ∈ (H 2 (Ω) ∩ V ) × H h divu = 0, in Ω (2.2.1) Assume that u L 2 (Ω) = O(1), f L 2 (Ω) = o(h). and u 0 in L 2 (Ω).
Let µ be a semiclassical defect measure associated with (u, h) (some subsequence of it). The precise definition and some properties of µ is given in the last chapter and we will not recall it here for simplicity.

In the interior, the full transport property of defect measure is proved. The following proposition illustrates that near a elliptic point on the boundary, there is no accumulate of singularity.

Proposition 2.3 µ1 E = 0. If we let ν be the semi-classical defect measure of the sequence

(h k ∂ ν u k | ∂Ω , h k ), then ν1 E = 0.
When a ray travels near a hyperbolic point or point in the glancing surface, the knowledge of the singularity is much less. Nevertheless, we have Theorem 2.2 Assume that Ω is a smooth, bounded star-shpaed domain with no infinite order of contact on the boundary. Suppose (u k ) is a sequence of solutions to the quasi-mode problem (2.1.1) with semi-classical parameters h = h k . Assume that f k ∈ H, f k L 2 (Ω) = o(h k ) and u k converges weakly to 0 in L 2 (Ω). Assume that µ is any semi-classical measure associated to some subsequence of (u k , h k ), then suppµ is invariant under Melrose-Sjöstrand flow.

Reduction to Semi-classical observability

This section is devoted to the proof of Proposition2.1. In fact, It is classical from [START_REF] Gearhart | Spectral theory for contraction semigroups on hilbert space[END_REF] that stabilization or observability of a self adjoint evolution system is equivalent to resolvent estimates. See also [START_REF] Burq | Stabilisation of wave equations on the torus with rough dampings[END_REF], [START_REF] Burq | Geometric control in the presence of a black box[END_REF].

Recall that the damped system is given by

         ∂ 2 t u -∆u + a(x)∂ t u + ∇p = 0, (t, x) ∈ R × Ω divu = 0, in Ω u(t, .)| ∂Ω = 0 (u(0), ∂ t u(0)) = (u 0 , v 0 ) ∈ V × H (2.3.1)
We always assume that Ω ⊂ R d is a bounded domain (open, connected set). We use ν to denote the outward normal vector on ∂Ω and the damping term a ∈ L ∞ (Ω) with a(x) ≥ 0.

We also consider the undamped system

         ∂ 2 t u -∆u + ∇p = 0, (t, x) ∈ R × Ω divu = 0, in Ω u(t, .)| ∂Ω = 0 (u(0), ∂ t u(0)) = (u 0 , v 0 ) ∈ V × H (2.3.2)

Some functional analysis preliminaries

We work with a Hilbert space H := V × H, equipped with the norm

(f, g) t 2 H := ∇f 2 L 2 (Ω) + g 2 L 2 (Ω) .
and denote Π : L 2 (Ω) N → H be the orthogonal projector(Leray-projector) and A = Π∆ be the Stokes operator. We consider the operator:

A = 0 Id A -Πa (2.3.3) with domain D(A) = (V ∩ H 2 (Ω)) × V.
In order to use semi-group theory, we first show that for some λ > 0, the operator (A -λ) is invertible: Take (f, g) ∈ V × H, and consider the system

v -λu = f Au -(Πa + λ)v = g (2.3.4)
We consider the bilinear form

B(u 1 , u 2 ) = Ω ∇u 1 • ∇u 2 dx + Ω (λ 2 + λa(x))u 1 • u 2 dx,
defined on V × V . We then conclude from Lax-Milgram that for λ > 0, there exists u ∈ V such that for any w ∈ V , we have

B(u, w) = - Ω (g • w + (a(x) + λ)f • w)dx.
Set v = λu + f , we have solved the system (2.3.4) in weak sense. Standard regularity argument gives that for λ > 0.

(A -λ) -1 : H → D(A),
is a bounded, and (A -λ) -1 : H → H is compact. Moreover, if λ ∈ Spec(A), we must have Re λ < 0. This will be clear in the proof of Proposition 2.6.

However, since the operator A is not maximal dissapative, the Hille-Yoshida theorem is not applicable. A slightly general modification ensures the existence of semi-group e tA which evolves the initial data in D(A) and solves the equation (2.3.1) with more regular data.

For solution u, ∂ t u to (2.3.1), we consider the energy functional

E[u](t) := 1 2 Ω (|∂ t u(t, x)| 2 + |∇u(t, x)| 2 )dx,
and we calculate

d dt E[u](t) = Ω ∂ t u • (∂ 2 t u -∆u)dx = - Ω ∂ t u • ∇pdx - Ω a(x)|∂ t u| 2 dx ≤ 0, thus E[u](t) ≤ E[u](s), ∀s ≤ t. (2.3.5)
By density argument, we can solve (2.3.1) with initial data in H such that the energy dissapation (2.3.5) still holds.

Observability and Stabilization

In this section, we will prove the stabilization for damped system is equivalent to observability for undamped system. For this part, we follow closely in the appendix of [START_REF] Burq | Stabilisation of wave equations on the torus with rough dampings[END_REF] in which the authors have sketched the standard argument for damped wave equation.

We first introduce the quantity

D[u](T ) = T 0 Ω a(x)|∂ t u(t, x)| 2 dxdt,
and it quantifies the dissipation of the energy:

E[u](T ) = E[u](0) -D[u](T ).

Proposition 2.4

The following assertions are equivalent:

1. Stabilization: There exists C 0 , α > 0, such that for every solution u ∈ C(R; V ∩ H 2 (Ω)) ∩ C 1 (R; V ) to the damped system (2.3.1), we have

E[u](t) ≤ C 0 E[u](0)e -αt , ∀t ≥ 0.
2. Observability: There exists C > 0 and T > 0, such that, for every solution v ∈ C(R; V ∩ H 2 (Ω)) ∩ C 1 (R; V ) to the undamped system (2.3.2), the observability inequality holds:

E[v](0) ≤ CD[v](T ).
Proof. We first claim that the stabilization of damped system is equivalent to the observability of damped system.

It is clear that

E[u](0) = E[u](t) -D[u](t).
Let us first assume the stabilization of damped system. Argue by contradiction, suppose the observability of damped system does not hold. We first choose T 0 > 0 large enough such that C 0 e -αT 0 < 1 2 . We can select a sequence of solutions (u k ) and such that

E[u k ](0) = 1, D[u k ](T 0 ) → 0, as k → ∞.
We thus have

1 2 > C 0 e -αT 0 ≥ E[u k ](T 0 ) = E[u k ](0) -D[u k ](T 0 ) = 1 + o(1), as k → ∞,
which leads to a contradiction.

Let us now assume the observability for damped system, i.e.

E[u](0) ≤ CD[u](T ),

We may assume that C > 1, from the energy dissapation and observability, we have

E[u](2T ) = E[u](0) -D[u](2T ) ≤ 1 - 1 C E[u](0).
For any t > 0, we write m = t 2T , therefore we have

E[u](t) ≤ E[u](m) ≤ 1 - 1 C m E[u](0),
after choosing C 0 , α appropriately, we have the stabilization of damped system.

Our second step is to justify the equivalence between observability of damped system (2.3.1) and undamped system (2.3.2). To do this, we denote u and v be solutions of the damped and of the undamped system, respectively, with the same initial data at t = 0. Let w = u -v, and simple calculations yield

∂ 2 t w -∆w = -a∂ t u -∇q, ∂ 2
t w -∆w + a∂ t w = -a∂ t v -∇q, with some pressure function q.

We calculate

d dt E[w](t) = - Ω a(x)|∂ t u| 2 dx + Ω a(x)∂ t u • ∂ t vdx - Ω ∂ t w • ∇qdx,
and the last term of left hand side vanishes, thanks to ∂ t w ∈ C(R; V ). Thus we can write

d dt E[w](t) = - Ω a(x)|∂ t u| 2 dx + Ω a(x)∂ t u • ∂ t vdx or equivalently d dt E[w](t) = - Ω a(x)∂ t u • ∂ t wdx.
Integrating the two expressions above and using the inequality of the type

|ab| ≤ |a| 2 + C( )|b| 2 , ∀ > 0, one easily get E[w](T ) ≤ B min (D[u](T ), D[v](T )), ∀T > 0, (2.3.6)
where B is another absolute constant. Now suppose we have observability for the damped system (2.3.1), if

D[u](T ) ≤ D[v](T ), the observability of undamped system (2.3.2) is trivial. Now assume that D[u](T ) > D[v](T ), we deduce from (2.3.6) that E[v](0) = E[u](0) ≤ CD[u](T ) ≤ CD[W ](T ) + CD[v](T ) ≤ C(E[w](T ) + D[v](T )) ≤ CD[v](T ).
The derivation of observability from undamped system to the damped follows in the same way, and we omit the details.

Remark 2.2

Since the domain D(A) is dense in H and the observability and energy decay only involves the L 2 norm of ∇u and ∂ t u, thus the same result of proposition 2.4 holds if we replace

u ∈ C(R; V ∩ H 2 (Ω)) ∩ C 1 (R; V ) to u ∈ C(R; V ) ∩ C 1 (R; H).

Resolvent estimates and stabilization

Recall that from the previous sections, the study of damped system (2.3.1) is equivalent to the project system

d dt u ∂ t u = 0 Id A -Πa u ∂ t u , u ∂ t u ∈ C(R; V × H).
(2.3.7)

We will use the notation U = (u, ∂ t u) t in the sequel.

In this part, we follows almost the same way as in the appendix of [START_REF] Burq | Stabilisation of wave equations on the torus with rough dampings[END_REF], only to pay attention to the changing of working spaces (appearance of the pressure term and divergence free structure). Moreover, we add some technical details which may seems standard to experts in analysis but not disposable for many applied people.

From last section, we know that the observability of undamped system (2.3.2) is equivalent to the stabilization of damped system (2.3.1), therefore we will concentrate ourselves to the study of stabilization of (2.3.1). The following result is standard in semigroup theory: Proposition 2.5 Consider a semi-group e tL on a Hilbert space X , with infinitesimal generator L defined on D(L). Then if there exists C > 0, δ > 0 such that the resolvent of L, (L -λ) -1 exists for Re λ ≥ -δ and satisfies ∀λ ∈ C δ := {z ∈ C : Re z > -δ}, (L -λ) -1 L(X ) ≤ C. Then there exists M > 0 such that for any t > 0,

e tL L(X ) ≤ M e -δt 2 .
We need a lemma from complex analysis. We temporarily use the convention of Fourier transform u(τ

) = 1 √ 2π ∞ -∞
e -itτ u(t)dt.

Lemma 2.1

Let u, v be two continuous functions with support in R + = (0, ∞). Assume moreover that u, v ∈ L 2 (R + ) and v has compact support. From Winer-Paley theory, we know that the Fourier transform v admits a holomorphic extension to C and of exponential type. Given a 0 > 0, suppose that the Fourier transform u is also holomorphic in

S a 0 = {z ∈ C : Im z < a 0 } and satisfies | u(z)| ≤ C| v(z)|, ∀z ∈ S a 0 .
Then for any a < a 0 , e at u(t) ∈ L 2 (R + ) and

∞ 0 e 2at |u(t)| 2 dt = ∞ -∞ | u(τ + ia)| 2 dτ.
Proof. We first claim that

∞ 0 e 2at |v(t)| 2 dt = ∞ -∞ | v(τ + ia)| 2 dτ, ∀a ∈ R. (2.3.8)
Indeed, since v is compactly supported,

v(τ + ia) = 1 √ 2π ∞ 0
e at e -itτ v(t)dt which is analytic in a and rapidly decreasing in τ for each fixed a ∈ R. Thus one easily deduce from the Plancherel (or calculate the integral directly) that (2.3.8) is true.

As a consequence, u(. + ia) ∈ L 2 (R) for each a < a 0 . Notice also that u ∈ L 2 (R + ), thus for each a with Re a < 0, the formula

∞ 0 e 2at |u(t)| 2 dt = ∞ -∞ | u(τ + ia)| 2 dτ (2.3.9)
holds true and analytic with respect to a. In particular, | u(z)| ≤ C| v(z)|,z ∈ S a 0 implies that u(τ + ia) is rapidly decreasing in τ for each fixed a < a 0 . For z = τ + ia with a < a 0 , consider the integral

F (a, t) = e at √ 2π ∞ -∞ e itz u(z)dτ = 1 √ 2π ∞ -∞ e itτ u(z)dτ ∈ L 2 (R).
From Cauchy integral theorem, we have that

F (a, t) = e at √ 2π ∞ -∞ u(τ )e itτ dτ = e at u(t) ∈ L 2 (R).
From this, we conclude that (2.3.9) follows for all a < a 0 .

Remark 2.3

In the previous lemma, the same results hold true if we replace u, v to be Hilbert-space valued functions.

proof of proposition 2.5. The basic tool to prove this proposition is the Fourier-Laplace transform in time variable. From the property of strongly continuous semi-group, we know that there exists ω 0 > 0 such that (see [START_REF] Michael | Partial Differential Equations I[END_REF])

e tL L(X ) ≤ e ω 0 t , ∀t ≥ 1.
Take u 0 ∈ D(L), and pick a nonnegative cut-off χ ∈ C ∞ (R) such that χ ≡ 0, ∀t ≤ 1 and χ ≡ 1, ∀t > 2. We define u(t) := χ(t)e tL-ωt u 0 for some ω > ω 0 and thus u ∈ L ∞ (R; X ). Moreover, we have the equation

(∂ t + ω -L)u = χ (t)e tL-ωt u 0 =: v(t).
By taking Fourier transform we get

(iτ + ω -L) u = v(τ ).
Since v is compactly supported in positive axis in time variable, the v(τ ) has a holomorphic and bounded extension in any domain of the form

S α = {τ ∈ C : Im τ < α}, α > 0.
From the assumption on the resolvent, we deduce that (iτ + ω -L) is invertible if τ ∈ S δ+ω and thus u(τ ) admits a bounded holomorphic extension to S δ+ω which satisfies

u(τ ) X ≤ C v(τ ) X .
Apply Lemma ??, we deduce that

∞ -∞ e (ω 0 +δ)t u 2 X dt = ∞ -∞ u(ξ + i(ω 0 + δ)) 2 X dξ ≤ C ∞ -∞ v(ξ + i(ω 0 + δ)) 2 X dξ ≤ C ∞ -∞ e (ω 0 +δ)t v 2 X dt ≤ C u 0 2 X .
We remark that one need use various types of Winer-Paley theorems to justify the above calculations, thanks to the fact that u(t), v(t) is supported on [1, ∞) and furthermore v(t) has compact support. Take ω < ω 0 + δ 2 in the definition of u, we have that

e δt 2 e tL u 0 L 2 (R + ;X ) ≤ C 1 u 0 X .
Thanks to the semi-group structure and uniform bound principal, we have that there exists M 0 > 0, such that for any interval I ⊂ (0, +∞) of length 1, sup t∈I,s>0,t+s∈I

|f (t + s)| |f (t)| ≤ M 0 .
with f (t) = e tL u 0 X . Therefore, for any T > 0,

T +1 T e δt |f (t)| 2 dt ≥ e δT min t∈[T,T +1] |f (t)| 2 .
Therefore,

|f (T + 1)| 2 ≤ M 2 0 min t∈[T,T +1] |f (t)| 2 ≤ e -δT T +1 T e δu |f (t)| 2 dt.
This implies the exponential decay

e tL u 0 X ≤ M e -δt 2 u 0 X .

Now we can introduce the semi-classical observability

Proposition 2.6 Assume that a ∈ L ∞ (Ω) ∩ C 0 (Ω) and Ω adx > 0. Then the stabilization of system (??) is implied by the following statement:

∃h 0 > 0, C > 0 such that ∀0 < h < h 0 , ∀(u, q, f ) ∈ H 2 (Ω) ∩ V × H 1 (Ω) × H solves the equation -h 2 ∆u -u + h∇q = f,
we have

u L 2 (Ω) ≤ C a 1/2 u L 2 (Ω) + 1 h f L 2 (Ω) . (2.3.10)
For the proof, we need two lemmas.

Lemma 2.2

Let L be a linear operator on Hilbert space X with a compact resolvent (L -Id) -1 . Suppose the spectrum Spec(L) ⊂ {z : Re z < 0} and satisfies that for any σ ∈ R, L-iσ is invertible and satisfies the uniform bound

sup σ∈R (L -iσ) -1 < ∞.
Then there exists δ > 0, such that

sup λ∈C δ (L -λ) -1 < ∞,
where C σ := {z ∈ C : Re z > -σ} for any σ ∈ R.

Proof. Write sup σ∈R (L -iσ) -1 = C We denote R(z) = (L -z) -1 for z ∈ ρ(L) := {z : z ∈ C \ Spec(L)}. Take z 0 ∈ ρ(L), we write L -z = (L -z 0 )(Id + (L -z 0 ) -1 (z 0 -z)),
and for |z -

z 0 | < 1 (L-z 0 ) -1 , we have R(z) ≤ R(z 0 ) ∞ n=0 |z -z 0 | n (L -z 0 ) -1 n ≤ R(z 0 ).
Therefore, for λ with | Re λ| ≤ δ, where 0 < δ < 1 2C , we have R(λ) ≤ C. To conclude, we only need show that there exists C 1 > 0, such that

sup Re z>δ (L -z) -1 ≤ C 1 .

Consider the holomorphic equivalence

ϕ : C 0 → D,ψ = ϕ -1 . ϕ(z) = z -1 z + 1 , ψ(ζ) = 1 + ζ 1 -ζ ,
where D := {ζ : |ζ| < 1} be the unit disk. One easily verifies that the operator-valued function

Φ(ζ) = R(ψ(ζ)) : D → L(X )
is analytic and satisfies the Cauchy integral formula

Φ(ζ 0 ) = 1 2πi |ζ|=1 Φ(ζ) ζ -ζ 0 dζ, ∀ζ 0 ∈ D.
Since dist (∂D, ϕ(C -δ )) ≥ 0 > 0 for some 0 depends only on δ, we deduce that for any

z ∈ C -δ , R(z) ≤ 1 2πi |ζ|=1 Φ(ζ) ζ -ϕ(z) dζ ≤ C 0 . Lemma 2.3 (Unique Continuation of Stoke Operator) Let σ > 0 and u ∈ V satisfies that Au = σ 2 u.
Then if u| ω ≡ 0, we must have u ≡ 0.

Proof. It is equivalent to write

-∆u + ∇p = σ 2 u, div u = 0, u ∈ V, Ω pdx = 0.
Take divergence of the equation, we have ∆p = 0. The vanishing of u in ω implies that p equals to a constant in a component of ω. Now since Ω is connected, the maximum principal implies that p ≡ 0 in Ω. Therefore we have reduced to unique continuation of eigenfunction of Laplace operator, and this implies that u ≡ 0 in Ω.

proof of proposition 2.1. We need show that the semi-classical observability implies the stabilization.

Note that the operator (A -λ) is invertible for any λ > 0. One write

A -z = (Id + (1 -z)(A -Id) -1 )(A -1), ∀z ∈ C. Since Id + (1 -z)(A -Id) -1
is Fredholm with index 0, we infer that A -z is invertible iff it is injective. In light of the previous lemmas and the Proposition 2.5, we only have to prove the fact that

∃C > 0, such that ∀σ ∈ R, U ∈ D(A), F ∈ V × H, (A -iσ)U = F implies U H ≤ C F H .
We argue by contradiction. If it is not true then we can find sequences (σ n ), (U n ), and

(F n ) such that (A -iσ n )U n = F n , U n H = 1, F n H < 1 n .
After extracting subsequences we may assume that σ n → σ, and we write

U n = (u n , v n ) t , F n = (f n , g n ) t .
We have several cases to analyse, according to the limit value σ.

1. σ = 0: In this case, we have AU n = o(1) H , which is equivalent to

v n = o(1) H 1 0 , Au n -Πav n = o(1) L 2 , thus Au n = o(1) L 2 .
Taking inner product with u n and integrating by part we have

Ω |∇u n | 2 dx = o(1).
This contradicts to U n H = 1.

2. 0 < |σ| < ∞: In this case we have AU n -iσU n = o(1) H , or equivalently, v n -iσu n = o(1) H 1 0 , Au n -(iσ + Πa)v n = o(1) L 2 .

Thanks to Poincaré inequality, we deduce that

Au n -iσ(iσ + Πa)u n = o(1) L 2 .
Applying Rellich compact embedding theorem followed by extracting to suitable subsequences, we may assume that

u n → u, in L 2 (Ω), u n u, in V.
Taking inner product with u n , we have

- Ω |∇u n | 2 dx = -σ Ω |u 2 n |dx + iσ Ω a(x)|u n | 2 dx + o(1),
which implies that au ≡ 0 in Ω. Thus we can conclude that u is an eigenfunction of Stokes operator A and vanishes in a non trivial open subset of Ω. The unique continuation property for the system

-∆u + ∇p = σ 2 u, div u = 0
implies that u ≡ 0. As a consequence, we have that

u n = o(1) H 1 0 , v n = o(1) L 2
. This contradicts to the original assumptions.

|σ| = ∞:

We only study the case σ n → +∞ (the other one is obtained by considering U n ).

Let h n = σ -1 n , and we deduce from the system AU n -iσ n U n = o(1) H :

h 2 n Au n + u n -ih n Πau n = h 2 n Πaf n + ih n f n + h 2 n g n = o L 2 (h n ) h n v n -iu n = h n f n = o(h n ) H 1 0 , h 2 n Av n + v n -ih n Πav n = ih n g n -h 2 n Af n = o L 2 (h n ) + o H -1 (h n ).
Define the operator P h = h 2 A + Id -ihΠa on H with domain H 2 (Ω) ∩ V , we have (dropping the subindex n for the moment)

(P h u|u) L 2 = u 2 L 2 (Ω) -h∇u 2 L 2 (Ω) -ih a 1/2 u 2 L 2 (Ω) .
Taking imaginary part, we have

a 1/2 u 2 L 2 (Ω) ≤ C P h u L 2 (Ω) u L 2 (Ω) h .
Applying the semi-classical observability to the equation

h 2 Au + u = ihΠau + f with f = o L 2 (h), we have u 2 L 2 (Ω) ≤C a 1/2 u 2 L 2 (Ω) + 1 h 2 ( f 2 L 2 (Ω) + h 2 a 1/2 u 2 L 2 (Ω) ) ≤ C h f L 2 (Ω) u L 2 (Ω) + C h 2 f 2 L 2 (Ω) .
( .3.11) This implies that

u n L 2 (Ω) ≤ C h n f n L 2 (Ω) = o(1).
To conclude, observe that v n satisfies

h 2 n Av n + v n = o H (h n ) + o H -1 (Ω) (h 2 n ),
and we claim that if

(h 2 A + 1)v = f 1 + f 2 , then v L 2 (Ω) + h∇v L 2 (Ω) ≤C a 1/2 v L 2 + f 1 L 2 (Ω) h + f 2 H -1 (Ω) h 2 .
(2.3.12)

Assume the claim for the moment, we thus have h n ∇v n L 2 (Ω) = o(1), and ∇u n L 2 (Ω) = o(1), thanks to u n + ih n v n = ih n f n . This contradicts to the original assumption. Now we turn to the proof of the claim. By density, (2.3.10) still valid when v ∈ V .

Taking inner product of v with P h v, we have

(P h v|v) L 2 = v 2 L 2 (Ω) -h∇v 2 L 2 (Ω) -ih a 1/2 v 2 L 2 (Ω) .
Therefore, by taking real part and injecting (2.3.10), we have

h∇v L 2 (Ω) + v L 2 (Ω) ≤ C a 1/2 v L 2 (Ω) + P h v L 2 (Ω) h . (2.3.13)
By taking real and imaginary part of (P h v|v) L 2 , we have

a 1/2 v 2 L 2 (Ω) ≤ P h v L 2 (Ω) v L 2 (Ω) h , ( 2
h∇v 2 L 2 (Ω) + v 2 L 2 (Ω) ≤ C P h v L 2 (Ω) v L 2 (Ω) h + P h v 2 L 2 (Ω) h 2 ,
and this implies that

h∇v L 2 (Ω) + v L 2 (Ω) ≤ C P h v L 2 (Ω) h .
Thus P h is bijective from H 2 (Ω) ∩ V to H and hence invertible. From the fact that

P h = (1 + (2 -ihΠa)(h 2 A -1) -1 )(h 2 A -1),
P h can be written as composition of a positive operator and a Fredholh operator of index 0. From the estimate above, we conclude that

P -1 h L 2 →L 2 ≤ C h , P -1 h L 2 →H 1 ≤ C h 2 .

Now come back to the equation (h

2 A + 1)v = f 1 + f 2 .
Taking g ∈ H, and letting w = P -1 h g, we have

(v|g) L 2 =((h 2 A + 1)v|w) L 2 + ih(v|Πaw) L 2 =(f 1 + f 2 |w) L 2 + ih(av|w) L 2 ≤ f 1 L 2 (Ω) P -1 h g L 2 ()Ω) + f 2 H -1 (Ω) P -1 h g V + h av L 2 (Ω) w L 2 (Ω) ≤C a 1/2 v L 2 (Ω) + f 1 L 2 (Ω) h + f 2 H -1 (Ω) h 2 g L 2 (Ω) .
(2.3.15)

This completes the proof.

Apriori Estimates for the quasi-mode system

Now we consider the quasi-modes of Stokes system

-h 2 k ∆u k -u k + h k ∇q k = f k , (u k , f k ) ∈ (H 2 (Ω) ∩ V ) × H, h k divu k = 0, in Ω (2.4.1)
To simplify the notation, we drop the sub index k and just keep the semi-classical parameter h everywhere. Note that the functions u, v, etc. should be understood as u(h), v(h), etc.

Assume that

u L 2 (Ω) = O(1), f L 2 (Ω) = o(h).
Taking inner product with u and integrate by part, we have

h∇u L 2 (Ω) = O(1).
One can always assume that Ω qdx = 0, since q ∈ L 2 (Ω)/R. From the regularity theory of steady Stokes system, (see [START_REF] Temam | Navier-Stokes equations: theory and numerical analysis[END_REF], page 33), and Poincaré inequality, we have

h 2 ∇ 2 u L 2 (Ω) = O(1), q L 2 (Ω) = O(h -1 ), h∇q L 2 (Ω) = O(1)
.

We now give some estimates on the trace. Write q 0 = q| ∂Ω ,

Lemma 2.4 q L 2 (Ω) = O(h -1 ), q 0 H 1/2 (∂Ω) = O(h -1 ), q 0 L 2 (∂Ω) = O(h -1 ).
Proof. Since q is harmonic function, then one can apply trace theorem H s (Ω) → H s-1/2 (∂Ω) for any s ∈ R. Hence the conclusions follows from these and interpolations. Proof. The first assertion follows from hdivu = 0 and Dirichlet boundary condition, while we apply a multiplier method to prove the second. From the geometric assumption on Ω, we can find a vector field L ∈ C 1 (Ω) such that L| ∂Ω = ν(see [START_REF] Manuel Milla | Introduction to exact control theory[END_REF], page 36). In global coordinate system, we write L = L j (x)∂ x j . By using the equation, we have

Ω Lu • f dx = Ω Lu • (-h 2 ∆u -u + h∇q)dx, - Ω Lu • udx = - Ω L j (x)∂ x j u i u i dx = - Ω ∂ x j L j (x)u i u i dx + Ω div L(x)|u| 2 dx = Ω L j (x)u i (x)∂ x j u i dx + Ω div L(x)|u| 2 dx = Ω Lu • udx + Ω div L(x)|u| 2 dx,
and thus

Ω Lu • udx = - 1 2 Ω div L(x)|u| 2 dx = O(1). We next calculate h Ω Lu • ∇qdx = -h Ω u i ∂ x j (L j ∂ x i q) dx = -h Ω u • L(∇q)dx -h Ω (div L(x))u • ∇qdx = -h Ω u • [L, ∇]qdx -h Ω div L(x)u • ∇qdx = O(1), -h 2 Ω Lu i ∆u i dx = -h 2 ∂Ω ∂ ν u i 2 dσ + h 2 Ω ∇L(∇u i , ∇u i )dx + h 2 Ω L j (x)∂ 2 x j x k u i ∂ x k u i = -h 2 ∂Ω ∂ ν u i 2 dσ + h 2 Ω ∇L(x)(∇u i , ∇u i )dx + h 2 Ω ∂ x j L j ∂ x k u i ∂ x k u i dx -h 2 Ω div L(x)∇u i • ∇u i (x)dx, h 2 Ω ∂ x j L j ∂ x k u i ∂ x k u i dx = h 2 ∂Ω L • ν ∂ ν u i 2 dσ -h 2 Ω L j (x)∂ x k u i ∂ 2 x j x k u i dx, -h 2 Ω Lu i ∆u i dx = - h 2 2 ∂Ω ∂ ν u i 2 dσ+ Ω ∇L(x)(h∇u i , h∇u i )dx- h 2 2 Ω div L(x)|∇u i | 2 dx.
Observing that Ω Lu • f dx = o(1), we have

∂Ω |h∂ ν u| 2 dσ = O(1).

Lemma 2.6

Under additional assumption that

a 1/2 u k L 2 (Ω) = o(1),
after extracting to subsequences, we have h k ∇q k 0 L 2 (Ω) and u k 0 weakly in L 2 (Ω). Therefore from Rellich theorem, we have hq → 0, strongly in L 2 (Ω).

Proof. We may assume that h∇q r, weakly in L 2 (Ω), and Rellich theorem implies that hq → P , strongly in L 2 (Ω), and thus ∇P = r, with the property Ω P = 0. Now we claim that ∆P = 0 in Ω.

Indeed, take any

ϕ ∈ C ∞ 0 (Ω), Ω ∇P • ∇ϕ = lim h→0 Ω h∇q • ∇ϕ = 0. Now suppose u k → U , weakly in L 2 (Ω), w k = h 2 k u k → W ,
weakly in H 2 (Ω), by taking the weak limit in the equation, we have

-∆W -U + ∇P = 0, in L 2 (Ω).
Notice that a 1/2 u k → 0, a 1/2 w k → 0, strongly in L 2 (Ω), and this implies that U | ω = W | ω = 0. Therefore, in a connect component ω of ω, we have ∇P ≡ 0. However, P is a harmonic function, then P ≡const., thanks to the fact that Ω is connected. Note that Ω P = 0, hence P ≡ 0. Moreover, from Rellich theorem that w k → W strongly in L 2 (Ω), and on the other hand

h 2 k u k L 2 (Ω) = o(1)
we must have W = 0. Therefore U = 0.

Proof of the Observability Estimates

In this part, we will prove the Proposition 2.1 under the assumption in Theorem 2.1 on Ω and ω.

We argue by contradiction, suppose (??) is not true, we can then choose a sequence

(u n , h n , q n , f n ) ∈ H 2 (Ω) ∩ V × R + × H 1 (Ω) × H satisfies equation -h 2 n ∆u n -u n + h n ∇q n = f n (2.5.1)
with the following properties:

u n L 2 (Ω) = 1, f n L 2 (Ω) = o(h n ), a 1/2 u n L 2 (Ω) = o(1), n → ∞.
Up to extracting to subsequence, we can associate (u n , h n ) with a semi-classical defect measure µ. From the h n -oscillating(see the previous chapter) of the measure µ and a 1/2 u k → 0 in L 2 (Ω) we deduce that µ, a = 0, namely ω ∩ π(supp(µ)) = ∅, where we denote π : T * Ω → Ω be the canonical projection. Denote φ(s, ρ) be the globally defined generalized bicharacteristic flow, thanks to the geometric assumption that Ω has no infinite contact. Pick any point ρ 0 with π(ρ 0 ) ∈ ω.For any time segment [0, s 0 ], there are several situations: Either φ([0, s 0 ], ρ 0 ) ⊂ Ω, or there exist π(φ([0, s 0 ], ρ 0 )) ∩ ∂Ω = ∅, then from the assumption on Ω, all points φ(s, ρ 0 ) with s ∈ [0, s 0 ] and π(φ(s, ρ 0 )) ∈ ∂Ω must lie in

H ∪ G 2,+ ∪ G 2,-∪ k≥3 G k . Now Theorem 2.2 implies that supp (φ(s, •) * µ) ⊂ supp (µ).
Therefore, we have φ([0, s 0 ], ρ 0 ) ∩ supp (µ) = ∅.

We now invoke the geometric control condition to deduce that

Ω ⊂ π ρ 0 ∈ω φ([0, T 0 ], ρ 0 )
for some T 0 > 0 and thus µ = 0. This contradicts to the assumption that

Ω |u n (x)| 2 dx = 1.

2.A Formal derivation of hyperbolic Stokes system

We will derive the hyperbolic Stokes system (2.1.2) from certain limit procedure of Lamé system from elastic theory:

     ∂ 2 t w -µ∆w -(λ + µ)∇divw = 0, (t, x) ∈ [0, T ] × Ω w(t, .)| ∂Ω = 0 (w(0), ∂ t w(0)) = (w 0 , z 0 ) ∈ (H 1 0 (Ω) × L 2 (Ω)) d (2.A.1)
where the solution w(t, x) is vector-valued.

Define u(t, x) := w(t/ √ µ, x), then we find that

∂ 2 t u -∆u - λ + µ µ ∇divu = 0.
We let = µ µ+λ 1, in the case that λ µ > 0. Thus we obtain a family of equations

     ∂ 2 t u -∆u + ∇p = 0, (t, x) ∈ [0, T ] × Ω u (t, .)| ∂Ω = 0 (u (0), ∂ t u (0)) = (u 0, , v 0, ) ∈ (H 1 0 (Ω) × L 2 (Ω)) d (2.A.2)
where p = -1 divu and satisfies Ω p dx = 0.

We make further assumption on the family of initial data (u 0, , v 0, ) so that

(u 0, , v 0, ) -(u 0 , v 0 ) H 1 ×L 2 ≤ C
for some divergence free data (u 0 , v 0 ) ∈ V × H. In particular, we have

divu 0, L 2 (Ω) ≤ C .
From the well-posedness of Lamé system, we have that u ∈ C([0, T ];

H 1 0 (Ω)), ∂ t u ∈ C([0, T ]; L 2 (Ω))
, and p ∈ C([0, T ]; L 2 (Ω)). Moreover, we have the conservation of energy

E[u ] = 1 2 Ω |∂ t u | 2 + |∇u | 2 + |p | 2 dx
and therefore

E[u ] = 1 2 Ω |u 0, | 2 + |v 0, | 2 + 1 |divu 0, | 2 dx.
From this, we have, up to some subsequence of (u , ∂ t u )

divu → 0, in L ∞ ([0, T ]; L 2 (Ω)), u * u, * weakly in L ∞ ([0, T ]; H 1 0 (Ω)), ∂ t u * ∂ t u, * weakly in L ∞ ([0, T ]; L 2 (Ω)).
From the uniform bound of ∂ t u L ∞ ([0,T ];L 2 (Ω)) , and apply Ascoli theorem, we have that (up to some subsequence)

u → u, in C([0, T ]; L 2 (Ω)).
Using the equation, we conclude that ∇p L ∞ ([0,T ];H -1 (Ω)) is uniformly bounded. Combine with the fact Ω p = 0, we have that p L ∞ ([0,T ];L 2 (Ω)) is uniformly bounded, thus up to some subsequence, we may assume that

p * p, * weakly in L ∞ ([0, T ]; L 2 (Ω)).
Now it is not difficult to verify that (u, p) is a weak solution to (2.1.1). Moreover, p satisfies the zero mean condition

Ω pdx = 0.
Part II

Control and stabilization of KP type equations

Chapter 3

Exact Controllability for linear KP-II equation

This chapter is based on a joint-work with Ivonne Rivas.

Introduction

The Kadomtsev-Petviashvili equation better known as KP is

∂ x (∂ t u + ∂ 3 x u + u∂ x u) ± ∂ 2 y u = 0 (3.1.1)
and it was introduced by Kodomtsev and Petviashvili (see [START_REF] Bb Kadomtsev | On the stability of solitary waves in weakly dispersing media[END_REF]) in 1970 from the study of transverse stability of the solitary wave solution of the Kortewed de-Vrie (KdV) equation.

The KP equations are completely integrable and can be solved by inverse scattering transform. Moreover, the equation (3.1.1) has been studied separatyly depending the on the sign is used, with negative sign is known as KP-I equation, otherwise is the KP-II equation, these propagation of the trajectories behave very differently from one equation to another one and do not allow us to study at the same time. In this paper, we concentrate on the KP-II equation.

Concerning about the Cauchy problem, the KP-II equation has been well studied. In the pioneering work of J.Bourgain [START_REF] Bourgain | On the cauchy problem for the kadomstev-petviashvili equation[END_REF], he proved the global well-posedness of KP-II equation in L 2 (T 2 ) by using the Fourier restriction norm introduced by himself. For non-periodic setting, Takaoka and Tzvetkov in [START_REF] Takaoka | On the local regularity of the kadomtsevpetviashvili-ii equation[END_REF] demonstrated local well-posedness in anisotropic Sobolev space H s 1 ,s 2 (R 2 ) with s 1 > - 1 3 and s 2 ≥ 0. Hadac, Kerr and Koch in [START_REF] Hadac | Well-posedness and scattering for the kp-ii equation in a critical space[END_REF] proved global well-posedness and scattering for small data in critical functional space H -1 2 ,0 (R 2 ). Molinet, Saut and Tzvetkov in [START_REF] Molinet | Global well-posedness for the kp-ii equation on the background of a non-localized solution[END_REF] showed the local and global well-posedness for partially periodic data.

We will address the exact controllability problem for KP-II equation. Before getting 111 into the problem, we observe that (3.1.1) can be written as

∂ t u + ∂ 3 x u + u∂ x u ± ∂ -1 x ∂ 2 y u = 0,
where the Fourier multiplier ∂ -1

x is defined by

∂ -1 x v(k, η) = 1 ik v(k, η) for all functions v ∈ D 0 (T 2 ) := {v ∈ D (T 2 ) : v(0, l) = 0 for all l ∈ Z}.
For any s ∈ R, we denote by

H s 0 (T 2 ) := H s (T 2 ) ∩ D 0 (T 2 ), a closed subspace of H s (T 2 ).
The internal control problem that we are interested in studying in this paper is as follows: Given T > 0 and u 0 , u 1 ∈ L 2 0 , does there exist a control input h ∈ L 2 ((0, T ); L 2 (T 2 ) in order to make the solution of

∂ t u + ∂ 3 x u + ∂ -1 x ∂ 2 y u + u∂ x u = G(h), (t, x, y) ∈ R × T × T, u| t=0 = u 0 ∈ L 2 0 (T 2 ), (3.1.2) satisfy u(T, •) = u 1 ?
The first step is to consider the internal control problem for linearized KP-II equation

∂ t u + ∂ 3 x u + ∂ -1 x ∂ 2 y u = G(h), (t, x, y) ∈ R × T × T, u| t=0 = u 0 ∈ L 2 0 (T × T). (3.1.3)
In order to keep the solution u(t) in L 2 0 , we need to define the control input Gh to keep it in the space D 0 (T 2 ). In this paper, we only consider the case where the control region ω is either a vertical strip or a horizontal strip.

For a vertical control region of the form ω = (a, b) × T, we fix a non-negative real

function g ∈ C 2 c (T 2 ) with T g = 1.
In this case, we define the control input by Our first result, gives a positive answer to the internal controllability of the linearized KP-II equation on vertical region: Theorem 3.1 Given T > 0. For any u 0 , u 1 ∈ L 2 0 (T), there exists a control h ∈ L 2 ((0, T ); L 2 (T)), such that the solution u of (3.1.3) with G = G ⊥ satisfies u(T ) = u 1 .

G(h)(x, y) = G ⊥ (h)(x, y) := g(x) h(x, y) - T g(x )h(x , y)dx , ( 3
For vertical region, once the exact controllability for linearized KP-II is established, we can adapt the technique in the Cauchy theory of KP-II equation to prove exact controllability for KP-II in local sense.

Theorem 3.2

Given T > 0. There exists R > 0 such that for any u 0 , u

1 ∈ L 2 0 (T 2 ) satisfying u 0 L 2 (T 2 ) ≤ R and u 1 L 2 (T 2 ) ≤ R, there exists a control h 2 ∈ L 2 ((0, T ); L 2 0 (T 2 ), such that the solution u of (3.1.2) with G = G ⊥ satisfies u(T ) = u 1 .

Remark 3.1

In [START_REF] Bourgain | On the cauchy problem for the kadomstev-petviashvili equation[END_REF], KP-II equation is globally well-posed in H s 0 (T 2 ) for all s ≥ 0. Our results Theorem 3.1 and 3.2 also hold for any data in H s 0 (T 2 ). The reason for working in L 2 (T 2 ) is that the quantity

T 2 |u(t, x, y)| 2 dxdy
is conserved along KP flow (3.1.1) and hence L 2 is the natural space to address the control problem.

On the contrary, for the controllability on horizontal region, we have a negative answer which shows that the exact controllability for linearized KP-II equation can not hold at any time T > 0 when the control region is a horizontal strip.

Theorem 3.3

Given T > 0 there exists u 1 ∈ L 2 (T 2 ) and there does not exist h ∈ L 2 ((0, T ); L 2 0 (T 2 )) such that the solution u of (3. 1.3)

with G = G satisfies u(T ) = u 1 .
The proof and disproof of controllability for linear equation rely on the propagation of singularity for KP-II flow. Because of the asymmetry in the horizontal,x, and vertical,y, coordinate, the waves described by the KP-II equation behave differently in the direction of propagation (x-direction) and transverse (y-direction). It turns out that the propagation on the horizontal direction is KdV like and much stronger than the propagation on the vertical direction. The heuristic is that any singularity will travel into some vertical control region in a very short time while the singularity cannot travel vertically into the horizontal control region in finite time. For this reason, we believe that the following formal criteria for the exact controllability is valid, although further efforts are needed to proved it:

In fact, the setting of the control problem, namely the good definition of the operator G, for general control region should different from what we have done for vertical and horizontal strip. It seems that there is no obvious way to keep the control input to be localized and simultaneously have zero horizontal mean. This observation suggests that we should look for the control problem directly for the equation

∂ x (∂ t u + ∂ 3 x u) + ∂ 2 y u = Gh instead of the non local version (3.1.3).
The paper is organized as follows. In section 2, some results of well-posedness are mentioned, they will recover importance in the proof of the controllability of the full control system. In section 3, the linear controllability is established by proving the observability inequality. In section 4, the local controllability of the nonlinear equation is proved by fixed point argument. In section 5, we construct a counterexample to complete the proof of Theorem 3.3.

Notations and Preliminaries

Throughout this article, we use the identification T = R/(2πZ) = [-π, π]/Z 2 . We need the following classical inequality Proposition 3.1 (Ingham inequality [START_REF] Edward | Some trigonometrical inequalities with applications to the theory of series[END_REF]) Suppose λ k+1 -λ k ≥ γ for all k ∈ Z. Then for all T > 2π γ , there exists two positive constants C 1 , C 2 depending only on γ and T such that

C 1 k∈Z |a k | 2 ≤ T 0 k∈Z a k e itλ k 2 dt ≤ C 2 k∈Z |a k | 2 .
Now we briefly review the Cauchy theory of KP-II and we mainly follow the material in [START_REF] Molinet | Global well-posedness for the kp-ii equation on the background of a non-localized solution[END_REF]. The initial value problem

∂ t u + ∂ 3 x u + ∂ -1 x ∂ 2 y u + u∂ x u = 0, (t, x) ∈ R × T 2 , u| t=0 = u 0 ∈ L 2 0 (T 2 ), (3.2.1)
is proved in [START_REF] Bourgain | On the cauchy problem for the kadomstev-petviashvili equation[END_REF] by Bourgain to be globally well-posed when u 0 ∈ H s (T 2 ) for s ≥ 0. Bourgain introduced a Fourier restriction norm by

u 2 X s,b,b 1 = R (k,l)∈Z 2 σ(τ, k, l) k 3 2b 1 σ(τ, k, l) 2b (k, l) 2s | u(τ, k, l)| 2 dτ where σ(τ, k, l) = τ -k 3 + l 2 k and • = 1 + | • | 2 . For T > 0, the norm in the localized time interval [0, T ] is defined by u X s,b,b 1 T := inf{ w X s,b,b 1 : w(t) = u(t)on(0, T )}.
Define by the linear evolution flow S(t) = e -it(∂ 3

x +∂ -1

x ∂ 2 y ) . We have the following linear estimate

Proposition 3.2 For s ≥ 0, -1 2 < b ≤ 0 < 1 2 < b ≤ b + 1, b 1 ∈ R and T ≤ 1, we have t 0 S(t -t )F (t )dt X s,b,b 1 T ≤ CT 1-(b-b ) F X s,b ,b 1 T .
The proposition above is false for the end points b = -1 2 and b = 1 2 . However, for periodic problem, it seems that we can not avoid to use these end points. To compromise, we need to use another norm

u Z s,b := σ b-1 2 (k, l) s u l 2 (k,l) L 1 τ
and the restricted spaces Z b,s T defined in the same manner. With these auxiliary norms, the linear estimate now holds true.

Proposition 3.3

Under the same conditions as in Proposition (3.2)

S(t)u 0 + t 0 S(t -t )F (t )dt X s, 1 2 ,b 1 T ∩Z s, 1 2 T ≤ C u 0 H s + C F X s,-1 2 ,b 1 T ∩Z s,-1 2 T .
The essential of the proof can be found in [START_REF] Tao | Nonlinear dispersive equations: local and global analysis[END_REF]. To show that the equation (3.2.1) is locally well-posed in the spaces with the Fourier restriction norm through the integral form of the solution

u(t) = S(t)u 0 + t 0 S(t -τ )u∂ x udτ, (3.2.2)
the following bilinear estimate is crucial

Proposition 3.4 ([51]) There exist 1 4 < b 1 < 3 8 and ν > 0 such that for all functions u, v ∈ X s, 1 2 ,b 1 with T u(t, x, y)dx = T v(t, x, y)dx = 0,
the following bilinear estimate holds

∂ x (uv) X s,-1 2 ,b 1 T ∩Z s,-1 2 T ≤ CT ν u X s, 1 2 ,b 1 T v X s, 1 2 ,b 1 T , provided that s ≥ 0.
We remark that this bilinear estimate is essentially established by J.Bourgain in [START_REF] Bourgain | On the cauchy problem for the kadomstev-petviashvili equation[END_REF]. We adapt to the statement in [START_REF] Molinet | Global well-posedness for the kp-ii equation on the background of a non-localized solution[END_REF] here, in which the authors dealt with partially periodic data.

Linear controllability on vertical strip

In this section, the study of the internal controllability of linear system (3.1.3) is addressed by defining a linear operator in Proposition 3.10, which characterize the control input of the linear system and drives the solution from an initial state u 0 to a final state u 1 . Notice that from reversability, the exact controllability is equivalent to null controllability: given any initial state u 0 ∈ L 2 0 , find a function h ∈ L 2 ((0, T ) × T 2 ) so that the equation satisfies u(0, •) = u 0 and u(T, •) = 0. Hence, we will study the null controllability.

The classical strategy to study the null controllability is to show the observability inequality for the adjoint system associated to the equation, in the KP-II case, it matches with the homogeneous linearized KP-II equation:

∂ t u + ∂ 3 x u + ∂ -1 x ∂ 2 y u = 0, (t, x, y) ∈ R × T × T, u| t=0 = u 0 ∈ L 2 0 (T), u| t=T = u 1 ∈ L 2 0 (T), (3.3.1)
From classical Hilbert Uniqueness Method (HUM), one can deduce that the null controllability is equivalent to the observability for its adjoint system. Proposition 3.5 Given T > 0, the system (3.1.3) is null controllable at T if and only if there exists a constant C = C(T ) > 0 such that

u 0 2 L 2 (T 2 ) ≤ C T 0 T 2 |Gu(t, x)| 2 dxdt, ∀u ∈ L 2 (T 2 ). (3.3.2)
The region where the control will be placed is a vertical strip given by ω := (a, b) × T and the operator G = G ⊥ is given by (3.1.4). The region ω will allow us to get a reduction of the KP-II equation (3.3.1) in one dimension. As it is stated in the following Remark:

Remark 3.2
Expanding the solution u(t, x) to (3.3.1) in Fourier series in y variable

u(t, x, y) = l∈Z a l (t, x)e ily ,
we find that for each l ∈ Z, a l satisfies the equation

∂ t a l + ∂ 3 x a l -l 2 ∂ -1
x a l = 0 Therefore, by changing notations, the equation (3.3.1) can be reduced to the study of following λ-dependent equation

∂ t u + ∂ 3 x u -λ 2 ∂ -1 x u = 0, (t, x) ∈ R × T, u| t=0 = u 0 ∈ L 2 0 (T), (3.3.3)

Observability inequality

Thanks to Proposition 3.5, the proof of Theorem 3.1 is reduced to the proof of (3.3.2).

From the previous remark and Plancherel's theorem, we can further reduce the observability (3.3.2) to the following uniform observability for the family of system (3.3.3).

Theorem 3.4

Given T > 0. There exists C = C(T ) > 0 such that for all λ > 0,

u 0 2 L 2 (T) ≤ C T 0 T |Gu(t, x)| 2 dxdt (3.3.4)
holds for all solution u of (3.3.3). Now we concentrate to the proof of this theorem. The strategy is as follows. First we reduce (3.3.4) to a weaker one, which on the one hand is the observability for high frequencies and on the other hand gets rid of the normalization part. Next we use timescaling and semi-classical reduction, inspired by the work of Lebeau in [START_REF] Lebeau | Contrôle de l'équation de schrödinger[END_REF], to reduce this weak observability for system (3.3.3) to an inequality of the same form but for another semi-classical system. The third step is to reduce the inequality in the previous step to a frequency-localized one. Finally, we use propagation argument to prove the frequencylocalized semi-classical observability.

Reduction to weak observability

The weak observability takes the form, uniformly in λ ≥ 0,

u 0 2 L 2 (T) ≤ C T 0 T |g(x)u(t, x)| 2 dxdt + C u 0 2 H -1 (T) . (3.3.5)
First, we prove a lemma concerning about the commutator of a high frequency cut-off and the operator G

Lemma 3.1 Take 1-χ ∈ C ∞ c (R) with supp(χ) ⊂ {|ξ| > 1} and χ| |ξ|≥2 = 1. Then, we have T 0 [χ(hD x ), G]u(t, •) 2 L 2 (T) dt ≤ Ch 2 u(0) 2 L 2 (T) . Proof. T 0 [χ(hD x ), G]u(t, •) 2 L 2 (T) dt ≤ C(I + II), I = T 0 T |[g(x), χ(hD x )]u(t, x, y)| 2 dxdt, II = T 0 T g(x) T g(x )χ(hD x )u(t, x )dx -χ(hD x ) g(x) T g(x )u(t, x )dx 2 dxdt,
Symbolic calculus yields (though g is not assumed to be smooth, the following estimate still valid)

[g(x), χ(hD x )] L 2 →L 2 ≤ Ch,
and hence due to the conservation of L 2 norm

I ≤ Ch 2 T 0 u(t) 2 L 2 (T) dt = Ch 2 T u(0) 2 L 2 (T) .
For II, we first calculate (for simplicity the variable t is omitted in here)

g(x) T g(x )(χ(hD x )u)(x )dx (l) -χ(hD x ) g(x) T g(x )u(x )dx (l) = g(l) l 1 =0 (χ(hl 1 ) -χ(hl)) g(l 1 ) u(l). Since, |χ(hl 1 ) -χ(hl)| ≤ χ L ∞ h|l 1 -l|,
we have

II ≤Ch 2 l | g(l)| 2 l 1 =0 |l 1 -l| g(l 1 ) u(l 1 ) 2 ≤Ch 2 l | g(l)| 2 l 1 =0 |l 1 -l| 2 | g(l 1 )| 2 l 1 =0 | u(l 1 )| 2 ≤Ch 2 u 2 L 2 (T) l,l 1 =0 |l 1 -l| 2 | g(l 1 )| 2 | g(l)| 2 =Ch 2 u 2 L 2 (T) , due to g ∈ C 2 c (T).
Proposition 3.6 (3.3.5) implies the following full observability inequality

u 0 2 L 2 (T) ≤ C T 0 T |Gu(t, x)| 2 dxdt. (3.3.6)
Proof. The proof is essentially a unique continuation argument. However, by the λdependence family of equations, we will divide the proof in two steps.

First, we prove that for any given λ > 0, (3.3.6) holds with constant C > 0 which may depend on λ. We argue by contradiction, assuming that (3.3.6) is not true, then we can select a sequence u n of solutions to (3.3.3) so that

u n (0) L 2 (T) = 1 and lim n→∞ T 0 T |Gu n (t, x)| 2 dxdt = 0.
After extracting to some subsequence, we may assume that u n (0) u 0 , weakly in L 2 (T). One can easily verify that u 0 ∈ L 2 0 . Moreover, from semi-group property, u n (t) u(t) weakly in C([0, T ]; L 2 (T)) and u(t) is the distributional solution to (3.3.3) with initial data u 0 . Since G : L 2 0 → L 2 0 is a bounded operator, we have that Gu(t, •) = 0 in L 2 0 (T) for a.e. t ∈ [0, T ]. This means that u(t, x)| ω = C(t) in D (ω) for a.e. t ∈ [0, T ]. Moreover, from the strong continuity of the semi-group on L 2 0 ,

C(t) = T g(x)u(t, x)dx, ∀t ∈ [0, T ].
and C(t) is a continuous function in t. Therefore we have that

g(x) (u(t, x) -C(t)) = 0, in C([0, T ]; L 2 (T))
and thus u(t, x)

| x∈ω = C(t) in D (ω) for all t ∈ [0, T ]. Now, if we rewrite the equation (3.3.3) as ∂ x (∂ t u + ∂ 3
x u) + λu = 0 and evaluate u for x ∈ ω, we have that u| ω = 0 in D (ω).

We claim that u ≡ 0. Indeed, consider the following set:

N := {u 0 ∈ L 2 0 (T) : u(t, •)| ω = 0, ∀t ∈ [0, T ]}.
Apply inequality (3.3.5) and we have that

u 0 L 2 ≤ C u 0 H -1 (T)
for all u 0 ∈ N . This implies that the subspace N in L 2 0 (T) is finite dimensional. Thus, for any u 0 ∈ N , we can write the solution in the form u(t, x) = 1≤|l|≤M a l e it(l 3 -λ 2 l -1 ) e ilx , this trigonometric polynomial is smooth and it vanishes in ω. From classical result (see for instance [START_REF] Lebeau | Introduction aux inégalités de carleman[END_REF]), a l ≡ 0 for all 1 ≤ |l| ≤ M . This implies that u ≡ 0.

Since the weak limit of u n (0) is 0, we have T g(x)u n (t, x)dx → 0 and hence gu n L 2 ([0,T ]×T) → 0. Moreover, up to a subsequence, we have u n (0) H -1 (T) → 0, due to Rellich theorem. This is a contradiction to the assumption that u n (0) L 2 (T) = 1.

For the second step, we need to prove (3.3.6) uniformly in λ. Again, we assume that (3.3.6) is not true. Then there are a sequence of positive numbers λ n > 0 and a sequence solutions u n to (3.3.3) with parameters λ n such that

u n (0) L 2 (T) = 1 and lim n→∞ T 0 T |Gu n (t, x)| 2 dxdt = 0.
Up to a subsequence, we may assume that λ n → λ ∞ ∈ [0, ∞]. Suppose λ ∞ < ∞, similar argument as in Step 1 will lead to contradiction.

The possibility that is left to study is λ ∞ = ∞. We write

u n (0) = l =0
a n,l e ilx and the corresponding solution to (3.3.3) satisfies

u n (t, x) = l =0 a n,l e it l 3 - λ 2 n l e ilx .
For any 0 > 0, we set

u ( 0 ) n := |l|≥ 1 0 a n,l e it l 3 - λ 2 n l e ilx , v ( 0 ) n = u n -u ( 0 ) n .
From Lemma 3.1, we have

T 0 Gu ( 0 ) n (t) 2 L 2 (T) dt ≤C 2 0 u n (0) 2 L 2 (T) + C T 0 (Gu n ) ( 0 ) (t) 2 L 2 (T) dt.
Thus, there exists C > 0 such that for any 0 > 0, we have

lim sup n→∞ T 0 Gu ( 0 ) n (t) 2 L 2 (T) dt ≤ C 2 0 , lim sup n→∞ T 0 Gv ( 0 ) n (t) 2 L 2 (T) dt ≤ C 2 0 . (3.3.7)
For any > 0 small, we can take 0 > 0 small enough such that

|l|≥ 1 0 | g(l)| 2 ≤ 2 ,
and then

g(x) T g(x )u ( 0 ) n (t, x )dx 2 L 2 (T) ≤ 2 g 2 L 2 (T) u ( 0 ) n (0) 2 L 2 (T) .
Thus, from (3.3.5),

u ( 0 ) n (0) 2 L 2 (T) ≤ C 2 + C 2 0 + u ( 0 ) n (0) 2 H -1 (T) ≤ C( 2 + 2 0 ),
holds true for n large enough.

On the other hand, direct calculation yields

T 0 Gv ( 0 ) n (t) 2 L 2 (T) dt = T 0 l 1≤|l 1 |≤1/ 0 ( g(l -l 1 ) -g(l) g(l 1 ))a n,l 1 e it l 3 1 - λ 2 n l 1 2 dt ≥C l 1≤|l 1 |≤1/ 0 | g(l -l 1 ) -g(l) g(l 1 )| 2 |a n,l 1 | 2 =C 1≤|l 1 |≤1/ 0 c l 1 |a n,l 1 | 2 with c l 1 = l | g(l -l 1 ) -g(l) g(l 1 )| 2
, where we have used the Ingham inequality, due to the assumption that λ n → ∞. Notice that the constant C can be chosen independent of n and 0 , provided that we choose n large enough such that

1≤|l 1 |≤1/ 0 (l 1 + 1) 3 -l 3 1 - λ 2 n l 1 + λ 2 n l 1 + 1 ≥ γ > 0 and T > 2π γ .
Note that c l 1 ≥ | g(0) -g(l 1 ) 2 | 2 and g(0) = 1, hence there exists a constant c 0 > 0, independent of 0 , and n, so that c l 1 ≥ c 0 for all 1 ≤ |l 1 | ≤ 1/ 0 . Thus, for sufficiently large n,

v ( 0 )(0) n 2 L 2 (T) ≤ C c 0 T 0 Gv ( 0 ) n (t) 2 L 2 (T) dt ≤ C 2 0 .
Therefore,

1 = lim sup n→∞ u n (0) 2 L 2 (T) = u 0 n (0) 2 L 2 (T) + v 0 n (0) 2 L 2 (T) ≤ C( 2 0 + 2 ) < 1,
which cannot happen.

Reduction to semi-classical observability

Now, we consider the semi-classical equation of the following form:

h∂ t u + (h∂ x ) 3 u -(h∂ x ) -1 u = 0, (t, x) ∈ R × T, u| t=0 = u 0 ∈ L 2 0 (T), (3.3.8)
Proposition 3.7 Assume that there exists T 0 > 0 such that the following semi-classical observability

u 0 2 L 2 (T) ≤ C T 0 0 T |g(x)u(t, x)| 2 dxdt + C u 0 2 H -1 (T) (3.3.9)
holds for any solution u to (3.3.8) with initial data u 0 ∈ L 2 0 , uniformly for 0 < h < 1. Then for any T > 0, observability inequality (3.3.5) holds true.

Proof. It sufficient to prove (3.3.5) when λ > 1 is large enough since for bounded λ ≥ 0, it can be viewed as a pertubation of linear KdV equation and the constant C in front of the right hand side can be chosen to be continuously depended on λ. For λ large enough, we write λ 2 = 1 h 4 and (3.3.3) becomes

h 3 ∂ t u + (h∂ x ) 3 u -(h∂ x ) -1 u = 0.
Consider the scaling in time variable w(t, x) = u(h 2 t, x), we have

h∂ t w + (h∂ x ) 3 w -(h∂ x ) -1 w = 0.
Now from (3.3.9), we have

w(0) 2 L 2 (T) ≤ C T 0 0 T |g(x)w(t, x)| 2 dxdt + C w(0) 2 H -1 (T) ,
changing back to the variable u(t, x),

u(0) 2 L 2 (T) ≤ C h 2 h 2 T 0 0 T |g(x)u(s, x)| 2 dxds + C u(0) 2 H -1 (T) .
Due to the time-translation invariant and conservation of H s -norm under the semi-classical flow, we have for any M ∈ N,

u(M h 2 T 0 ) 2 L 2 T = u(0) 2 L 2 (T) ≤ C h 2 (M +1)h 2 T 0 M h 2 T 0 T |g(x)u(s, x)| 2 dxds + C u(M h 2 T 0 ) 2 H -1 (T) = C h 2 (M +1)h 2 T 0 M h 2 T 0 T |g(x)u(s, x)| 2 dxds + C u(0) 2 H -1 (T) Summing M from 0 to 0 h -2 with 0 T 0 ≤ T , u(0) 2 L 2 (T) ≤ C 0 T 0 T |g(x)u(t, x)| 2 dxdt + C 0 u(0) 2 H -1 (T) .
This completes the proof.

Reduction to frequency localized semi-classical observability

We use a standard homogeneous Littlewood-Paley decomposition. Take

χ ∈ C ∞ c (R) with support suppψ ⊂ {1/2 ≤ |ξ| ≤ 2} and ψ k ∈ C ∞ c (R) such that k∈Z ψ k (ξ) = 1, ∀ξ = 0,
where ψ k (ξ) = ψ(2 k ξ).

Linear controllability on vertical strip 123 Proposition 3.8

There exists 0 > 0, h 0 > 0, small and T 0 > 0, C 0 = C 0 ( 0 ) > 0 such that for all k ∈ Z, with 2 k h ≤ 0 ,

ψ k (hD x )u(0) 2 L 2 (T) ≤ C 0 T 0 0 T |g(x)ψ k (hD x )u(t, x)| 2 dxdt (3.3.10)
holds true for all solutions u(t, x) of (3.3.8), uniformly in h ∈ (0, h 0 ).

We will prove this proposition in the next subsection. In fact, from the proof, we can deduce that if Proposition 3.8 holds true for some 0 > 0, h 0 > 0, it is also true for any other parameter 1 , h 1 such that 1 < 0 and h 1 < h 0 without change of the constant C 0 = C 0 ( 0 ). Lemma 3.2 Proposition 3.8 implies (3.3.9). Indeed, applying Lemma (3.1), we have

gψ k (hD x )u 2 L 2 (T) ≤2 ψ k (hD x )(gu) 2 L 2 (T) + 2 [ψ(2 k hD x ), g]u 2 L 2 (T) ≤2 ψ k (hD x )(gu) 2 L 2 (T) + C 0 (2 k h) 2 u(t) 2 L 2 (T) ,
where the constant C 0 = C 0 ( 0 ) for some fixed but small 0 > 0. Now for any 1 > 0, we have

k≤log 2 ( 1 /h) ψ k (hD x )u(0) 2 L 2 (T) ≤ C 0 k≤log 2 ( 1 /h) T 0 0 ψ k (hD x )(gu(t)) 2 L 2 (T) + C 0 T 0 k≤log 2 ( 1 /h) (2 k h) 2 u(0) 2 L 2 (T) ≤ C 0 T 0 0 gu(t) 2 L 2 (T) dt + C 0 T 0 2 1 u(0) 2 L 2 (T) .
Thus, from Littlewood-Paley decomposition,

u(0) 2 L 2 (T) ≤ C 0 T 0 0 T |g(x)u(t, x)| 2 dxdt + C 0 T 0 2 1 u(0) 2 L 2 (T) + C 0 u(0) 2 H -1 (T) .
We then choose 2 1 < C 0 T 0 2 to obtain (3.3.9).

In summary, we have showed that in order to prove the uniform observability inequality (3.3.6) for all solutions of (3.3.3), it suffices to prove the observability (3.3.10) for all solutions of (3.3.8), uniformly in 0 < h 1 and k ∈ Z such that 2 k h < 0 holds.

Propagation estimate with parameter dependence symbol

This section is devoted to the proof of Proposition 3.8. We need some preparation about h-pseudo-differential calculus. For m ∈ R, let S m be the set of h-dependent functions a(x, ξ, h) with parameter h ∈ (0, 1) such that for any indices α, β, sup (x,ξ, h)∈R 2d ×(0,1)

|∂ α x ∂ β ξ a(x, ξ, h)| ≤ C α,β (1 + |ξ|) m-|β| .
For a ∈ S m , we denote by Op h(a) the h-pseudo-differential operator acting on Schwartz functions via

Op h(a)f (x) := 1 (2π h) d R 2d e i(x-y)•ξ h a(x, ξ, h)f (y)dydξ.
We refer [START_REF] Zworski | Semiclassical analysis[END_REF] for symbolic calculus and another basic properties about h-pseudo-differential operator. For functions on a compact Riemannian manifold, hwe can also define h-pseudo-differential operator by using local coordinate and partition of unity. Now let us consider the following -dependence symbols:

p (x, ξ) = 4 ξ -ξ 3 χ(ξ), q (x, ξ) = 1 ξ -4 ξ 3 χ(ξ), where χ ∈ C ∞ c (R) with supp(ξ) ⊂ {α < |ξ| < β} for some 0 < α < 1 2 , β > 2 and χ ≡ 1 in a neighborhood of {1/2 ≤ |ξ| ≤ 2}. Denote P = Op h (p ) and Q = Op h (q ). Denote U (t), V (t) solution operators to the equations h i ∂ t U (t) + U (t)P = 0, U (0) = I, (3.3.11)
h i ∂ t V (t) + V (t)Q = 0, V (0) = I (3.3.12)
The flow associated to the vector fields H p , H q is explicitly given by

φ ,t (x 0 , ξ 0 ) = x 0 - 4 ξ 2 0 + 3ξ 2 0 χ(ξ 0 )t + 4 ξ 0 -ξ 3 0 χ (ξ 0 )t, ξ 0 , ϕ ,t (x 0 , ξ 0 ) = x 0 - 1 ξ 2 0 + 3 4 ξ 2 0 χ(ξ 0 )t + 1 ξ 0 -4 ξ 3 0 χ (ξ 0 )t, ξ 0
with respectively. From Egorov's theorem (see [START_REF] Zworski | Semiclassical analysis[END_REF]), we know that for any symbol a(x, ξ)

∈ C ∞ c (T * M ), U (-t)Op h (a)U (t) = Op h (a • φ ,t ) + O L 2 →L 2 ( h), V (-t)Op h (a)V (t) = Op h (a • ϕ ,t ) + O L 2 →L 2 ( h).
We remark that the bound O L 2 →L 2 ( h) is independent of ≤ 1 since all the semi-norms of the symbol p , q can be chosen continuously depending on . Now we prove the following localized observability estimates:

Proposition 3.9 There exists C 0 > 0, T 0 > 0, h 0 > 0 such that for all u 0 ∈ L 2 0 (T), all h ≤ h 0 ψ( hD x )u 0 2 L 2 (T) ≤ C 0 T 0 0 gU (t)ψ( hD x )u 0 2 L 2 (T) dt, (3.3.13)
ψ( hD x )u 0 2 L 2 (T) ≤ C 0 T 0 0 gV (t)ψ( hD x )u 0 2 L 2 (T) dt. (3.3.14)
Proof. Here we only prove the first inequality, and the second one will follow in the same manner. Consider the symbol a(x, ξ) = g(x) 2 ψ(ξ) (strictly speaking, g is not smooth and we need approximate it by smoothing functions) and its quantization Op h (a) = (g(x)) 2 ψ( hD x ), where ψ is a slight enlargement of ψ wo that ψψ = ψ and supp ψ ⊂ {α < |ξ| < β}. From Egorov's theorem, we have

U (-t)Op h (a)U (t) = Op h (a • φ ,t ) + O L 2 →L 2 ( h), uniformly in ≤ 1.
Note that on the support of a, χ (ξ) = 0, and thus we have

ϕ ,t (x 0 , ξ 0 ) = x 0 - 4 ξ 2 0 + 3ξ 2 0 t, ξ 0 .
Notice that 4 ξ 2 0 + 3ξ 2 0 ≥ c 0 > 0, uniformly in , on the ξ-support of ψ. Therefore, for some T 0 = T 0 (c 0 ) > 0, and c 1 > 0 , we have

T 0 0 a • φ ,t dt ≥ c 1 > 0.

Now we calculate

T 0 0 gU (t)ψ( hD x )u 0 2 L 2 (T) dt = T 0 0 gU (t)ψ( hD x )u 0 , gU (t) ψ( hD x )ψ( hD x )u 0 L 2 (T) dt = T 0 0 U (-t) ψ( hD x )g 2 U (t)u 0 , ψ( hD x )u 0 L 2 (T) dt = Op h (b T 0 )ψ( hD x )u 0 , ψ( hD x )u 0 L 2 (T) , with b T 0 (x, ξ) = T 0 0 a • φ ,t dt modulo hS 0 . Thus, from Sharp Gårding inequality, Op h (b T 0 )ψ( hD x )u 0 , ψ( hD x )u 0 L 2 (T) ≥ c 1 2 ψ( hD x )u 0 2 L 2 (T) -C h ψ( hD x )u 0 2 L 2 (T) .
To conclude the proof, we just need choose h 0 < min{ c 1 4C , 1}.

Now we prove Proposition 3.8

Proof. Fox fixed h 1, we divide k ∈ Z into three regimes:

Regime I: |k| ≤ N 0 for some large natural number N 0

This regime corresponds to the case |ξ| ∼ 1. Let u k = ψ k (hD x )u, the equation satisfied by u k is simple (3.3.8). In this case, we can either use (3.3.13) or (3.3.14) with parameter = 1 to obtain that (note that h = 2 k h ∼ h in this regime)

ψ k (hD x )u 0 2 L 2 (T) ≤ C 0 T 0 0 T 0 0 gψ k (hD x )u(t) 2 L 2 (T) dt.
Regime II: k ≤ -N 0 for some large constant N 0 Look back to our first micro-localization, this case corresponds to |ξ| ∼ 2 -k 1. Define a new semi-classical parameter h k = 2 k h 1 and rescale the time variable by setting w k (t, x) := ψ( h k D x )u(2 2k t, x), u k = ψ( h k D x )u, we find the equation satisfied by

w k : h k ∂ t w k + ( h k ∂ x ) 3 w k + 2 4k ( h∂ x ) -1 w k = 0.
then by applying (3.3.13) to w k with = 2 k 1 and h = h k we obtain

w k (0) 2 L 2 (T) ≤ C T 0 0 gw k (t) 2 L 2 (T) dt.
From conservation of L 2 -norm along the flow, we apply the inequality above 2 -2k -1 times to obtain

1 2 2k u k (0) 2 L 2 (T) ≤ C 2 2k 2 -2k -1 M =0 (M +1)2 2k T 0 M 2 2k T 0 gu k (t) 2 L 2 (T) dt = C 2 2k T 0 0 gu k (t) 2 L 2 (T) dt,
and this is exactly

ψ k (hD x )u(0) 2 L 2 (T) ≤ C T 0 0 gψ k (hD x )u(t) 2 L 2 (T) dt. Regime III: k ≥ N 0 This case corresponds to |ξ| ∼ 2 -k 1.
Define the new small semi-classical parameter hk = 2 k h, thanks to the restriction that 2 k h ≤ 0 1.

Denote u k = ψ( h k D x )u and define v k (t, x) = u k (2 -2k t, x). v k solves the equation h k ∂ t v k + 2 -4k ( h k ∂ x ) 3 v k + ( h k ∂ x ) -1 v k = 0. Applying (3.3.14) with h = h k , = 2 -k , we obtain that v k (0) 2 L 2 (T) ≤ C T 0 0 gv k (t) 2 L 2 (T) dt.
Again by conservation of L 2 -norm as in the argument of regime II, we finally have

u k (0) 2 L 2 (T) ≤ C T 0 0 gu k (t) 2 L 2 (T) dt.
Once the observability inequality (3.3.5) has been established the internal controllability for the linear KP II is obtained, we conclude this section by summarizing it in the following proposition: Proposition 3.10 For T > 0 given. There exists a bounded linear operator

Υ : (L 2 (T 2 )) 2 → L 2 (0, T ; L 2 (T 2 ))
such that for any u 0 , u T ∈ L 2 (T 2 ), the control defined by h := Υ(u 0 , u 1 ) drives the solution of

∂ t u + ∂ 3 x u + ∂ -1 x ∂ 2 y u = Gh, (t, x) ∈ R × T 2 , u| t=0 = u 0 , (3.3.15)
to u(T ) = u 1 . Moreover, there exists C > 0 such that for any u 0 , u 1 ∈ L 2 (T 2 ), we have

Υ(u 0 , u 1 ) L 2 (0,T ;L 2 (T 2 )) ≤ C (u 0 , u 1 ) (L 2 (T 2 )) 2 .

Local controllability of Nonlinear equation

For the full KP-II control system

∂ t u + ∂ 3 x u + ∂ -1 x ∂ 2 y u + u∂ x u = Gh, (t, x) ∈ R × T 2 , u| t=0 = u 0 , u| t=T = u 1 , (3.4.1)
in order to prove the existence of a u ∈ L 2 (0, T ; L 2 (T 2 )) solving u| t=0 = u 0 , u| t=T = u 1 , we will reduce it to a fix point problem by standard argument. The solution defined by (3.4.1) with control input h is given by

u(t) = S(t)u 0 + υ(t, u) + t 0 S(t -t )Gh(t )dt with υ(t, u) = t 0 S(t -t )u∂ x udt .
It must satisfy

u 1 = S(T )u 0 + v(T, u) + T 0 S(T -t )Gh(t )dt.
Choosing the control input of the form h = Υ(u 0 , w), this implies that

S(T )u 0 + T 0 S(T -t )Gh(t )dt = w.
This indicates that w = u 1 -υ(T, u). In summary, define the nonlinear map Γ by

Γ(u) = S(t)u 0 + υ(t, u) + t 0 S(t -t )Gh u (t )dt with h u = Υ(u 0 , u 1 -v(T, u)),
and we need to find a fix point of Γ.

From (3.2.2) we define the map Γ :

X 0, 1 2 ,b 1 T ∩ Z 0, 1 2 T → X 0, 1 2 ,b 1 T ∩ Z 0, 1 2 T as Γ(u) = S(t)u 0 + t 0 S(t -τ )Gh u dτ + t 0 S(t -τ )u∂ x udτ. (3.4.2)
From the bilinear estimates and linear estimate, we have

Γ(u) X 0, 1 2 ,b 1 T ∩Z 0, 1 2 T ≤ C u 0 L 2 (T) + Gh u X 0,-1 2 ,b 1 T + u 2 X 0, 1 2 ,b 1 T ≤ C u 0 L 2 (T 2 ) + u 1 L 2 (T 2 ) + υ(T, u)(T ) L 2 (T) + u 2 X 0, 1 2 ,b 1 T ≤ C u 0 L 2 (T 2 ) + u 1 L 2 (T 2 ) + u 2 X 0, 1 2 ,b 1 T
and C > 0 does not depend on u 0 . For R > 0, let B R = B R (0) be the ball center at cero with radio R, that is

B R := {u ∈ X 0, 1 2 ,b 1 T ∩ Z 0, 1 2 T : u X 0, 1 2 ,b 1 T ∩Z 0, 1 2 T < R} and Γ(u) X 0, 1 2 ,b 1 T ∩Z 0, 1 2 T ≤ C u 0 L 2 (T 2 ) + u 1 L 2 (T 2 ) + R 2 .
(

Additionally, for u, v ∈ B R we have

Γ(u) -Γ(v) X 0, 1 2 ,b 1 T ∩Z 0, 1 2 T ≤C t 0 S(t -τ )(Gh u -Gh v )dt X 0, 1 2 ,b 1 T ∩Z 0, 1 2 T (3.4.4) + t 0 S(t -t )(u∂ x u -v∂ x v)dt X 0, 1 2 ,b 1 T ∩Z 0, 1 2 T ≤C Υ(u 0 , u 1 -υ(T, u)) -Υ(u 0 , u 1 -υ(T, v))) X 0, 1 2 ,b 1 T ∩Z 0, 1 2 T +C t 0 S(t -t )(u∂ x u -v∂ x v)dt X 0, 1 2 ,b 1 T ∩Z 0, 1 2 T ≤C υ(T, u) -υ(T, v) X 0, 1 2 ,b 1 T ∩Z 0, 1 2 T (3.4.5) + u -v X 0, 1 2 ,b 1 T u + v X 0, 1 2 ,b 1 T ≤C u -v X 0, 1 2 ,b 1 T u + v X 0, 1 2 ,b 1 T (3.4.6) ≤ 1 2 u -v X 0, 1 2 ,b 1 T (3.4.7)
by using properties of the bounded linear operator Υ. Choosing δ > 0 and R > 0 such that 2Cδ + CR 2 ≤ R and CR < 1 2 with u 0 L 2 (T 2 ) < δ and u 1 L 2 (T 2 ) < δ. We can conclude from (3.4.3) that the image of B R through Γ stays in the ball B R and from (3.4.4) that Γ is a contraction.

Non Controllability in horizontal strip

In this section, we will address the exact control problem of linearized KP-II equation (3.1.3) with G = G and prove Theorem 3.3.

We first construct a counterexample of observability inequality for 1D semi-classical Schrödinger equation for short time.

Lemma 3.3

Assume that ω = (-π, α) ∪ (α, π]. Then for any T > 0, there exists a sequence of solutions u n to

ih n ∂ t u n + h 2 n ∂ 2 x u n = 0, u n | t=0 = u n,0 ∈ L 2 (T) (3.5.1) such that lim inf n→∞ u n,0 L 2 (T) > 0 and lim n→∞ T 0 ω |u n (t, x)| 2 dxdt = 0.
Proof. Take G(x) = e -x 2 4 and define

G n (x) = 1 √ n G x n . Denote the Fourier coefficient of G n by g n (k) = 1 2π π -π G n (x)e -ikx dx = √ n 2π π n -π n G(s)e -i nks ds.
The coefficient function g n (z) satisfies the following estimates:

g n L ∞ (R) = O( 1/2 n ), (g n ) L ∞ (R) = O( 3/2 n ), (g n ) L ∞ (R) = O( 5/2 n ). (3.5.2)
Take an even cut-off functionψ

∈ C ∞ c (R) with supp ψ ⊂ [-B, B] and 0 ≤ ψ ≤ 1, ψ| [-b,b] ≡ 1. We define u n,0 = k∈Z g n (k)ψ(h n k)e ikx ,
and then the corresponding solution to (3.5.1) is given explicitly by

u n (t, x) = k∈Z g n (k)ψ(h n k)e i(kx-k 2 hnt) .
We need estimate the mass of initial data. Firstly,

G n 2 L 2 (T) = k∈Z |g n (k)| 2 ∼ 1
holds from Plancherel theorem and the definition of g n (k). We next estimate the mass away from the frequency scale h

-1 n , that is k∈Z |(1 -ψ(h n k))g n (k)| 2 ≤ |k|>h -1 n b |g n (k)| 2 ≤ |k|>h -1 n b n 4π 2 R G(z)e -ik nz dz 2 = |k|>h -1 n b n 4π 2 R G(z) 1 -ik n d dz e -ik nz dz 2 ≤ |k|>h -1 n b 1 4k 2 π 2 n G 2 L 1 (R) . By setting n = √ h n 1, we have (1 -ψ(h n D x ))G n L 2 (T)
1 and then u n,0 L 2 (T) ∼ 1. It remains to estimate the term on the right hand side of observability inequality.

Observe that u n,0 is localized by |k| ≤ B hn in frequency and by |x| ≤ n in space obeying uncertain principal ( n h -1 n 1). Since the wave packet of the frequency scale smaller than Bh -1 n moves at velocity ≤ 2Bh -1 n , it will remain small for |t| < T in ω. More precisely, we need dispersive estimate for |u n (t, x)| when x ∈ ω and |t| < T . Now we choose B > 0 so that |x -2Bt| ≥ c 0 > 0 mod 2π for all x ∈ ω and |t| ≤ T .

Write

u n (t, x)

= k∈Z K (n) t,x (k) with K (n)
t,x (z) = g n (z)ψ(h n z)e i(zx-hnz 2 t) . From Poisson summation formula, we have

u n (t, x) = m∈Z K (n) t,x (2πm).
For fixed m ∈ Z,

K (n) t,x (2πm) = R g n (z)ψ(h n z)e iϕt,x(z) dz = R g n (z)ψ(h n z)L 2 (e iϕt,x(z) )dz with L = 1 iϕ t,x (z) d dz
and ϕ t,x (z) = (x -2πm)z -h n z 2 t. We then integrate by part to get

K (n) t,x (2πm) = R d dz 1 iϕ t,x (z) d dz g n (z)ψ(h n z)
iϕ t,x (z) e iϕt,x(z) dz.

After tedious calculation, we have d dz

1 iϕ t,x (z) d dz g n (z)ψ(h n z) iϕ t,x (z) = (g n ) ψ(h n z) + 2h n (g n ) ψ (h n z) + h 2 n ψ (h n z)g n (ϕ t,x ) 2 - 3((g n ) ψ(h n z) + h n ψ (h n z)g n )ϕ t,x (ϕ t,x ) 3 - 3g n ψ(h n z)(ϕ t,x ) 2 (ϕ t,x ) 4 .
From (3.5.2) and the choice n = √ h n , we have

| K (n) t,x (2πm)| ≤ sup |hnz|≤B C 1/2 n ψ W 2,1 (R) |(x -2h n zt) -2πm| 2 . For any x ∈ 2πp + (-π, -α) ∪ (α, π], |x -2h n zt| ≥ c 0 > 0 module 2π, it holds m∈Z | K (n) t,x (2πm)| ≤C m∈Z C 1/2 n |c 0 -2π(m -p)| 2 ≤C 1/2 n .
Therefore, Suppose ω = T × ((-π, α) ∪ (α, π]) ⊂ T 2 . Then for any T > 0, the observability inequality

T 0 ω |u n (t, x)| 2 dxdt ≤ C 1/2 n T |ω| → 0, as n → ∞.
u(0) 2 L 2 (T 2 ) ≤ C T 0 T 2 |Gu(t, x)| 2 dxdydt
can not hold.

Proof. For any T > 0, we will construct a sequence of solutions u n to the linearized KP-II equation such that

u n (0) L 2 (T 2 ) ∼ O(1) and lim n→∞ T 0 T 2 |Gu n (t, x, y)| 2 dxdydt = 0.
Denote by v n (t, y) the sequence of solutions to the semi-classical Schrodinger equation which satisfies the conditions in Lemma 3.3. Define

u n (t, x, y) = v n (t, y)e it h 3
n e ix hn = k∈Z v n (k)e i(ky-hnk 2 t) e i x hn + t h 3 n .
Then u n solves linearized KP-II equation. Moreover, |v n (t, y)| 2 dtdy → 0, as n → ∞.

u n (0) L 2 (T 2 ) = v n (0) L 2 (T) ∼ O(1),

Now we claim that

lim n→∞ T g(y )v n (t, y )dy → 0 in L ∞ ([0, T ]; L 2 (T)). Indeed, T g(y )v n (t, y )dy = k∈Z g(k)g n (k)ψ(h n k)e -ik 2 t =   |k|≤M + |k|>M   g(k)g n (k)ψ(h n k)e -ik 2 t ≤ 1/2 n g L 2 (T) M 1/2 + G n L 2 (T)   |k|>M | g(k)| 2   1/2
and the right hand side tends to 0 as n → ∞ since we can choose M to be arbitrarily large before taking the limit in n. The validaty of the claim implies that g(y)

T g(y )u n (t, x, y )dy → 0 in L 2 ([0, T ] × T 2
). This completes the proof.

Chapter 4

Exact controllability of linear KP-I equation

Introduction

The precise model considered here is the linear KP-I equation

∂ t u + ∂ 3 x u -∂ -1 x ∂ 2 y u = 0, (4.1.1)
where the Fourier multiplier ∂ -1

x is defined by

∂ -1 x v(k, ) = 1 ik v(k, )
for all functions v ∈ D 0 (T 2 ) := {v ∈ D 0 (T 2 ) : v(0, l) = 0 for all l ∈ Z}.

We denote by L 2 0 (T 2 ) = L 2 (T 2 ) ∩ D 0 (T 2 ). For a vertical control region of the form ω = (a, b) x × T y , we fix a non-negative real function g ∈ C 2 c (T) with T g = 1. In this case, we define the control input by

G(h)(x, y) = G ⊥ (h)(x, y) := g(x) h(x, y) - T g(x )h(x , y)dx . (4.1.2)
The main result of this note is the observability from a vertical region.

Theorem 4.1 For any T > 0, there exists C T > 0, such that for any solution u ∈ C(R; L 2 0 (T 2 )) of (4.1.1), we have

u(0) 2 L 2 (T 2 ) ≤ C T T 0 T 2 |Gu(t, x, y)| 2 dxdydt. (4.1.3)
As explained in chapter 3, a consequence of HUM method of Lions [START_REF] Lions | On some hyperbolic equations with a pressure term, partial differential equations and related subjects: Proceedings of the conference dedicated do l. nirenberg[END_REF] is the exact controllability of linear KP-I equation from vertical domain. Theorem 4.2 Given any T > 0 and u 0 ∈ L 2 0 (T 2 ), u 1 ∈ L 2 0 (T 2 ), there exists f ∈ L 2 ((0, T ); L 2 (T 2 ) such that the solution of the equation Then for any given time T > 0, the similar observability (4.1.3), replacing Gu by G u, does not hold true. The argument is the same as the treatment for linear KP-II in chapter 3.

∂ t u + ∂ 3 x u -∂ -1 x ∂ 2 y u = Gf, u| t=0 = u 0 ( 4
The main part of this note is devoted to the proof of Theorem 4.2. In appendix, we discuss the validity of the observability for fractional linear KP I of the form

∂ t u -|D x | α ∂ x u -∂ -1
x ∂ 2 y u = 0. (4.1.6)

We will prove the following dichotomy result which asserts the least dispersion needed for the observability.

Theorem 4.3 1. If α ≥ 1, then for any T > 0, there exists C T > 0, such that

u(0) 2 L 2 (T 2 ) ≤ C T T 0 T 2 |Gu(t, x, y)| 2 dxdydt
holds for any solution u of (4.1.6).

2. If 0 < α < 1, then for any T > 0, there exist a sequence of solution (u n ) of (4.1.6), such that

lim n→∞ T 0 T 2 |Gu n (t, x, y)| 2 dtdxdy u n (0) 2 L 2 (T 2 ) = 0.

Notations and Preliminaries

Notation

We identify T d = R d /(2πZ d ) with fundamental domain [-π, π] d . The Fourier transform on T d is denoted by

f (ξ) = (2π) -d T d f (z)e -iz•ξ dz, ξ ∈ Z d .
In the case where there is no risk of confusing, we will also use f to note the Fourier transform of one variable. For the derivative, we sometimes use the notation

D t = 1 i ∂ t , D x = 1 i ∂ x .
We will only use L 2 based norms for this linear problem, hence we denote by

v := v L 2 (T d ) , v s := v H s (T d ) , f T := f L 2 (0,T ;L 2 (T d )) .
We will also use the inner product notations

(u, v) := T d u(x) • v(x)dx, (f, w) T := T 0 T d f (t, x)w(t, x)dxdt,
where d = 1 or 2, which will be clear in the context.

Symbols and quantization on Torus

We briefly review the h pseudo-differential calculus on torus. For m ∈ R, let S m be the set of h-dependent functions a(x, ξ, h) with parameter h ∈ (0, 1) such that for any indices α, β, sup We refer [START_REF] Zworski | Semiclassical analysis[END_REF] for symbolic calculus and another basic properties about h pseudo-differential operator. For functions on a compact Riemannian manifold, we can define h pseudodifferential operator by using local coordinate and partition of unity. On the torus, we can also use the global definition of pseudo-differential calculus. Denote by S m per be symbols in

S m (R 2d ) which are 2π-periodic in x ∈ R d , namely a(x + 2πk, ξ) = a(x, ξ), ∀(x, ξ) ∈ R 2d , k ∈ Z d .

Symbols in S m

per can depend on h with uniform estimate (4.2.1),though the dependence is not displayed in our notation. We quantize a ∈ S m per as an operator on S (T d ) via the formula

Op h (a)f (x) := k∈Z d 1 (2πh) d [-π,π] d R d a(x, ξ)e i(x-y+2kπ)•ξ h f (y)dydξ (4.2.2)
From Poisson summation formula, we have

Op h (a)f (x) = k∈Z d a(x, h 1 k) f (k)e ik•x . (4.2.
3)

The globally defined quantization via (4.2.3) is equivalence to (modulo hS m-1 ) the usual definition via partition of unity, see the exercise in the book [START_REF] Alinhac | Opérateurs pseudo-différentiels et théorème de Nash-Moser[END_REF].

Quick review of 1D semi-classical reduction

Expanding the solution u(t, x, y) to (4.1.1) in Fourier series in y variable u(t, x, y) = l∈Z a l (t, x)e ily , we find that for each l ∈ Z, a l satisfies the equation

∂ t a l + ∂ 3 x a l -l 2 ∂ -1
x a l = 0 Therefore, by changing notations, the equation (4.1.1) can be reduced to the study of the following λ dependent equation

∂ t u + ∂ 3 x u + λ 2 ∂ -1 x u = 0, (t, x) ∈ R × T, u| t=0 = u 0 ∈ L 2 0 (T), (4.2.4)
We take λ = 1 h 2 and rewrite (4.2.4) as

h 3 D t u -(hD x ) 3 u -(hD x ) -1 u = 0, (t, x) ∈ R × T, u| t=0 = u 0 ∈ L 2 0 (T), (4.2.5)
The solution u depends on the parameter h and we will drop the dependence in the sequel.

From the same proof of Proposition 3.6 in chapter 3, we reduce the proof of Theorem 4.1 to the following weak observability.

Theorem 4.4 T > 0 be given. There exist a constant C T > 0 and a sufficiently small number h 0 > 0, such that for all h ∈ (0, h 0 ), the solution u of the h dependent equation (4.2.5) satisfies

u 0 2 ≤ C T T 0 T |g(x)u(t, x)| 2 dxdt + C T u 0 2 -1 . (4.2.6)
We use a standard homogeneous Littlewood-Paley decomposition. Take

ψ ∈ C ∞ c (R) with support suppψ ⊂ {1/2 ≤ |ξ| ≤ 2} and ψ n ∈ C ∞ c (R) such that n∈Z ψ n (ξ) = 1, ∀ξ = 0,
where ψ n (ξ) = ψ(2 n ξ). With this notation, we further reduce the proof of Theorem 4.4 to the following frequency-localized estimate.

Proposition 4.1 Let T > 0 and 0 > 0 be given. There exist h 0 > 0, small and C 0 = C 0 ( 0 , T ) > 0 such that for all n ∈ Z which subject to 2 n h ≤ 0 ,

ψ n (hD x )u(0) 2 ≤ C 0 T 0 0 T |g(x)ψ n (hD x )u(t, x)| 2 dxdt (4.2.7)
holds true for all solutions u(t, x) of (4.2.5).

The derivation from Proposition 4.1 to Theorem 4.4 is simple and can be found in chapter 3. The remaining part of this note is devoted to the proof of (4.2.7). We summarize the path of the proof as follows:

• Regimes n ≥ N 0 and n ≤ -N 0 : n ≤ -N 0 corresponds to the very low frequency regime in which the term (hD x ) -1 dominates the dispersion. n ≥ N 0 corresponds to the very high frequency regime in which the term (hD x ) 3 dominates the dispersion. The arguments are similar as for linear KP-II.

• Regime |n| ≤ N 0 : This is the essential difference between KP-I and KP-II. The group velocity of KP-I could be very small in this regime.

4.3

The proof of Proposition 4.1

Regimes far from critical points

Let us consider the following -dependence symbols:

p (x, ξ) = 4 ξ + ξ 3 χ(ξ), q (x, ξ) = 1 ξ + 4 ξ 3 χ(ξ),
where χ ∈ C ∞ c (R) with supp(χ) ⊂ {µ < |ξ| < ν} for some 0 < µ < 1 2 , ν > 2 and χ ≡ 1 in a neighborhood of {1/2 ≤ |ξ| ≤ 2}. Denote by P = Op h (p ) and Q = Op h (q ).We use the notations U (t), V (t) to represent solution operators to the following two equations

h i ∂ t U (t) + U (t)P = 0, U (0) = I, (4.3.1)
h i ∂ t V (t) + V (t)Q = 0, V (0) = I (4.3.2)
The flows associated to the vector fields H p , H q are explicitly given by φ ,t (x 0 , ξ 0 ) = x 0 + -

4 ξ 2 0 + 3ξ 2 0 χ(ξ 0 )t + 4 ξ 0 + ξ 3 0 χ (ξ 0 )t, ξ 0 , ϕ ,t (x 0 , ξ 0 ) = x 0 + - 1 ξ 2 0 + 3 4 ξ 2 0 χ(ξ 0 )t + 1 ξ 0 + 4 ξ 3 0 χ (ξ 0 )t, ξ 0
with respectively.

From Egorov's theorem (see [START_REF] Zworski | Semiclassical analysis[END_REF]), we know that for any symbol a(x, ξ)

∈ C ∞ c (T * M ), U (-t)Op h (a)U (t) = Op h (a • φ ,t ) + O L 2 →L 2 ( h), V (-t)Op h (a)V (t) = Op h (a • ϕ ,t ) + O L 2 →L 2 ( h).
We remark that the bound O L 2 →L 2 ( h) is independent of ≤ 1 since all the semi-norms of the symbol p , q can be chosen continuously depending on . Now we prove the following localized observability estimates:

Proposition 4.2
There exists C 0 > 0, T 0 > 0, h 0 > 0 and δ 0 > 0 such that for all u 0 ∈ L 2 0 (T), and all h ≤ h 0

ψ( hD x )u 0 2 ≤ C 0 T 0 0 gU (t)ψ( hD x )u 0 2 dt, (4.3.3)
ψ( hD x )u 0 2 ≤ C 0 T 0 0 gV (t)ψ( hD x )u 0 2 dt, (4.3.4)
uniformly in < δ 0 .

Proof. Here we only prove the first inequality, and the second one will follow in the same manner. Consider the symbol a(x, ξ) = g(x) 2 ψ(ξ) (strictly speaking, g is not smooth and we need approximate it by smoothing functions) and its quantization Op h (a) = (g(x)) 2 ψ( hD x ), where ψ is a slight enlargement of ψ so that ψψ = ψ and suppχ| supp( ψ) = 1. From Egorov's theorem, we have

U (-t)Op h (a)U (t) = Op h (a • φ ,t ) + O L 2 →L 2 ( h), uniformly in ≤ 1.
Note that on the support of a, χ (ξ) = 0, and thus we have

ϕ ,t (x 0 , ξ 0 ) = x 0 + - 4 ξ 2 0 + 3ξ 2 0 t, ξ 0 .
We choose δ 0 = δ 0 (µ, ν), sufficiently small, such that -4 ξ 2 0 + 3ξ 2 0 ≥ c 0 > 0, uniformly in < δ 0 on the ξ-support of ψ. Therefore, for some T 0 = T 0 (c 0 ) > 0, and c 1 > 0 , we have

T 0 0 a • φ ,t dt ≥ c 1 > 0.

Now we calculate

T 0 0 gU (t)ψ( hD x )u 0 2 dt = T 0 0 gU (t)ψ( hD x )u 0 , gU (t) ψ( hD x )ψ( hD x )u 0 dt = T 0 0 U (-t) ψ( hD x )g 2 U (t)u 0 , ψ( hD x )u 0 dt = Op h (b T 0 )ψ( hD x )u 0 , ψ( hD x )u 0 , with b T 0 (x, ξ) =
T 0 0 a • φ ,t dt modulo hS 0 . Thus, from Sharp Gårding inequality,

Op h (b T 0 )ψ( hD x )u 0 , ψ( hD x )u 0 ≥ c 1 2 ψ( hD x )u 0 2 -C h ψ( hD x )u 0 2 .
To conclude the proof, we just need choose h 0 < min{ c 1 4C , 1}.

As a consequence, we can proof Proposition 4.1 in the easy regimes:

Corollary 4.1 There exist h 0 > 0 and a integer N 0 which depends on δ 0 in Proposition 4.2, such that for all h < h 0 , |n| ≥ N 0 , and 2 n h ≤ 0 , the inequality (4.2.7) holds true.

Proof. Take N 0 ∈ N such that 2 -N 0 < 2 -10 δ 0 . Fix h 0 < min{2 -N 0 0 , h 0 } > 0. We first consider the case n ≤ -N 0 . Define a new semi-classical parameter h n = 2 n h 1 and rescale the time variable by setting w n (t, x) := ψ( h n D x )u(2 2n t, x). w n satisfies the following equation:

h n ∂ t w n + ( h n ∂ x ) 3 w n + 2 4n ( h n ∂ x ) -1 w n = 0.
Applying (4.3.3) to w n with = 2 n ≤ δ 0 and h = h n we obtain that

w n (0) 2 ≤ C T 0 0 gw n (t) 2 dt.
From conservation of L 2 norm along the flow, we apply the inequality above 2 -2n -1 times to obtain

1 2 2n u n (0) 2 ≤ C 2 2n 2 -2n -1 M =0 (M +1)2 2n T 0 M 2 2n T 0 gu k (t) 2 dt = C 2 2n T 0 0 gu n (t) 2 dt,
and this is exactly

ψ n (hD x )u(0) 2 ≤ C T 0 0 gψ n (hD x )u(t) 2 dt.
We next consider the case n ≥ N 0 and 2 n h ≤ 0 . Define the new small semi-classical parameter h n = 2 n h, thanks to the restriction that 2 n h ≤ 0 1. Denote by u n = ψ( h n D x )u and define v n (t, x) = u n (2 -2k t, x). Thus v n solves the equation

h n ∂ t v n + 2 -4n ( h n ∂ x ) 3 v n + ( h n ∂ x ) -1 v n = 0.
Applying (4.3.4) with h = h n , = 2 -n , we obtain that

v n (0) 2 ≤ C T 0 0 gv n (t) 2 dt.
Again from conservation of L 2 norm as in the previous argument, we finally have

u n (0) 2 ≤ C T 0 0 gu n (t) 2 dt.

Near the critical points

Now we prove inequality (4.2.7) for |n| ≤ N 0 . Since N 0 only depends on µ, ν > 0 which is chosen in a priori, it would be suffices to prove the inequality for n = 0 only. Rewriting (4.2.5) as h 3 D t u -φ(hD x )u = 0, u = ψ(hD x )u, with Fourier multiplier φ(ξ) = ξ 3 + 1 ξ . There are only two zeros of φ (ξ) = 3ξ 2 -1 ξ 2 , say ξ 0 = ± 1 4 √ 3 . Splitting ψ(ξ) = ψ + (ξ) + ψ -(ξ) with ψ + = ψ1 ξ>0 , and ψ -= ψ1 ξ<0 , it would be sufficient to prove (4.2.7) for u = ψ + (hD x )u. For δ > 0, we take another cut-off

χ δ ∈ C ∞ c (R) such that χ δ (ξ)| |ξ-ξ 0 |≤δ ≡ 1, χ δ (ξ)| |ξ-ξ 0 |>2δ ≡ 0.
Taking δ > 0 smaller, we may assume that χ δ (ξ)ψ(ξ) = χ δ (ξ). On the support of (1-χ δ )ψ + , we have |ψ (ξ)| ≥ c δ > 0, thus the same propagation argument as in the previous subsection yields

(1 -χ δ (hD x ))ψ + (hD x )u(0) 2 ≤ C δ T δ 0 g(1 -χ δ (hD x ))ψ + (hD x )u(t) 2 dt (4.3.5)
for some C δ , T δ depending on δ > 0. To complete the proof, it remains to prove the similar inequality for χ δ (hD x )u. Indeed, the sum of the two frequency pieces on the right hand side can be bounded by gψ + (hD x )u 2 T δ +C δ T h u(0) 2 in which the error term comes from the commutator [g, χ δ (hD x )].

Before treating the term χ δ (hD x )u, we make a further simplification. Denote by v = χ δ (hD x )u, v = e i ξ 0 h x w, and then w(k) = v k + ξ 0 h . We see that

h 3 D t w -Φ(hD x )w = 0, w(0) = θ δ (hD x )w 0 , (4.3.6)
with

Φ(ξ) = φ ξ + h ξ 0 h , θ δ (ξ) = χ ξ + h ξ 0 h .
Note that the support of θ δ is now near the origin and

φ (σ h ) = 0, Φ (σ h ) = 12 4 √ 3 > 0, σ h = h ξ 0 h - ξ 0 h .
We are now ready to close the demonstration of Proposition 4.1 by proving the following, which is the main ingredient of this note:

Proposition 4.3
There exist constants δ > 0, h 0 > 0 small and C T > 0 such that for all 0 < h < h 0 and w = θ δ (hD x )w, h dependent solution to (4.3.6), we have

w(0) 2 ≤ C T T 0 T g(x)w(t) 2 dt.
The proof is down by splitting the frequency into two parts. One part contains cluster of relatively low frequencies and we control it by spectral inequality. The other part contains relatively high frequencies and will be controlled by propagation estimate after rescaling the time variable. First we notice that the inequality would not change if we replace w by w exp iΦ(σ h )t h 3

. We may assume that Φ(σ h ) = 0. Denote by r h = h -1 σ h ∈ [0, 1). For any n 0 ∈ N, we define the sharp frequency truncation

Π ≥n 0 f := |k|≥n 0 f (k)e ikx .
We divide the proof into several lemmas. Lemma 4.1 Given T > 0, there exist N 0 ∈ N, h 0 > 0, C T > 0, such that for any integer n 0 ≥ N 0 , h < h 0 and T > 0,

Π ≥n 0 w(0) 2 ≤ C T T 0 g(x)w(t) 2 dt (4.3.7)
holds true for all solutions of (4.3.6).

Proof. We rewrite

Π ≥n 0 w(0) = l 0 ≤l≤L 0 ψ(2 l hD x )Π ≥n 0 w(0), with 2 -l 0 = 4δ, 2 -L 0 = n 0 h 4 .
From almost orthogonal inequality

Π ≥n 0 w(0) 2 ≤ 4 l≤l≤L 0 ψ(2 l hD x )w(0) 2 ,
we need estimate each term in the summation. By choosing n 0 ≥ N 0 large, we denote by h 1 = 2 l h ≤ 4 n 0 a new semi-classical parameter. We put ω = ψ(h 1 D x )w, and then ω solves

2 -l h 3 1 D t ω -Φ l (h 1 D x )ω = 0, with Φ l (ξ) = 2 2l Φ(2 -l ξ).
Note that Φ l is a symbol with uniform bound in l for |ξ| ≤ 2 as well as all of its derivatives. We rescale the time by setting v(t, x)

:= ω(2 -l h 2 1 t, x), hence h 1 D t v -Φ l (h 1 D x )v = 0. Notice that |∂ ξ Φ l (ξ)| = |2 l Φ (2 -l ξ)| ∼ 1 for ξ ∈ suppψ.
From the same argument as in the proof of Proposition 4.2, there exist T 0 > 0 and C T 0 > 0 such that

ψ(h 1 D x )v(0) 2 L 2 ≤ C T 0 T 0 0 T |g(x)ψ(h 1 D x )v(t, x)| 2 dxdt
holds true for all h 1 = 2 l h, provided that h < h 0 is small enough and n 0 ≥ N 0 is large(while keeping the relation hn 0 1). Back to the function w, we have

ψ(h 1 D x )w(0) 2 L 2 = ψ(h 1 D x )v(0) 2 L 2 ≤C T 0 T 0 0 T |g(x)ψ(h 1 D x )v(t, x)| 2 dxdt = C T 0 h 1 h h 1 hT 0 0 T |g(x)ψ(h 1 D x )w(t, x)| 2 dxdt.
Thanks to the conservation of L 2 norm, we have for all q ∈ N,

ψ(h 1 D x )w(0) 2 L 2 = ψ(h 1 D x )w(qh 1 hT 0 ) 2 L 2 ≤ C T 0 h 1 h (q+1)h 1 hT 0 qh 1 hT 0 T |g(x)ψ(h 1 D x )w(t, x)| 2 dxdt.
Summing q from 0 to T h 1 hT 0 , we have

ψ(h 1 D x )w(0) 2 L 2 ≤ C T T 0 T |g(x)ψ(h 1 D x )w(t, x)| 2 dxdt. (4.3.8)
Thus

l 0 ≤l≤L 0 ψ(2 l hD x )w(0) 2 L 2 ≤C T l 0 ≤l≤L 0 T 0 T |g(x)ψ(2 l hD x )w(t, x)| 2 dxdt ≤ l 0 ≤l≤L 0 C T T 0 T |ψ(2 l hD x )(gw)(t, x)| 2 + 2 2l h 2 |w(t, x)| 2 dxdt ≤C T T 0 T |gw(t, x)| 2 dxdt + C T n 2 0 w(0) 2 L 2 ,
where we have used the simple commutator estimate [g, ψ(2 l hD x )] L 2 →L 2 ≤ C2 l h in the second inequality. This completes the proof by choosing n 0 sufficiently large.

We need the following spectral inequality, and the proof is classical and can be found in [START_REF] Lebeau | Introduction aux inégalités de carleman[END_REF]. (4.3.9)

The following elementary lemma is needed in the final argument.

Lemma 4.3

For any r ∈ [0, 1), there exist µ 1 , µ 2 ∈ 1 8 , 7 8 , such that µ 1 + µ 2 = 2r mod Z.

(4.3.10)

Proof. We denote by {x} := x -x , the fractional part of a real number. If

Put M 0 := h -1 Φ(h(n 0 + µ 2 )).
Recall that Φ is strictly increasing for ξ ∈ [σ h , δ) and strictly decreasing for ξ ∈ (-δ, σ h ]. Thus there exist ξ * < r h , such that Φ(ξ * ) = h 2 M 2 0 . We fix δ > 0 small such that 2 < Φ (ξ) < 12 for all |ξ| ≤ δ. Thus (n 0 +µ 2 ) 2 ≤ 9. We claim that for sufficiently small h > 0, we have

ξ * h = m 0 , ξ * h -µ 1 < 1 16 .
Indeed, Taylor expansion gives , w H = w -w L .

Φ (σ h )h 2 2 ξ * h -r h 2 = Φ (σ h )h 2 2 (n 0 + µ 2 -r h ) 2 + O(h 3
From (4.3.9) and the property of Φ, we have We next calculate

w L (0) 2 ≤ κ(n 0 + 2)T T 0 T |g(x)w L (t, x)| 2 dxdt. ( 4
T 0 T g(x)w H (t, x) • g(x)w L (t, x)dxdt = T 0 T g(x) 2 |Φ(hk 1 )|≤h 2 M 2 0 |Φ(hk 2 )|>h 2 M 2 0 w 0 (k 1 ) w 0 (k 2 )e -i(k 1 -k 2 )x e it h 3 (Φ(hk 2 )-Φ(hk 1 )) dxdt = |Φ(hk 1 )|≤h 2 M 2 0 |Φ(hk 2 )|>h 2 M 2 0 2π G(k 1 -k 2 ) w 0 (k 1 ) w 0 (k 2 )h 3 e iT h 3 (Φ(hk 2 )-Φ(hk 1 )) -1 Φ(hk 2 ) -Φ(hk 1 ) ≤Ch 3 |Φ(hk 1 )|≤h 2 M 2 0 |Φ(hk 2 )|>h 2 M 2 0 | G(k 1 -k 2 ) w 0 (k 1 ) w 0 (k 2 )| 1 Φ(hk 2 ) -Φ(hk 1 ) with G(x) = g(x) 2 . If (k 1 -r h )(k 2 -r h ) ≥ 0, we have |Φ(hk 2 ) -Φ(hk 1 )| = hk 2 hk 1 Φ (t)dt ≥ h|Φ (hk 1 )||k 2 -k 1 | ≥ ch 2 |k 2 -k 1 |, in the case |k 1 -r h | ≥ 1. If otherwise |k 1 -r h | < 1, we directly estimate |Φ(hk 2 )| -|Φ(hk 1 )| ≥ h 2 M 2 0 -sup ξ∈[-2h,2h] |Φ (ξ)| > h 2 M 2 0 2
by taking n 0 reasonable. There are two possibilities in the case of (k 1 -r h )(k 2 -r h ) < 0:

either k 2 ≥ n 0 + 1, ξ * h ≤ k 1 < r h , or k 2 < ξ * h , r h < k 1 ≤ n 0 .
For the first case, we have

Φ(hk 2 ) -Φ(hk 1 ) ≥Φ(hk 2 ) -Φ(ξ * ) = Φ (σ h )h 2 2 k 2 - ξ * h k 2 + ξ * h -2r h + O(h 3 M 3 0 ) ≥ Φ (σ h )h 2 2 (k 2 -k 1 )(k 2 + m 0 + µ 1 -2r h ) + O(h 3 M 3 0 ) ≥ Φ (σ h )h 2 2 |k 2 -k 1 |(n 0 + 1 + m 0 + µ 1 -2r h ) + O(h 3 M 2 0 ) ≥ |k 2 -k 1 |h 2 16
by choosing h small enough, thanks to (4.3.11), (4.3.10) and (4.3.11). In the case that

k 2 < ξ * h , we have Φ(hk 2 ) -Φ(hk 1 ) ≥ Φ (σ h )h 2 2 (k 1 -k 2 )(-k 2 -k 1 + 2r h ) + O(h 3 M 3 0 ) ≥h 2 |k 2 -k 1 |(-m 0 -n 0 + 2r h ) + O(h 3 M 2 0 ) ≥ |k 1 -k 2 |h 2 16 .
This implies that

T 0 T g(x) 2 w H (t, x) • w L (t, x)dxdt ≤Ch k =k 2 | G(k 1 -k 2 )|| w 0 (k 1 )|| w 0 (k 2 )| ≤Ch k∈Z | G(k)| w 0 2 ,
where we have used Young's convolution inequality. From this, we could improve the estimate of w L (0) 2 as follows.

w L (0) 2 ≤κ(n 0 + 2)T T 0 T |gw L (t, x)| 2 dxdt =κ(n 0 + 2)T T 0 T |g(x)w(t, x)| 2 dxdt -κ(n 0 + 2)T T 0 T |g(x)w H (t, x)| 2 dxdt -2κ(n 0 + 2)T Re T 0 T g(x)w H (t, x) • g(x)w L (t, x)dxdt ≤κ(n 0 + 2)T T 0 gw(t) 2 dt + Chκ(n 0 + 2)T w(0) 2 ,
and

w(0) 2 = w L (0) 2 + w H (0) 2 ≤ (C T + κ(n 0 + 2)T ) T 0 gw(t) 2 dt + Chκ(n 0 + 2) w(0) 2 .
The last term on the right hand side can be absorbed to the left, and this completes the proof.

4.A On the observability of fractional linear KP I

In this appendix, we will give a proof of Theorem 4.3 for fractional KP I equation

∂ t u -|D x | α ∂ x u -∂ -1 x ∂ 2 y u = 0. (4.A.1)
When α ≥ 1, the proof of observability can be reduced to the 1D uniform observability of

h α+1 D t v -Φ α (hD x )v = 0, v = χ δ (hD x )v, with Φ δ (ξ) = φ α ξ + h ξ 0 h , φ α (ξ) = |ξ| α ξ + 1 ξ
, after doing the same reduction as for the linear KP-I. Thus it would be sufficient to prove Proposition 4.3 for solutions of (4.A.1). Actually, the proof of Proposition 4.3 works also in the case α > 1. For α = 1, we need a little more argument.

Taylor expansion gives

Φ 1 (ξ) = Φ 1 (σ h ) 2 (ξ -σ h ) 2 + Φ 1 (θ) 6 (ξ -σ h ) 3 .
Note that Φ 1 (σ h ) = φ 1 (ξ 0 ) = 2A 0 is independent of h, and we have

Φ 1 (hD x ) = h 2 A 0 (D x -r h ) 2 + O( Φ 1 L ∞ ))((hD x -σ h ) 3 ). with r h = σ h h ∈ [0, 1). For 0 < δ 1 , we decompose v = v 1 + v 2 , v 1 = χ Aδ (h 1/3 D x )χ δ (hD x )v. Then v 1 solves D t v 1 -A 0 (D x -σ h ) 2 v 1 = O L 2 →L 2 (Aδ)v 1 .
We denote by S σ h (t) the semi-group associated with the evolution Schrödinger operator D t -A 0 (D x -σ h ) 2 . From observability for classical Schrödinger equation, we have

v 1 (0) 2 ≤ C T T 0 gS σ h (t)v 1 (0) 2 dt
with constant C T independent of σ h ∈ (0, 1). For this independence assertion, we refer to Lemma 2.4 of [START_REF] Burq | Control for schrödinger operators on tori[END_REF]. Therefore, we have from Duhamel formula that

v 1 (0) 2 ≤C T T 0 gv 1 (t) -g t 0 O L 2 →L 2 (δ)v 1 (t )dt 2 dt ≤C T T 0 gv 1 (t) 2 dt + AC T δ v 1 2 T ≤C T T 0 gv 1 (t) 2 dt + AC T δ v 1 (0) 2 , (4.A.2)
where we have used the conservation of L 2 norm in the last step. For given T > 0, we take δ > 0 sufficiently small in a priori, and thus

v 1 (0) 2 ≤ C T T 0 gv 1 (t) 2 dt.
The estimate of v 2 follows in the same way as in the proof of Lemma 4.1. Therefore we have

v 2 (0) 2 ≤ C T T 0 gv 2 (t) 2 dt.
Finally, from the commutator estimate χ Aδ (h 1/3 D x )χ δ (hD x ), g L 2 →L 2 ≤ Ch 1/3 , the proof is complete.

We now construct the conterexample of observability for the case α < 1. The construction is in the same spirit as in [START_REF] Sun | Internal controllability of non-localized solution for the kodomtsev-petviashvili ii equation[END_REF]. Proposition 4.4 Suppose 0 < α < 1. Then for any T > 0, there exists a sequence v n , solutions of

h 1+α n D t v n + Φ 1 (h n D x )v n = 0, such that lim n→∞ T 0 ω |v n (t, x)| 2 dxdt T |v n (0, x)| 2 dx = 0.
Proof. We may assume that ω

= (-π, -β) ∪ (β, π]. Take G(x) = e -x 2 2 and define G n (x) = 1 √ n G x n . Denote the Fourier coefficient of G n by g n (k) = 1 2π π -π G n (x)e -ikx dx = √ n 2π π n -π n G(z)e -i nkz dz.
The coefficient function g n (z) satisfies the following estimates:

g n L ∞ (R) = O( 1/2 n ), (g n ) L ∞ (R) = O( 3/2 n ), (g n ) L ∞ (R) = O( 5/2 n ). (4.A.3) Take an even cut-off function ψ ∈ C ∞ c (R) with supp ψ ⊂ [-B, B] and 0 ≤ ψ ≤ 1, ψ| [-b,b] ≡ 1. We define v n,0 = k∈Z g n (k)ψ( h n k)e ikx , with h n = h 1-α n .
The corresponding solution is given explicitly by

v n (t, x) = k∈Z g n (k)ψ( h n k) exp   ikx - itΦ 1 ( h 1 1-α n k) h 1+α 1-α n   .
We first estimate the lower bound of the mass of initial data.

G n 2 L 2 (T) = k∈Z |g n (k)| 2 ∼ 1
holds from Plancherel theorem and the definition of g n (k). We next estimate the mass away from the frequency scale h

-1 n , that is k∈Z (1 -ψ( h n k))g n (k) 2 ≤ | hnk|>b |g n (k)| 2 ≤ | hnk|>b n 4π 2 R G(z)e -ik n zdz 2 = | hnk|>b n 4π 2 R G(z) 1 -ik n d dz e -ik n zdz 2 ≤ | hnk|>b 1 4k 2 π 2 n G 2 L 1 (R)
.

By setting

n = h n 1, we have (1 -ψ( h n D x ))G n L 2 (T) 1 and then v n,0 L 2 (T) ∼ 1. Now we choose B > 0 so that |x -Φ 1 L ∞ ([-δ,δ]) Bt| ≥ 2c 0 > 0 mod 2π for all x ∈ ω = (-π, -β) ∪ (β, π) and |t| ≤ T . Write v n (t, x) = k∈Z K (n) t,x (k) with K (n) t,x (z) = g n (z)ψ( h n z) exp izx -i h -1+α 1-α n Φ 1 ( h 1 1-α n z)t .
From Poisson summation formula, we have

v n (t, x) = m∈Z K (n) t,x (2πm). For fixed m ∈ Z, K (n) t,x (2πm) = R g n (z)ψ(h n z)e iϕt,x(z) dz = R g n (z)ψ(h n z)L 2 (e iϕt,x(z) )dz with L = 1 iϕ t,x (z) d dz and ϕ t,x (z) = (x -2πm)z -h -1+α 1-α n Φ 1 ( h 1 1-α n z)t. Thus K (n) t,x (2πm) = R d dz 1 iϕ t,x (z) d dz g n (z)ψ( h n z) iϕ t,x (
z) e iϕt,x(z) dz.

After tedious calculation, we have d dz

1 iϕ t,x (z) d dz g n (z)ψ( h n z) iϕ t,x (z) = (g n ) ψ( h n z) + 2 h n (g n ) ψ ( h n z) + h 2 n ψ ( h n z)g n (ϕ t,x ) 2 - 3((g n ) ψ( h n z) + h n ψ ( h n z)g n )ϕ t,x (ϕ t,x ) 3 - 3g n ψ( h n z)(ϕ t,x ) 2 (ϕ t,x ) 4 .
From (4.A.3), we have

| K (n) t,x (2πm)| ≤ sup | hnz|≤B C 1/2 n ψ W 2,1 (R) (x -2πm) -h -α 1-α n Φ 1 ( h 1 1-α n z)t 2 .
Note that from Taylor expansion,

h -α 1-α n Φ 1 ( h 1 1-α n z)t = Φ 1 (θ n ) h n zt -Φ 1 (θ n )σ hn h -α 1-α n t for some θ n ∈ (σ hn , h 1 1-α n z).
Therefore, for sufficiently large n, and for any

x ∈ 2πZ + (-β, -α) ∪ (β, π], x -h -α 1-α n Φ 1 ( h 1 1-α n z)t ≥ c 0 > 0 module 2π, thus m∈Z | K (n) t,x (2πm)| ≤C m∈Z C 1/2 n |c 0 -2π(m -p)| 2 ≤C 1/2 n holds for any p ∈ Z. Therefore, T 0 ω |v n (t, x)| 2 dxdt ≤ C 1/2 n T |ω| → 0, as n → ∞.
Corollary 4.2 Suppose 0 < α < 1, then for any T > 0, the observability for u n , solutions of (4.A.1) does not hold true.

Proof. We take h n , v n as in Proposition 4.4. Define

u n (t, x, y) = w n (t, x) exp iyh -α+2 2 n
, where (h

-α+2 2 n
) is a sequence of positive integers which converges to infinity. u n solves (4.A.1) means that This is guranteed by

h α+1 n ∂ t w n -|h n D x | α h n ∂ x w n -h -1 n ∂ -1 x w n = 0.
T g(x )u n (t, x , y)dx → 0, in L 2 ((0, T ) × T y )
from our construction. This completes the proof.

Chapter 5

Control and Stabilization of Nonlinear KP-II type equations

Introduction

The precise models considered here are KP-II equation

∂ x (∂ t u + ∂ 3 x u + u∂ x u) + ∂ 2 y u = 0, (t, x, y) ∈ R × T 2 (5.1.1)
and fifth order KP-II equation, or 5KP-II in short,1

∂ x (∂ t u -∂ 5 x u + u∂ x u) + ∂ 2 y u = 0, (t, x, y) ∈ R × T 2 . (5.1.2)
These equations arise naturally in the study of transverse stability of the solitary wave solutions of the Kortewed de-Vries (KdV) equation and its higher order analogues. They have been widely studied in many different contexts( [START_REF] Bb Kadomtsev | On the stability of solitary waves in weakly dispersing media[END_REF], [START_REF] Molinet | Global well-posedness for the kp-ii equation on the background of a non-localized solution[END_REF], [START_REF] Rousset | Transverse nonlinear instability for twodimensional dispersive models[END_REF])

In the periodic setting, both KP-II and 5KP-II are known to be global well-posed for L 2 initial data (see [START_REF] Bourgain | On the cauchy problem for the kadomstev-petviashvili equation[END_REF], [START_REF] Saut | The cauchy problem for the fifth order kp equations[END_REF]). The two models are infinite dimensional Hamiltonian system and the first conserved quantity for them are

M [u](t) = T 2 u(t, x, y) 2 dxdy.
Moreover, the solutions to (5.1.1) and (5.1.2) must satisfy T u(t, x, y)dx = 0 in the distributional sense, we may rewrite the equations in a non-local version

∂ t u + (-1) l-1 ∂ 2l+1 x u + u∂ x u + ∂ -1 x ∂ 2 y u = 0,
where the Fourier multiplier ∂ -1

x is defined by

∂ -1 x v(k, η) = 1 ik v(k, η)
for all functions v ∈ D 0 (T 2 ) := {v ∈ D (T 2 ) : v(0, m) = 0 for all m ∈ Z}.

(5.1.3)

We will study (5.1.3) in control point of view with a forcing term f (t, x, y) added as control input:

∂ t u + (-1) l-1 ∂ 2l+1 x u + u∂ x u + ∂ -1 x ∂ 2 y u = f, (5.1.4)
where f is assumed to be supported in a given region ω ⊂ T 2 . In control theory, the following two problems are essential: Exact controllability: Given initial state u 0 and final state u 1 , can one find a control input f such that the equation (5.1.4) admits a solution u which satisfies u(0) = u 0 and u(T ) = u 1 ? Stabilization: Can one find a feedback control law f = Ku such that the resulting system

∂ t u + (-1) l-1 ∂ 2l+1 x u + u∂ x u + ∂ -1 x ∂ 2 y u = Ku
is asymptotically stable as t → ∞? Those problems are firstly studied by Russel-Zhang for KdV equation (see [START_REF] Russell | Exact controllability and stabilizability of the korteweg-de vries equation[END_REF]) and widely extended to other dispersive models in last decades. Most notably, [START_REF] Dehman | Stabilization and control for the subcritical semilinear wave equation[END_REF], [START_REF] Dehman | Stabilization for the nonlinear Klein Gordon equation with critical exponent[END_REF], [START_REF] Laurent | On stabilization and control for the critical klein-gordon equation on a 3-d compact manifold[END_REF], for nonlinear Klein-Gordon(wave) equation, [START_REF] Dehman | Stabilization and control for the nonlinear schrödinger equation on a compact surface[END_REF], [START_REF] Laurent | Global controllability and stabilization for the nonlinear schrödinger equation on some compact manifolds of dimension 3[END_REF] for nonlinear Schrödinger equation, [START_REF] Laurent | Control and stabilization of the korteweg-de vries equation on a periodic domain[END_REF] for KdV equation and [START_REF] Linares | Control and stabilization of the benjamin-ono equation on a periodic domain[END_REF] for Benjamin-Ono equation .

The set up of control problems for KP type equations is a little different. We observe that there is no obvious way to keep the control input f to be localized and simultaneously have zero horizontal mean, unless the region ω ⊂ T 2 is a vertical strip of the form (a, b) × T or a horizontal strip T × (a, b). When ω is prescribed as a horizontal strip, it is proved in [START_REF] Sun | Internal controllability of non-localized solution for the kodomtsev-petviashvili ii equation[END_REF] that the observability estimate can not hold for linear KP-II equation, due to the weak propagation phenomenon in y direction. Therefore, we address the problems of exact controllability and stabilization only for ω = (a, b) × T in this paper. In this case, the control input f should be of the form

f = Gh := g(x) h(x, y) - T g(x )h(x , y)dx , where g ∈ C ∞ c (T), {x ∈ T : g(x) > 0} = (a, b) and T g(x)dx = 1.
(5.1.5)

We fix the assumption above throughout this paper. For the stabilization, the feedback control law is given by f = -G * Gu.

Corollary 5.2

Let s ≥ 0, R 0 > 0 be given. There exist T > 0,

δ 0 > 0, such that if u 0 , u 1 ∈ H s 0 (T 2 ) ∩ S δ 0 with u 0 H s (T 2 ) ≤ R 0 , u 1 H s (T 2 ) ≤ R 0 ,
then one can find a control input Gh with h ∈ L 2 ((0, T ); H s (T 2 )) such that the system (5.1.4) with l = 2 admits a solution u ∈ C([0, T ]; H s 0 (T 2 )) satisfying

u(0) = u 0 , u(T ) = u 1 .
Let us make some comments on the results. Firstly, we call the condition that data in S δ 0 the smallness assumption of low horizontal frequency( in x variable ). For both KP-II and 5KP-II, we need this smallness assumption because in our argument in section 5, we do not prove the observability for the regime where the frequency in x remains bounded while the frequency in y is high. Heuristically, we can view this regime as a coupled system of nonlinear Schrödinger equation in y variable. In this case, the propagation in x-direction is negligible and the propagation estimate in y variable could not help us to observe the solution from the vertical control region. In chapter 3, for linear KP-II, we exploit certain dispersive effect (Ingham inequality) in y variable to resolve this difficulty. However, since H -1 2 ,0 is the scaling-critical functional space for KP-II equation, the non-linear effect can no longer be ignored when we treat it as a system of nonlinear Schödinger equation in y variable. Therefore, How to remove this low horizontal frequency condition is a challenging nonlinear problem.

We also mention that the result we have obtained for KP-II equation is slightly weaker than 5KP-II equation. It is attributed to a priori estimate for Sobolev norms of solutions below L 2 . In general framework of proving stabilization, we need semi-group property of the equation to iterate. In each step of iterating, our argument need the smallness of low horizontal frequency portion of the solution. However, low frequency portion, or equivalently, Sobolev norm with negative index in x, is not conserved along the nonlinear flow. This gives rise to the following problem: Problem 5.1 Does there exist s < 0 such that for any given T > 0, any global solution u of equation (5.1.6) satisfies sup

t∈[0,T ] u H s,0 (T 2 ) ≤ C(T, u(0) H s,0 (T 2 ) )?
Any positive answer of the problem above with the weaker bound C(T, u(0) L 2 (T 2 ) ) on the right hand side will improve the Theorem 5.1 to all s ≥ 0. We believe that Problem 5.1 has its own interest since it relates to the existence of weak solution of KP-II below L 2 (T 2 ). Similar problems in the context of nonlinear Schrödinger equation, KdV equation without damping term have been studied intensively, see for example [START_REF] Koch | Conserved energies for the cubic nls in 1-d[END_REF], [START_REF] Killip | Low regularity conservation laws for integrable pde[END_REF] and the references therein. Nevertheless, partially due to the local-well posedness of periodic 5KP-II equation below L 2 (see [START_REF] Saut | The cauchy problem for the fifth order kp equations[END_REF]), we are able to apply almost-conservation trick, or I-method, to achieve a priori estimate of negative Sobolev norms of solutions of 5KP-II equation.

We organise the paper as follows. In the second section, we introduce some notations and state several basic propositions of Bourgain type functional spaces related to both KP-II and 5KP-II equation. In the third section, we prove the global well-posedness of damped KP-II and damped 5KP-II equation. Next, we use energy estimate to prove that the smallness condition of low horizontal frequencies of KP-II and 5KP-II equation is persistent along the nonlinear flow. In the fifth section, we prove the observability estimate for high horizontal frequencies of both inhomogeneous KP-II type equations by adapting the dyadic propagation estimate for linear equation in chapter 3. The final section devotes to the proof of stabilization. We also add an appendix to give a proof of stabilization of linear KP-II type equations by using observability inequality established in chapter 3.

Notations and Preliminaries

Notations

We begin by some notations which are standard in Partial Differential Equations. We use the notation A B or A = O(B) to denote the estimate that |A| ≤ CB for certain uniform constant. We also use the notation

|A| ≤ C A 1 ,A 1 ,••• B of |A| ≤ C(A 1 , A 2 , • • • )B to denote various constants C, depending on the quantities A 1 , A 2 , • • • ,
which may change from line to line. We will use Japanese bracket

• := 1 + | • | 2 .
We identify T = R/2πZ. For 2π-periodic function f on R, we define its Fourier transform on Z by

F x f (k) = f (k) = 1 2π 2π 0 f (x)e -ikx dx.
We identify function f on T 2 by 2π-periodic function f on R 2 . The Fourier transform on Z 2 is denoted by

f (n) = 1 (2π) 2 T 2 f (z)e -iz•n dxdy with z = (x, y) ∈ T 2 and n = (k, m) ∈ Z 2
We introduce some notations about one-dimensional Little-wood Paley projection. Fix a cutoff function η such that

η ∈ C ∞ c (R), 0 ≤ η ≤ 1, η| [-1,1] = 1, supp(η) ⊂ [-2, 2].
We define

φ(ξ) = η(ξ) -η(2ξ) and φ N (ξ) = φ ξ N
Throughout the paper, capitalized variables such as N, M, N 1 , M 1 , • • • are assumed to be dyadic and large or equal than 1. Summations over these capitalized variables are presumed to be dyadic summation. We define Littlewood-Paley projector

P N f := F -1 x φ N F x f, Q M f := F -1 y φ N F y f for N ≥ 1 and P 0 f = F -1 x η(2•)F x f, Q 0 f = F -1 y η(2•)F y f. We have P 0 + N P N = 1, Q 0 + M Q M = 1.
We will also use

P >N ( or Q >N ) := K>N P N ( or Q N ), P ≤N ( or Q ≤N ) = 1 -P >N ( or Q >N ).
We need the notations of non-dyadic spectral projection. We denote by

Π k 0 0 f = F -1 x 1 |k|=k 0 0 F x f, Π =0 := F -1 x 1 k =0 F x , π m 0 f = F -1 y 1 |m|=m 0 F y f, Π ≥k 0 = k≥k 0 Π k , Π >k 0 = k>k 0 Π k , Π ≤k 0 = 1 -Π >k 0 , Π <k 0 = 1 -Π ≥k 0
and similarly for π ≥m 0 , π >m 0 , π ≤m 0 , π <m 0 .

Functional spaces

Anisotropic Sobolev spaces H s 1 ,s 2 (T 2 ) is defined via the norm

f 2 H s 1 ,s 2 (T 2 ) := (k,m)∈Z 2 k 2s 1 m 2s 2 | f (k, m)| 2
The free groups associated with the linearized KP-II and 5KP-II equation are defined as

S(t)f (k, m) = e it k 3 -m 2 k f (k, m), U (t)f (k, m) = e it k 5 -m 2 k f (k, m)
with respectively. Anisotropic Bourgain spaces adapted to periodic KP-II and 5KP-II equations are defined via the norms

u 2 X s 1 ,s 2 ;b,b 1 3 := (k,m)∈Z 2 R τ -p 3 (k, m) k 3 2b 1 τ -p 3 (k, m) 2b k 2s 1 m 2s 2 | u(τ, k, m)| 2 dτ, u 2 X s 1 ,s 2 ;b,b 1 5 := (k,m)∈Z 2 R τ -p 5 (k, m) k 4 2b 1 τ -p 5 (k, m) 2b k 2s 1 m 2s 2 | u(τ, k, m)| 2 dτ,
with respectively, where

p 3 (τ, k, m) = k 3 - m 2 k , p 5 (τ, k, m) = k 5 - m 2 k .
We make some conventions here. If there is no risk of confusing, we denote by X s 1 ,s 2 ;b,b 1 the Bourgain spaces of KP-II or 5KP-II. We also denote by

X s 1 ,s 2 ,b if b 1 = 0 and X s,b simply if s 1 = s 2 = s, b 1 = 0.
The norm in the localized time interval I is defined by

u X s 1 ,s 2 ,b,b 1 (I) := inf{ w X s 1 ,s 2 ,b,b 1 : w(t)| I = u(t)| I }.
When I = [0, T ] we also denote by X s 1 ,s 2 ;b 1 ,b 2 (T ). In general, we estimate the norm X s 1 ,s 2 ;b,b 1 (I) by estimating ψ T u X s 1 ,s 2 ;b,b 1 for time localization version of u, where

ψ T = ψ(T -1 •), ψ ∈ C ∞ c (R), ψ(t) ≡ 1, |t| ≤ 1 2 and ψ(t) ≡ 0 for |t| ≥ 1. Proposition 5.1 For -1 2 < b ≤ b < 1 2 and 0 < T < 1, we have ψ T u X s 1 ,s 2 ;b ≤ CT b-b u X s 1 ,s 2 ;b .
We have the following linear estimate for 5KP-II.

Proposition 5.2 For -1 2 < b ≤ 0 < 1 2 < b ≤ b + 1, T ≤ 1, we have 1. ψ T (t)U (t)u 0 X s 1 ,s 2 ;b 5 ≤ CT 1 2 -b u 0 H s (T 2 ) .
2.

ψ T t 0 U (t -t )F (t )dt X s 1 ,s 2 ,b 5
≤ CT 1-(b-b ) F X s 1 ,s 2 ,b 5
For KP-II, it seems that we can not avoid to use the end points b = -1 2 and b = 1 2 . However, the analogue of the linear estimate above is false. We need an auxiliary norm

u Y s,b := τ -p 3 (n) b-1 2 n s u l 2 n L 1 τ
and the restricted spaces Y b,s (I) defined in the same manner. We let

Z s,b (I) := X s,s;b,b 1 3 (I) ∩ Y s,b (I).
With this axillary norm, the linear estimate now holds true.

Proposition 5.3 1. ψ T (t)S(t)u 0 Z s, 1 2 ≤ C u 0 H s (T 2 ) . 2. ψ T (t) t 0 S(t -t )F (t )dt Z s, 1 2 ≤ C F Z s,-1 2 . 3. sup t∈[0,T ] t 0 S(t -t )F (t )dt H s (T 2 ) ≤ C T F Z s,-1 2 .
All the constants C are independent of T .

We summarize several basic properties of these spaces:

1. X s 1 ,s 2 ;b,b 1 is a Hilbert space. For b ≤ b, s 1 ≤ s 1 , s 2 ≤ s 2 , b 1 ≤ b 1 , X s 1 ,s 2 ;b,b 1 is continuously imbedded in the space X s 1 ,s 2 ;b ,b 1 . 2. Z s, 1 2 (I) ⊂ C(I; H s (T 2
)) for any s ∈ R.

3. X s,b+ ⊂ Y s,b holds for any > 0.

Given any finite interval

I, if b < b, b 1 < b 1 , s 1 < s 1 , s 2 < s 2 , then the space X s 1 ,s 2 ;b,b 1 (I) is compactly imbedded into X s 1 ,s 2 ;b ,b 1 (I).
We note that bounded sequence may not have weak limit in the space Z s, 1 2 (I) since it is L 1 based space and Banach-Alaoglu theorem is not applicable.

We now recall the bilinear estimates needed for Cauchy problem and energy estimate. For KP-II, we have Proposition 5.4 ([6], [START_REF] Molinet | Global well-posedness for the kp-ii equation on the background of a non-localized solution[END_REF]) There exist 1 4 < b 1 < 3 8 and ν > 0 such that for all functions u, v ∈ X s, 1 2 ,b 1 with T u(t, x, y)dx = T v(t, x, y)dx = 0, the following bilinear estimate holds

∂ x (uv) Z s,-1 2 ≤ C u X s,s; 1 2 ,b 1 3 v X s,b 3 + C v X s,s; 1 2 ,b 1 3 u X s,b 3
, with some 0 < b < 1 2 , for all s ≥ 0. As a consequence,

ψ 1 (t) t 0 S(t -t )ψ T (t )∂ x (uv(t ))dt Z s, 1 2 ≤ CT ν u X s,s; 1 2 ,b 1 3 v X s,s; 1 2 ,b 1 3
for some ν > 0. All the constants C are independent of T .

For 5KP-II, we have

Proposition 5.5 ([55])

There exist 1 4 ≤ b

1 < 3 8 b > 1 2 , b < 1 2 , b + b < 1, then ∂ x (uv) X s 1 ,s 2 ;-b , 1 4 5 ≤ C u | X s 1 ,s 2 ;b,b 1 5 v | X s 1 ,s 2 ;b,b 1 5 . We have that v Z 0, 1 2 (T ) ≤ C 0 v 0 L 2 (T 2 ) + C 0 T ν 1 + u Z 0, 1 2 (T ) v Z 0, 1 2 (T ) . For T < T 1 ( u 0 L 2 (T 2 ) ) = min 2T 0 ( u 0 L 2 (T 2 ) ), 2C 0 (1 + u 0 L 2 (T 2 ) ) -1 ν , we have v L ∞ ([0,T ];L 2 (T 2 )) ≤ C v Z 0, 1 2 (T ) ≤ C 1 v 0 L 2 (T 2 ) . Now since u(t) L 2 (T 2 ) ≤ u 0 L 2 (T 2
) for all t ≥ 0, we can extend v globally. The same argument holds true for w = ∂ y u. Taking w 1 = ∂ 2 y u, w 2 = ∂ 3 y u, we have that u ∈ C(R + ; H 0,3 0 (T 2 )). From the equation

∂ 3 x u = -v -∂ -1 x ∂ 2 y u -u∂ x u -G * Gu,
we obtain that for any T 0 > 0 and 0 ≤ t < T 0 ,

∂ 3 x u(t) L 2 (T 2 ) ≤ v(t) L 2 (T 2 ) + ∂ -1 x ∂ 2 y u(t) L 2 (T 2 ) +C u(t) L 2 (T 2 ) + u(t) L 2 x (T;L ∞ y (T)) ∂ x u(t) L ∞ x (T;L 2 y (T)) .
Using one-dimensional Sobolev imbedding and interpolation, we finally have

sup t∈[0,T 0 ] ∂ 3 x u(t) L 2 (T 2 ) ≤ C(T 0 , u 0 H 3 (T 2 ) )
and u ∈ C(R + ; H 3 (T 3 )). From induction, if u 0 ∈ H 3k 0 (T 2 ) for k ∈ N, we have that u ∈ C(R + ; H 3k (T 2 )). Other values of s can be obtained from interpolation result. This completes the proof. The assertion that α(T, R 0 ) is polynomial in T follows from inductio and partition of [0, T ] for large T > 0.

For damped 5KP-II Proof. First we note that from Cauchy theory,

∂ t u -∂ 5 x u + ∂ -1 x ∂ 2 y u + u∂ x u = -G * Gu, (t, x) ∈ R × T 2 , u| t=0 = u 0 ∈ H s 0 (T 2 ), (5.3
u Z s, 1 2 T ≤ C(T 0 , R 0 ). Since for any N 0 , u 2 H -1,0 (T 2 ) ≤ P ≤N 0 u 2 L 2 (T 2 ) + 1 N 2 0 u 2 L 2 (T 2 ) ,
it would be suffice to estimate

P ≤N 0 u(t) 2 L 2 (T 2 )
. We perform a simple energy estimate. Let v = P ≤N 0 u, and w = u -v. v solves the equation

(∂ t + ∂ 3 x + ∂ -1 x ∂ 2 y )v = P ≤N 0 - 1 2 ∂ x (v 2 ) -∂ x (vw) - 1 2 ∂ x (w 2 ) -G * G(v + w) .
We have

1 2 d dt v(t) 2 L 2 (T 2 ) = -Gv(t) 2 L 2 (T 2 ) - T 2 v∂ x (vw) - 1 2 T v∂ x (w 2 ) + 1 2 T 2 P >N 0 v∂ x (v 2 ) - T 2 Gv • Gw.
From bilinear estimate, we have

v(t) 2 L 2 (T 2 ) -v(0) 2 L 2 (T 2 ) ≤C v X 0, 1 2 T 0 v X 0,0; 1 2 ,b 1 w X 0,0; 1 2 ,b 1 + w 2 X 0,0; 1 2 ,b 1 +C v L ∞ ([0,T 0 ];L 2 (T 2 )) w L ∞ ([0,T 0 ];L 2 (T 2 )) ≤C u X 0, 1 2 T 0 N s 0 u X 0,0; 1 2 ,b 1 w X s,s; 1 2 ,b 1 + w 2 X s,s; 1 2 ,b 1 + C N s 0 u L ∞ ([0,T 0 ];L 2 (T 2 )) w L ∞ ([0,T 0 ];H s (T 2 )) ≤ C(T 0 , R 0 ) N s 0 .
We complete the proof by choosing N 0 large enough and δ 0 small such that

δ 2 0 + C(T 0 , R 0 )N -s 0 < 2 .

5KP-II

Fix s 0 ∈ 0, 1 4 . The Stronger dispersion of 5KP-II enable us to prove a finer energy estimate in H -s 0 ,0 (T 2 ) for bounded L 2 solutions. Proposition 5.7 Assume that R 0 > 0, T 0 > 0 are given. Then there exists a positive increasing function β : R + × R + → R + , such that for all solutions u of (5.3.2) with u(0) L 2 (T 2 ) ≤ R 0 , we have

sup t∈[0,T 0 ] u(t) H -s 0 ,0 (T 2 ) ≤ β(T 0 , R 0 ) 8s 0 1+8s 0 u(0) 1 1+8s 0 H -s 0 ,0 (T 2 ) .
Furthermore, for any fixed R 0 > 0, β(T 0 , R 0 ) depends polynomial on T 0 .

Corollary 5.3

Assume R 0 > 0, T 0 > 0 are given. For any > 0, there exists a small number δ 0 > 0, such that for all solutions u of (5.3.2) with

u(0) L 2 (T 2 ) ≤ R 0 , u(0) H -1,0 (T 2 ) ≤ δ 0 , the solution satisfies sup t∈[0,T 0 ] u(t) H -1,0 (T 2 ) ≤ . Remark 5.1
The choice of δ 0 will be the power of divided by power of T 0 . One can also use local Cauchy theory in H -s 0 ,0 (T 2 ) for 5KPII to prove Corollary 5.3 directly, which will be enough for the final proof. However, the small number δ 0 depend exponentially on T 0 in this way. In what follows, we choose to prove Proposition 5.7 via energy estimate, both for relaxing the restriction on δ 0 and its own interest.

For the proof, we will perform a slightly different energy estimate as in the last section. Inspired by the "I-method", firstly introduced by C-K-S-T-T in [START_REF] Colliander | Global well-posedness for schrödinger equations with derivative[END_REF], we define the operator

I N u = F -1
x θ(k)F x u, where the Fourier multiplier

θ(ξ) = 1, |ξ| ≤ N, θ(ξ) = N s 0 |ξ| s 0 , |ξ| > 2N.
for some 0 < s 0 < 1 4 and a large dyadic number N 1 to be chosen later. In what follows, we denote by I the operator I N . Lemma 5.1

[I, G] L 2 →L 2 ≤ C N ,
where the constant C only depend on the operator G.

Proof. Since all the operators are acting on functions of L 2 (T), it suffices to estimate for functions of one variable. For any f ∈ L 2 (T), we calculate

[I, G]f 2 L 2 (T) = k∈Z k 1 ∈Z (g k-k 1 -g k g k 1 )(θ(k) -θ(k 1 )) f (k 1 ) 2 .
We estimate the sum according to kk 1 ≥ 0 and kk 1 < 0. If k and k 1 have the same sign, say positive, we use the fundamental theorem of calculus

|θ(k) -θ(k 1 )| = (k -k 1 ) 1 0 θ (λk 1 + (1 -λ)k)dλ ≤ C|k -k 1 | N . Thus k∈Z k 1 =0,k 1 k≥0 (g k-k 1 -g k g k 1 )(θ(k) -θ(k 1 )) f (k 1 ) 2 ≤ C 2 N 2 k∈Z k 1 =0 |k -k 1 |(g k-k 1 -g k g k 1 ) f (k 1 ) 2 ≤ C 2 N 2
from Young's convolution inequality and Cauchy-Schwartz, since g ∈ C ∞ c (T) yields (kg k ) k∈Z ∈ l 1 k . If k and k 1 have different sign, the only contributions of the sum comes from:

1. -N ≤ k ≤ 0, k 1 > N or 0 ≤ k ≤ N, k 1 < -N . 2. k < -N, 0 ≤ k 1 ≤ N or k > N, -N ≤ k 1 ≤ 0. 3. |k| > N, |k 1 | > N, kk 1 < 0.
In both situations we could bound the summation by

g k 2 l 1 (|k>N |) f (k) 2 l 2 (k) + g k 2 l 2 g k 2 l 2 (|k>N |) f (k) 2 l 2 (k) ≤ C N 2 ,
thanks to Young's convolution inequality, Cauchy-Schwartz and the fact that g ∈ C ∞ c (T).

We need a dyadic version of bilinear estimate proved in [START_REF] Saut | The cauchy problem for the fifth order kp equations[END_REF].

Lemma 5.2 ([55])

Let u 1 and u 2 be two functions defined on R × T 2 such that supp( u j ) = (τ j , k j , m j ) :

|k j | ∼ N j , τ j -k 5 j + m 2 j j j ∼ L j , j = 1, 2. Then P N (u 1 u 2 ) L 2 (R×T 2 ) ≤C(L 1 ∧ L 2 ) 1 2 (L 1 ∨ L 2 ) 1 4 (N 1 ∧ N 2 ) 1 2 (N 1 N 2 ) 1 4 N 1 4 u 1 L 2 (R×T 2 ) u 2 L 2 (R×T 2 ) . N -100 max and define a [N 1 ,N 2 ,N 3 ] (t) = 1 [0,T 0 ] * G [N 1 ,N 2 ,N 3 ] and b [N 1 ,N 2 ,N 3 ] (t) = 1 3 [0,T 0 ] -a [N 1 ,N 2 ,N 3 ] (t) 3 . We first estimate the contribution of b [N 1 ,N 2 ,N 3 ] (t) to I [N 1 ,N 2 ,N 3 ] . From Corollary 5.4, we have R×T 2 b [N 1 ,N 2 ,N 3 ] (t)I∂ x (P N 1 uP N 2 u)(t)IP N 3 u(t) = b [N 1 ,N 2 ,N 3 ] (t)P N 3 u L 2 (R×T 2 ) I∂ x (P N 1 uP N 2 u) L 2 (R×T 2 ) ≤CN max P N 1 uP N 2 u L 2 (R×T 2 ) b [N 1 ,N 2 ,N 3 ] L 2 (R) P N 3 u L ∞ (R;L 2 (T 2 )) ≤C N -48 max b [N 1 ,N 2 ,N 3 ] L 2 (R) u 3 X 0,0; 1 2 + ,0 ,
where to the last inequality we have used the property that b [N 1 ,N 2 ,N 3 ] L 2 ≤ N -50 max . We thus write

I [N 1 ,N 2 ,N 3 ] = J [N 1 ,N 2 ,N 3 ] + r [N 1 ,N 2 ,N 3 ] , with J [N 1 ,N 2 ,N 3 ] := (N 1 ,N 2 ,N 3 )∈[N 1 ,N 2 ,N 3 ] J N 1 ,N 2 ,N 3 , r [N 1 ,N 2 ,N 3 ] = I [N 1 ,N 2 ,N 3 ] -J [N 1 ,N 2 ,N 3 ] and J N 1 ,N 2 ,N 3 = R×T 2 I∂ x (P N 1 vP N 2 v)(t)IP N 3 v(t), v = a [N 1 ,N 2 ,N 3 ] (t)u.
since a [N 1 ,N 2 ,N 3 ] is uniformly bounded in L ∞ with respect to the equivalence class [N 1 , N 2 , N 3 ]. We next investigate the contribution a [N 1 ,N 2 ,N 3 ] (t). From the trivial estimate for the approximation of identity and the Sobolev estimate

(f 1 f 2 ) H 1 2 + (R) ≤ C f 1 H 1 2 + (R) f 2 H 1 2 + (R) ,
we have that P N j v X 0,0; 1 2 + ,-≤ C P N j u X 0,0; 1 2 + ,0 N 100 max .

To estimate J [N 1 ,N 2 ,N 3 ] , we use Plancherel in space and time to calculate where to the last step, we have used the fact that the permutation of indices of (τ j , n j ) keeps the symmetric summation

J [N 1 ,N 2 ,N 3 ] = τ 3 +τ 4 =0 n 3 +n 4 =0 ik 4 θ(k 3 )θ(k 4 ) (N 1 ,N 2 ,N 3 )∈[N 1 ,N 2 ,N 3 ] v N 1 v N 2 (τ 4 , n 4 ) v N 3 (τ 3 , n 3 ) = τ 1 +τ 2 +τ 3 =0 n 1 +n 2 +n 3 =0 i(k 1 + k 2 )θ(k 1 + k 2 )θ(k 3 ) (N 1 ,N 2 ,N 3 )∈[N 1 ,N 2 ,N 3 ] 3 j=1 v N j (τ j , n j ) = -i
(N 1 ,N 2 ,N 3 )∈[N 1 ,N 2 ,N 3 ] 3 j=1
v N j (τ j , n j ) invariant. For fixed [N 1 , N 2 , N 3 ], it would be suffice to estimate the term in which N 1 ≤ N 2 ≤ N 3 . We further decompose the modulation of each v, and we have

J [N 1 ,N 2 ,N 3 ] = L 1 ,L 2 ,L 3 J L 1 ,L 2 ,L 3 N 1 ,N 2 ,N 3 ,
where

J L 1 ,L 2 ,L 3 N 1 ,N 2 ,N 3 = τ 1 +τ 2 +τ 3 =0 n 1 +n 2 +n 3 =0 3 j=1 k j θ(k j ) 2 3 j=1
v L j N j (τ j , n j ), and v L j N j (τ j , n j ) = φ L j ( τ j -p 5 (n j ) ) φ N j (k j ) v(τ j , n j ).

Case 1: N 1 ∼ N 2 ∼ N 3 : In this case, we may assume that L 1 ≤ L 2 ≤ L 3 . From Lemma 5.2, we estimate

J L 1 ,L 2 ,L 3 N 1 ,N 2 ,N 3 ≤C N max N 2s 0 N 2s 0 max N 3 4 max L 1 L 1 4
2 L 1 2 3 3 j=1 v L j N j X 0,0; 1 2 + ,0 .

(

The modulation condition for fifth order KP-II We apply Lemma 5.2 to estimate

L 1 ,L 2 ,L 3 ,L 1 =Lmax J L 1 ,L 2 ,L 3 N 1 ,N 2 ,N 3 ≤ L 1 ,L 2 ,L 3 ,L 1 =Lmax N 4 2 N 1 C N 1 N 2 L 2 L 3 L 1 2
1 N Similar estimate yields

N N 1 ∼N 2 ,N 1 N L 1 ,L 2 ,L 3 ,L 3 =Lmax J L 1 ,L 2 ,L 3 N 1 ,N 2 ,N 3 ≤ C N 3
4 -100 max u 3 X 0,0; 1 2 ,0 .

In summary, we have

I [N 1 ,N 2 ,N 3 ] ≤ C N 3 4 -100 max u 3 X 0, 1 2 + .
. We fix < 1 400 and thus Notice that u(t) H -s 0 ,0 (T 2 ) ≤ Iu(t) L 2 (T 2 ) ≤ N s 0 u(t) H -s 0 ,0 (T 2 ) ,

we have that for any 0 < t ≤ T 0 , u(t) H -s 0 ,0 (T 2 ) ≤ N s 0 u(0) H -s 0 ,0 (T 2 ) + β(T 0 , R 0 )

N 1 4
, with β(T 0 , R 0 ) = C (α(T 0 , R 0 ) 3 R 3 0 + 1). We complete the proof by optimally choosing N = β(T 0 , R 0 ) u(0) H -s 0 ,0 (T 2 )

8 1+8s 0 .
Taking the difference of the two expressions, we have (note that v(0) = 0)

F Π =0 ( 2l+2 (h 1 D x ) 2l+1 -(h 1 D x ) -1 )Π =0 , ϕ v (k) =1 k =0 k 1 =0 2l+2 h 2l+1 1 (k 2l+1 -k 2l+1 1 ) - 1 h 1 1 k - 1 k 1 ϕ(k -k 1 ) v(k 1 )
We conclude by the algebraic manipulation

2l+2 (h 1 k) 2l+1 - 1 h 1 k -2l+2 (h 1 k 1 ) 2l+1 - 1 h 1 k 1 =h 1 (k -k 1 ) 2l j=0 2l + 1 j ( 2l+2 (h 1 k 1 ) j (h 1 (k -k 1 )) 2l-j + 1 h 2 1 1 k 2 1 + (k 1 -k) kk 2
1 since all the factors (k -k 1 ) j will fall on ϕ as derivatives and all the power (h 1 k 1 ) α is a bounded Fourier multiplier.

For dyadic number N 1, we denote by w N = P N w, f N = P N f . Pick ψ ∈ C ∞ c (R) with 0 ≤ ψ ≤ 1, ψ| [-1/2,1/2] ≡ 1, ψ(t) = 0, |t| > 1. For any T > 0, we define ψ T (•) := ψ(T -1 •).

Lemma 5.4

Given T > 0, there exist a small parameter 0 < h 0 1, such that for all 0 < h < h 0 , where the constant C T depends only on T > 0.

Proof. In the calculation below, all the implicit constants depend on T . Let A(x, hD x ) = ϕ(x)p (hD x ) -1 χ(hD x )

where the cutoff χ = χ 2 -100 . Denote by v N = ψ T (t)w N and P h = h 2l+1 ∂ t -ip(hD x ). On the one hand, we calculate

1 h [A(x, hD x ), P h ]v j , v j L 2 t,x = - i h [A(x, hD x ), p(hD x )]v N , v N L 2 t,x =(ϕ (x)v N , v N ) L 2 t,x + O(h) v N 2 L 2 t,x .
(5.5.7)

In the calculus above, we have used symbolic calculus and the Poisson bracket(see [START_REF] Zworski | Semiclassical analysis[END_REF]) that ϕ(x)χ(ξ)p (ξ) -1 , p(ξ) = ϕ (x)χ(ξ). On the other hand, we calculate the commutator (5.5.8)

From Cauchy-Schwartz, we have

h 2l (A(x, hD x )ψ T w N , ψ T f N ) L 2 t,x ≤ h 2l Π =0 ϕ(x)p (hD x ) -1 χ(hD x )ψ T w N X 0, 1 2 h ψ T f N X 0,-1 2 h .
(5.5.9) We need estimate the two terms on the right hand side. For the first term, though the Fourier multiplier p (hD x ) -1 χ(hD x ) is uniformly bounded on X s 1 ,s 2 ;b,b 1 h , multiplication by ϕ(x) is not. We observe that for any space-time function

G N = P N G N , Π =0 ϕ(x)G N 2 X s,0 h G N 2 X s,0 h , Π =0 ϕ(x)G N 2 X s,1 h = Π =0 ϕ(x)G N 2 X s,0 h + h -2(2l+1) Π =0 (h 2l+1 D t -p(hD x ))(ϕ(x)G N ) 2 X s,0 h G N 2 X s,0 h + h -2(2l+1) | Π =0 ϕ(x)(h 2l+1 D t -p(hD x ))G N 2 X s,0 h +h -2(2l+1) Π =0 (h 2l+1 D t -p(hD x ))Π =0 , ϕ G N 2 X s,0 h G N 2 X s,1 h + 1 h 2(2l+1) Π =0 [p(hD x )Π =0 , ϕ] G N 2 X s,0 h 1 h 4l G N 2 X s,1
h , thanks to Lemma 5.3. From interpolation, we have

Π =0 ϕ(x)G N X s, 1 2 h 1 h l G N X s, 1 2 h . Therefore, h 2l (A(x, hD x )ψ T w N , ψ T f N ) L 2 t,x h l ψ T w N X 0, 1 2 h ψ T f N X 0,-1 2 h . Similarly, h 2l (A(x, hD x )ψ T f N , ψ T w N ) L 2 t,x h l ψ T w N X 0, 1 2 h ψ T f N X 0,-1 2 h
. in two ways. From symbolic calculus, we have (5.5.12)

We remark that the implicit constant in big O is independent of . Using equation satisfied by w N , we have

1 h 1 A(x, h 1 D x ), h 2l+1 1 2l+2 ∂ t -ip (h 1 D x ) ψ T w N , ψ T w N L 2 t,x =h 2l 1 2l+2 (A(x, h 1 D x )ψ T f N , ψ T w N ) L 2 t,x + h 2l 1 2l+2 (A(x, h 1 D x )ψ T w N , ψ T f N ) L 2 t,x +O(h 2l 1 2l+2 ) ψ T w N 2 L 2 t,x .
(5.5.13) Combining (5.5.12),(5.5.13), we have

(ϕ (x)ψ T w N , ψ T w N ) L 2 t,x h 2l 1 2l+2 (A(x, h 1 D x )ψ T f N , ψ T w N ) L 2 t,x +h 2l 1 2l+2 (A(x, h 1 D x )ψ T w N , ψ T f N ) L 2 t,x (5.5.14)
The two terms on the right hand side are of the same type. Hence we only detail the estimate for the second term. For N ∼ 1 h 1 and any frequency localized space-time function . Therefore,

G N = P N G N , we have Π =0 ϕ(x)G N 2 X s,0 h G N 2 X s,0 h , and
Π =0 ϕ(x)G N 2 X s,1 h G N 2 X s,0 h + Π =0 ϕ(x) D t -D 2l+1 x + h -(2l+2) D -1 x G N 2 X s,0 h + Π =0 (D t -D 2l+1 x + h -(2l+2) D -1 x )Π =0 , ϕ G N 2 X s,0 h G N 2 X s,1 h + Π =0 (D 2l+1 x -h -(2l+2) D -1 x )Π =0 , ϕ G N 2 X s,0 h Π =0 (D 2l+1 x -h -(2l+2) D -1 x )Π =0 , ϕ G N X s,0 h = 1 h 2l+1 1 Π =0 ((h 1 D x ) 2l+1 --(2l+2) (h 1 D x ) -1 )Π =0 , ϕ G N X s,0 h = 1 h 2l+1 1 2l+2 Π =0 ( 2l+2 (h 1 D x ) 2l+1 -(h 1 D x ) -1 )Π =0 , ϕ G N X s,0 h = 1 h 2l 1 2l+2 G N X s,
h 2l 1 2l+2 (A(x, h 1 D x )ψ T w N , ψ T f N ) L 2 t,x h l 1 l+1 ψ T w N X 0, 1 2 h ψ T f N X 0,-1 2 h
The rest argument is the same as in the proof of Lemma 5.4.

Lemma 5.6

Given T > 0. There exists 0 < 0 1 such that for any h > 0(not necessarily small) and any N ≥ 1 0 ∨ 2 100 h , all solution w to (5.5.3) satisfies (5.5.15) where the constant C 0 ,T only depends on 0 , T .

ψ T (t)w N 2 L 2 t,x ≤C 0 ,T N -l ψ T (t)w N X 0, 1 2 h ψ T (t)f N X 0,-1 2 h +C 0 ,T ψ T (t)gw N 2 L 2 t,x ,
Proof. We first choose 0 > 0 as in Lemma 5.5. Set h 1 = h ∼ 1 N ≤ 0 and we rewrite the equation 5. Define A(x, h 1 D x ) = ϕ(x)q (h 1 D x ) -1 χ(h 1 D x ) and we calculate on the one hand that

1 h 1 [A(x, h 1 D x ), h 2l+1 1 ∂ t -iq (h 1 D x )]ψ T w N , ψ T w N L 2 t,x = (ϕ (x)ψ T w N , ψ T w N ) L 2 t,x + O(h 1 ) ψ T w N 2 L 2 t,x
where the implicit constant in the big O does not depend on h 1 , h and . On the other hand, we use the equation to write

1 h 1 [A(x, h 1 D x ), h 2l+1 1 ∂ t -iq (h 1 D x )]ψ T w N , ψ T w N L 2 t,x =h 2l 1 (A(x, h 1 D x )ψ T f N , ψ T w N ) L 2 t,x + h 2l 1 (A(x, h 1 D x )ψ T w N , ψ T f N ) L 2 t,x +O(h 2 1 ) ψ T w N 2 L 2 t,x .
We estimate

h 2l 1 (A(x, h 1 D x )ψ T w N , ψ T f N ) L 2 t,x h 2l 1 Π =0 A(x, h 1 D x )ψ T w N X 0, 1 2 h ψ T f N X 0,-1 2 h .
For G N = P N G N , we have Π =0 ϕ(x)G N X s,0 h G N X s,0 h , and

Π =0 ϕ(x)G N X s,1 h G N X s,1 h + Π =0 (D t + D 2l+1 x -h -2l+2 D -1 x )Π =0 , ϕ G N X s,0 h G N X s,1 h + 1 h 2l+1 1 Π =0 [q (h 1 D x )Π =0 , ϕ] G N X s,0 h 1 h 2l 1 G N X s,1 h .
Therefore, from interpolation and Lemma 5.3, we have

ϕ(x)G N X s, 1 2 h 1 h l 1 G N X s, 1 2 h
. Applying the estimate above to G N = q (h 1 D x ) -1 χ(h 1 D x )w N , we have

h 2l 1 (A(x, h 1 D x )ψ T w N , ψ T f N ) L 2 t,x h l 1 ψ T w N X 0, 1 2 h ψ T f N X 0,-1 2 h
. The rest argument is the same as in the proof of Lemma 5.4 and we omit the details. Proposition 5.8 Fix 0 > 0, T > 0. There exists 0 < h 0 1, C 0 ,T > 0, θ > 0, and a dyadic number N 0 ≥ 1, such that for all h < h 0 , all N 1 ≥ N 0 , and all solution w of (5. Proof. Denote by w ≥N 1 ,≤N 2 = P ≥N 1 P ≤N 2 w. Fix any 0 1 as in Lemma 5.5 and 5.6. We also denote by N 0 = A -1 0 with the big constant A > 0 to be chosen later such that hA < 2 -100 0 1.

(5.5.17)

From Lemma 5.5, we have

A 0 ≤N ≤ 1 2 100 h ψ T (t)w N 2 L 2 t,x ≤C 0 ,T A 0 ≤N ≤ 1 2 100 h N -l ψ T (t)w N X 0, 1 2 h ψ T (t)f N X 0,-1 2 h +C 0 ,T A 0 ≤N ≤ 1 2 100 h ψ T (t)gw N 2 L 2 t,x .
We note that here and in the sequel, the summations taken over N, M and another captical alphabets are all dyadic summations. Using the commutator estimate (5.5.20) Finally, by choosing A 2 > 2 100 2 0 C 0 ,T + 2 100 and h 0 < 0 2 100 A , we have that for all h < h 0 ,

ψ T w ≥N 0 2 L 2 t,x ≤C 0 ,T 1 N θ 0 ψ T w ≥N 0 /2 X 0, 1 2 h ψ T f ≥N 0 /2 X 0,-1 2 h + C 0 ,T gw ≥N 0 /2 2 L 2 t,x + C 0 ,T N 2 0 ψ T w ≥N 0 /2 2 L 2
t,x . for some θ > 0. The estimate still holds true for any N 1 ≥ N 0 . which is a contradiction. case 2 : α n → α = 0.

In this case, we make the substitution u n := un αn . u n satisfies Note that in this case, the nonlinear term α n u n ∂ x u n → 0 strongly in E(T ) and we do not need the smallness of u n H -1,0 (T 2 ) (it is not necessarily small). We need the following lemma from linear analysis.

(∂ t + ∂ 3 x + (-1) l-1 ∂ 2l+1 x ∂ 2
Lemma 5.7

Denote by S 2l+1 (t) = e -t(∂ 2l+1

x -∂ -1

x ∂ 2 y ) . Then

P ≤N 0 Q ≥M 0 v(0) 2 L 2 (T 2 ) ≤ C T T 0 GP ≤N 0 S 2l+1 (t)v(0) 2 L 2 (T 2 ) dt,
in which M 0 N 0 and the constant C T only depends on T > 0.

Proof. The proof is explicit calculation. We write v(t, x, y) = P ≤N 0 Q ≥M 0 S 2l+1 (t)v(0) =

(g k-k 1 -g k g k 1 )a k 1 ,m e -it k 2l+1 1 -m 2 k 1 T 0 Gv(t) 2 L 2 (T 2 ) dt = k,|m|≥M 0 /2 T 0 0<|k 1 |≤2N 0 (g k-k 1 -g k g k 1 )a k 1 ,m e -it k 2l+1 1 -m 2 k 1 2 dt ≥ C 1 k 1 ,|m|≥M 0 /2 k |g k-k 1 -g k g k 1 | 2 |a k 1 ,m | 2 ≥ C 2 v(0) 2 L 2 (T 2 )
. For the first inequality, we have used Ingham inequality since for M 0 N 0 and the spectral gap

inf k 1 =k 2 ,0<|k 1 |,|k 2 |≤N 0 k 2l+1 1 -k 2l+1 2 + m 2 1 k 2 - 1 k 1 ≥ γ 0 > 0 (5.6.4)
and γ 0 can be chosen independent of N 0 , provided that we fix M 0 N 0 large enough. This implies that the constant C 1 is independent of N 0 .

For the second inequality, we use the fact that We apply this lemma to u n (0). Take M 0 N 0 and we have Integrating in time, bilinear estimate yield

P ≤N 0 Q ≥M 0 u n (0)
P ≥N 0 u n (0) L 2 (T 2 ) ≤ C T 1 N θ 0 + δ n ,
thanks to α n → 0. By choosing N 0 large enough and M 0 N 0 , for sufficiently large n, we have P ≤2N 0 Q ≤2M 0 u n (0) L 2 (T 2 ) ≥ 1 2 . Up to a subsequence, we assume that u n (0) converges to weakly to u 0 in L 2 (T 2 ). Thus from Rellich theorem, we have that Thus C (t) = 0, a.e. and u| ω ≡ C 0 in the distributional sense. The unique continuation argument for linear KP-II (see [START_REF] Sun | Internal controllability of non-localized solution for the kodomtsev-petviashvili ii equation[END_REF]) yields u = C 0 = 0 since u ∈ D 0 . This contradicts to (5.6.6).

P ≤2N 0 Q ≤2M 0 u 0 L 2 (T 2 ) ≥ 1
As a consequence, we have the L 2 stabilization.

Corollary 5.5 Let s > 0, R 0 > 0 be given. There exist C > 0, γ > 0 and δ 0 > 0 such that the inequality u(t) L 2 (T 2 ) ≤ Ce -γt u 0 L 2 (T 2 ) , t > 0 holds for every solution u of damped KP-II equation (5.1.6) with initial data in S δ 0 such that u 0 H s (T 2 ) ≤ R 0 .

Proof. The proof follows from a standard argument by using semi-group property. Let T > 0, 0 > 0 as in Proposition 5.10. Let

n 0 = log 0 -log R 0 log(C T -2) -log C T + 1, T 0 = n 0 T.
From Corollary 5.6, there exists δ 0 > 0 such that if u(0) ∈ S δ 0 , we have sup t∈[0,T 0 ] u(t) H -1,0 (T 2 ) ≤ 0 .

In particular, given any T > 0 and > 0, there exists T 1 = T 1 ( , u 0 L 2 (T 3 ) ) > 0 such that for any t > T 1 , u Z 0, 1 2 ([t,t+T ]) ≤ . Let v = ∂ t u or ∂ y u as in the proof of Cauchy problem, thus

∂ t v + ∂ 3 x v + ∂ -1 x ∂ 2 y v = -∂ x (uv) -G * Gv.
We first assume that Suppose that u 0 ∈ H 3 0 (T 2 ). Fixing t * T 1 , we write

v(t) = S G (t)v(0) - t 0 S G (t -t )∂ x (uv)(t )dt .
From stabilization result for linear KP-II equation (see Appendix), we have that for any T > 0, S G (t)v(0) L 2 (T 2 ) ≤ C 0 e -c 0 t v(0) L 2 (T 2 )

where the constants C 0 , c 0 are independent of T and t * . From the proof of Cauchy problem, we have v((n + 1)T ) L 2 (T 2 ) ≤C 0 e -c 0 T v(nT

) L 2 (T 2 ) + C T u Z 0, 1 2 ([nT,(n+1)T ]) v Z 0, 1 2 ([nT,(n+1)T ])
≤C 0 e -c 0 T v(nT

) L 2 (T 2 ) + C T v Z 0, 1
2 ([nT,(n+1)T ]) .

We also have Then we fix T and choose > 0 small enough so that

C T < 1 2 , C 2 T < 1 4 e -c 1 T .
Thus we have

v((n + 1)T ) L 2 (T 2 ) ≤ C 0 e -c 0 T + 2C 2 T v(nT ) L 2 (T 2 ) ≤e -c 1 T v(nT ) L 2 (T 2 ) .
Therefore, v(t) L 2 (T 2 ) ≤ C( v(0) L 2 (T 2 ) )e -c 1 t v(0) L 2 (T 2 ) , for all t ≥ 0.

By taking v = ∂ 2 y u, ∂ 3 y u, the same arguments yields ∂ t u(t) L 2 (T 2 ) + u(t) H 0,3 (T 2 ) ≤ C( u 0 H 3 (T 2 ) )e -c 1 t u(0) H 3 (T 2 ) .

From the equation -∂ t u = ∂ 3 x u + ∂ -1

x ∂ 2 y u + u∂ x u + G * Gu, we have that

∂ 3 x u(t) L 2 (T 2 ) ≤ ∂ t u(t) L 2 (T 2 ) + ∂ -1 x ∂ 2 y u(t) L 2 (T 2 ) +C u(t) L 2 (T 2 ) 1 + u(t) 1 2
L 2 (T 2 ) ∂ 3

x u(t)

1 2
L 2 (T 2 ) .

This implies that u(t) H 3 (T 2 ) ≤ C( u 0 H 3 (T 2 ) )e -c 1 t for some 0 < c 1 < c 1 . Now by induction, the stabilization holds for any s ∈ 3N. To complete the proof, it would be suffice to deal with the case s ∈ (0, 3). Observe that w = u -v satisfies

∂ t w + ∂ 3 x w + ∂ -1 x ∂ 2 y w = -∂ x (w(u + v)) -G * Gw,
we can repeat the argument above to obtain

u(t) -v(t) L 2 (T 2 ) ≤ C R 0 e -c 2 t u(0) -v(0) L 2 (T 2 ) ,
for all u(0) L 2 (T) 2 ≤ R 0 , v(0) L 2 (T 2 ) ≤ R 0 . Applying interpolation theorem 5.5, the H s stabilization holds true.

Finally, the proof of Theorem 5.2 follows the same way as for KP-II by changing the functional spaces adapted to the Cauchy problem of 5KP-II equation. We thus omit the details. Proof. First we assume that s = 0. From semi-group property, it suffices to show that there exists T > 0 and C T > 0 such that for all solutions u 0 ∈ L 2 0 (T 2 ),

5.A Stabilization for linear equation

u 0 2 L 2 (T 2 ) ≤ C T T 0 GS G (t)u(0) 2 L 2 (T 2 ) dt, (5.A.2)
where we denote by S §G (t) the linear semi-group associated to (5.A.1). In chapter 32 , we know that the observability for undamped linear KP-II holds true, namely We denote by u(t) = S G (t)u 0 and v(t) = S(t)u 0 and w(t) = v(t) -u(t). w solves

(∂ t + (-1) l-1 ∂ 2l+1 x + ∂ -1 x ∂ 2 y )w = G * Gu.
Multiplying by w and integrating on T 2 , we have and this gives (5.A.2). To finish the proof of general s ≥ 0, we first observe that u(t) H 0,s 2 (T 2 ) ≤ C e -γt u 0 H 0,s 2 (T 2 ) for any integer s 2 . Next we replace u by ∂ t u and deduce that the stabilization holds for any u 0 ∈ H 3s 1 ,s 2 , s 1 , s 2 ∈ N. Other values of s follows from interpolation. This completes the proof.

5.B Nonlinear Interpolation result

Let (A 0 , A 1 ) and (B 0 , B 1 ) are two pair of Banach spaces. A 0 ⊂ A 1 , B 0 ⊂ B 1 . Suppose that the map (nonlinear) T : A 0 → B 0 and T : A 1 → B 1 satisfies

• T a -T b B 1 ≤ f ( a A 1 , b A 1 ) a -b α A 1 , ∀a, b ∈ A 1 ,
• T a B 0 ≤ g( a A 1 ) a β A 0 , ∀a ∈ A 0 , where f, g are continuous functions.

Theorem 5.5 ([59])

For any 0 < θ < 1, 1 ≤ p ≤ ∞, we have T a (B 0 ,B 1 )η,q ≤ Cg(2 a A 1 ) 1-η f ( a A 1 , 2 a A 1 ) η a

(1-η)β+ηα (A 0 ,A 1 ) θ,p , where η and q are given by

1 -η η = 1 -θ θ α β , q = max 1, p (1 -η)β + ηα .
In our applications, we take A 0 = B 0 = L 2 (T 2 ), A 1 = B 1 = H 3 (T 2 ), θ = s 3 ,p = 2. Thus (H 3 , L 2 ) θ,2 = H s for 0 < s < 3. T be the solution map u(0) → u(t) and α = β = 1, η = θ.

1

 1 Vue d'ensemble de la théorie du contrôle d'EDP

T

  g(x )h(x , y)dx , (0.2.10) Si le domaine de contrôle est une bande horizontale ω = T × (a, b), on définit le contrôle via Gh(x, y) := G (h)(x, y) = g(y) h(x, y) -T h(x , y)dx . (0.2.11)

Proposition 1. 5

 5 For any real-valued scalar function a ∈ C ∞ c (Ω × R d ) vanishing near ξ = 0, we have d ds µ, a • γ(s, •) = 0.

  we define S δ, := supp (ã δ ) ∩ {y ≤ , r ≥ } for any 0 < δ . Note that near S δ, (thus near ρ k ∈ G k , k ≥ 3) we have |r 1 | ≤ δ k , and this implies that ζ 1

  s) -z (s)) = O(y(s)), d ds (ζ(s) -ζ(s)) = O((y(s)), dy ds = 2η(s). Note that |η| 2 = |r| = O(1) and y(s 0 ) = 0, z(s 0 ) = z(s 0 ), ζ(s 0 ) = ζ(s 0 ), we have

δ 1

 1 we have γ(s 1 , ρ) ∈ G k and γ(s, ρ) / ∈ G k for any 0 > s > s 1 . In this case we have |y(s)| ≤ C|s -s 1 | for all s ∈ [s 1 , 0]. Therefore we must have d ds Θ k (z (s), ζ(s)) ≤ C|s -s 1 |.

5 .

 5 Integrating for y > 0 yields (1.C.1).

1 0∂

 1 -x )ξ+i(y-y )η h ϕ(y , y, x, η, ξ)u(y , x )dy dx dξdη, whereϕ(y , y, x, η, ξ) = 1 (2πh) d-1 e i(x-z)(ξ -ξ) ha(y, x, η, ξ )b(y , z, ξ)dξ dz.Talor expansion gives ϕ(y , y, x, η, ξ) = ϕ(y, y, x, η, ξ) + (y -y) y ϕ(ty + (1 -t)y, y, x, η, ξ)dt.

  y, x, η, ξ) has compact support in (η, ξ) and |∂ m η ∂ α ξ H t (y -hw, y, x, η, ξ)| ≤ C m,α for any m ∈ N and α ∈ N d-1 . Thus, integrating by part in the expression of k h yields sup (y,x) |k h (y, x, w, v)| ≤ C(1 + |w| + |v|) -(d+1) .

Proposition 2. 2

 2 For any real-valued scalar function a ∈ C ∞ c (Ω × R d ) vanishing near ξ = 0, we have d ds µ, a • γ(s, •) = 0.

Lemma 2. 5

 5 h∂ ν u| ∂Ω = (h∂ ν u , 0), and h∂ ν u| ∂Ω L 2 (∂Ω) = O(1).

.1. 4 )

 4 when the control region is a horizontal strip of the form ω = T × (a, b), we put the control input as Gh(x, y) := G (h)(x, y) = g(y) h(x, y) -T g(y )h(x, y )dy .(3.1.5)

Remark 3. 3

 3 Uniform observability estimate for semi-classical Schrödinger equation only holds for frequency scale 1 h . Now we prove Theorem 3.3 by disproving the horizontal observability. Theorem 3.5

  |u n (t, x, y)| 2 dxdydt = T 0 (-π,α)∪(α,π]

.1. 4 )u| t=T = u 1 . 4 . 1

 4141 satisfies Remark When the control region is a horizontal strip of the form ω = T x × (a, b) y and we put the control input as G h(x, y) := g(y) h(x, y) -T g(y )h(x, y )dy . (4.1.5)

(

  x,ξ,h)∈R 2d ×(0,1)|∂ α x ∂ β ξ a(x, ξ, h)| ≤ C α,β (1 + |ξ|) m-|β| . (4.2.1)For a ∈ S m , we denote by Op h (a) the h pseudo-differential operator acting on Schwartz functions viaOp h (a)f (x) := 1 (2πh) d R 2d e i(x-y)•ξ ha(x, ξ, h)f (y)dydξ.

Lemma 4. 2

 2 There exists an positive increasing function κ : R + → R + , such that|k|≤m 0 |c k | 2 ≤ κ(m 0 )

m 3 0 2 0w 0

 020 ), with implicit constant in big O depending only on sup ξ∈(-δ,δ) |Φ (ξ)|. As a consequence, we havem 0 + µ 1 -ξ * h = O(hM 2 0 ). (4.3.12)The claim follows easily by choosing h small enough.Define a slightly different frequency truncationw L (t, x) = k:|Φ(hk)|≤h 2 M (k)e ikx+ itΦ(hk) h 3

Now we set w n = v n e iξ 0 0 T 2 T 2

 0022 hn x , and we have from Proposition 4.4 that lim n→∞ T |g(x)u n (t, x, y)| 2 dxdydt T 2 |u n (0, x, y)| 2 dxdy )u n (t, x, y)| 2 dxdydt by T 0 |Gu n (t, x, y)| 2 dxdydt.

4 5 ( 2 < b < 5 8 and 1 4 ≤

 4524 Fix any s ≥ 0. For any given T > 0 and any u 0 ∈ H s 0 (T 2 ), there exists a unique solution u ∈ X s,s;b,b 1 T ) ∩ C([0, T ]; H s 0 (T 2 )) of (5.4), where 1 b 1 < 3 8 . Moreover, we have the following estimate u X s,s;b,b 1

τ 1 +τ 2 +τ 3 =0 n 1 +n 2 +n 3 =0k 3 θ(k 3 ) 2 (N 1 ,N 2 ,N 3 )∈[N 1 ,N 2 ,N 3 ] 3 j=1v 3 j=1v

 33321231233 N j (τ j , n j ) = -i 3 τ 1 +τ 2 +τ 3 =0 n 1 +n 2 +n 3 =0 3 j=1 k j θ(k j ) 2 (N 1 ,N 2 ,N 3 )∈[N 1 ,N 2 ,N 3 ] N j (τ j , n j ),

3 j=1((k 1 m 3 -k 3 m 1 ) 2 k 1 k 2 k 3 implies that L 3 ≥ 1 ,L 2 ,L 3 J L 1 ,L 2 ,L 3 N 1 ,N 2 ,N 3 1 N 2 ∼ N 3 : 2 N 1 .

 3333131312312321 τ j -p 5 (n j )) = -5k 1 k 2 k 3 (k 2 3 + k 3 k 1 + k 2 1 ) -CN 4 max N min . Thus L By symmetry, we only need estimate J L 1 ,L 2 ,L 3 N 1 ,N 2 ,N 3 separately according to L 1 = L max or L 1 L 3 = L max .From modulation condition, we have L max N 4 Using mean value theorem, we observe that3 j=1 k j θ(k j ) 2 =k 1 θ(k 2 1 ) + k 2 θ(k 2 ) 2 + k 3 θ(k 2 ) 2 + 2k 1 θ(k 2 + λk 1 )θ (k 2 + λk 1 ) =O (N 1 )

2 I∂ x (u 2 )P,Nmax> N 4 IP,Nmax> N 4 C u 3 X 2 .

 224432 (t)Iu(t)dtdxdy = [N 1 ,N 2 ,N 3 ] ≤ fixed dyadic number N 0 , #{[N 1 , N 2 , N 3 ] : N max = N 0 } ≤ 3 log 2 (N 0 )Combining with Lemma 5.1, we havesup t∈[0,T 1 ] Iu(t) 2 L 2 (T 2 ) ≤ Iu(0) 2 L 2 (T 2 )

.

1 2 2 t

 12 100 h ≤ N ≤ 2 100 h , any solution w to (5.5.3) satisfiesψ T (t)w N 2 L ,x ≤C T h l ψ T (t)w N X

1 h 1 A 2 t

 112 (x, h 1 D x ), h 2l+1 1 2l+2 ∂ t -ip (h 1 D x ) ψ T w N , ψ T w N L 2 t,x = (ϕ (x)ψ T w N , w N ) L 2 t,x + O(h 1 ) ψ T w N 2 L ,x .

5 .3 as h 2l+1 1 ∂

 51 t w N -iq (h 1 D x )w N = h

2 L 2 t 2 L 2 t

 2222 5.3),ψ T P ≥N 1 w +C 0 ,T (gP ≥N 1 /2 w)

+

  [g, P N ] L 2 →L 2 ≤ C N , C T h 2 ψ T w ≥N 0 /2Finally from Lemma 5.6 and similar arguments, we haveψ T w ≥ 2

5 (T ) if l = 2 . 0 G

 520 y ) u n = -α n u n ∂ x u n -G * G u n . The classical boot-strap argument yields ( u n ) is a bounded sequence in E(T ), where E(T ) := Z 0, 1 2 (T ) if l = 1 and E(T ) = X 0,0;b,b 1 Observe that u n (0) L 2 (T 2 ) = 1,andT u n (t) 2L 2 (T 2 ) dt → 0.

0<|k|≤2N 0 ,|m|≥M 0 / 2 a k,m e ikx+imy e -it k 2l+1 -m 2 k

 22 Gv(t, x, y) = k,|m|≥M 0 /2 e ikx+imy 0<|k 1 |≤2N 0

  )e -ikx dx which satisfies g 0 = 1 and |g k | < 1 for all k = 0. Therefore,inf 0<|k 1 |≤N 0 k |g k-k 1 -g k g k 1 | 2 ≥ c 0 > 0 independent of N 0 .Hence the constant C 2 does not depend on N 0 .

2 L 2 (G u n (t) 2 L 2 (T 2 )n u n 2 EPP≥N 0 ũn 2 L 2 (T 2 ) = -α n T 2 P 2 n 2 dx - T 2 P

 2222222222222 dt + C T α 2 ≤2N 0 Q ≥M 0 /2 u n (0) 2 L 2 (T 2 ) ≥N 0 u n L 2 ([0,T /2]×T 2 ) ≤ C 2 (T, R 0 ) ≥N 0 u n • P ≥N 0 ∂ x u ≥N 0 u n P ≥N 0 G * G u n dx.

2 ( 5 ∂

 25 .6.6) by taking N 0 and M 0 large enough. Now since α n u n ∂ x u n → 0, strongly in L 2 ([0, T ] × T 2 ) as well as G * G u n , there exists a subsequence of u n (still denoted by u n ) which converges to u weakly in L 2 ([0, T ] × T 2 ) and u is the solution of linear KP-II equation t u + (-1) l-1 ∂ 2l+1x u + ∂ -1 x ∂ 2 y u = 0 u| t=0 = u 0 ∈ L 2 0 (T 2), G u| ≡ 0, in D 0 (T 2 ).

( 5 . 6 . 7 ) 2 χ 0 C

 56720 On the support of g, we have u(t, x, y) = C(t, y). Rewriting the equation as∂ x (∂ t u + (-1) l-1 ∂ 2l+1 x u) + ∂ 2 y u = 0,we have that ∂ 2 y C(t, y) = 0 and thus u [0,T ]×ω = C(t), in the distributional sense. From the equation, we also have∂ -1 x ∂ 2 y u| [0,T ]×ω = -C (t) in D ([0, T ] × ω). Take non-zero function χ ∈ C ∞ c (ω), depending only on x variable and χ ≥ 0, we have-(x)∂ -1x ∂ 2 y u(t, x, y)dxdy = 0 from integrating by part on y variable. This implies that T (t)ψ(t)dt = 0 for any ψ ∈ C ∞ c ((0, T )).

v Z 0, 1 2Z 0, 1 2

 11 ([nT,(n+1)T ]) ≤ C T v(nT ) L 2 (T 2 ) + C T v ([nT,(n+1)T ]) .Now fix any c 1 ∈ (0, c 0 ), we take T > 0 large enough such that C 0 e -c 0 T < 1 2 e -c 1 T .

( 5 .A. 1 )

 51 Consider the damped general linear KP-II∂ t u + (-1) l-1 ∂ 2l+1 x u + ∂ -1 x ∂ 2 y u = -G * Gu, (t, x) ∈ R × T 2 , u| t=0 = u 0 ∈ L 2 0 (T 2 ), Proposition 5.11Let s ≥ 0. There exists constants C = C s , γ = γ s > 0 such that for all solutions u to (5.A.1) with initial data u 0 ∈ H s 0 (T 2 ), u(t) H s (T 2 ) ≤ C e -γt u 0 H s (T 2 ) .

u 0 2 L 2 (T 2 ) ≤ C T T 0 GS 2l+1 (t)u 0 2 L 2 (

 222022 T 2 ) dt.(5.A.3)

2 L 2 (T 2 ) dt, then u 0 2 L 2 (T 2 ) 0 Gv(t) 2 L 2 (T 2 ) 0 Gu(t) 2 L 2 (Otherwise, we have u 0 2 L 2 (T 2 ) ≤ C T T 0 Gv(t) 2 L 2 (T 2 ) 0 Gu(t) 2 L 2 ( 0 Gw(t) 2 L 2 ( 2 T 0 Gu L 2 (

 22222202220222220222022022202 = v(0) 2 L 2 (T 2 ) ≤ C T T dt ≤ C T T T 2 ) dt. dt ≤ C T T T 2 ) dt + T T 2 ) dt .From (5.A.4) we haveGw L 2 ([0,T ]×T 2 ) ≤C w L 2 ([0,T ]×T 2 ) ≤ C( )T 1 [0,T ]×T 2 ) ,

  by using the equation Ph w N = h 2l+1 f N that 1 h ([A(x, hD x ), P h ]v N , v N ) L 2 t,x = 1 h (A(x, hD x )P h (v N ), v N ) L 2 t,x -(P h A(x, hD x )(v N ), v N ) L 2 t,x = h 2l A(x, hD x )f N ψ T , w N L 2 t,x + h 2l A(x, hD x )ψ T w N , v N L 2 t,x + h 2l A(x, hD x )v N , ψ T w N L 2 t,x + h 2l A(x, hD x )ψ T w N , ψ T f N L 2 t,x =h 2l (A(x, hD x )ψ T f N , ψ T w N ) L 2 t,x + h 2l (A(x, hD x )ψ T w N , ψ T f N ) L 2

		t,x
	+O(h 2l ) ψ T w N	2 L 2 t,x .

  GP ≤N 0 u n (t) 2 L 2 (T 2 ) dt + C T α 2 n GP ≤N 0

	2
	L 2 (T 2 )
	T
	≤C T
	0

t 0 S 2l+1 (t -t ) u n ∂ x u n (t )dt 2 L 2 ([0,T ]×T 2 ) +C T GP ≤N 0 t 0 S 2l+1 (t -t )G * G u n (t )dt

  ≤C( )Gv(t) 2 L 2 (T 2 ) -(1 -) Gu(t) 2 L 2 (T 2 ) .

	1 2	d dt	w(t) 2 L 2 (T 2 ) =	T 2	Gv(t) • Gu(t)dxdy -Gu(t) 2 L 2 (T 2 )
	Thus
			w(t) 2 L

2 (T 2 ) ≤ C( ) t 0 Gv(t ) 2 L 2 (T 2 ) dt .

More generally, one could add the third order linear term a∂ 3 x u to the equation.

(T ) ≤ C(T, u 0 L 2 (T 2 ) ) u 0 H s (T 2 ) .

Though the we only treat KP-II equation, namely l = 1, in chapter

3, the generalization to any integer l > 1 is trivial.

Remerciements

superbe cours de M2. Merci encore à Nikolay Tzvetkov qui m'a offert un post-doc à

The resulting closed-loop system is called damped KP-II equation

and damped 5KP-II equation

(5.1.7)

To state our main results, we define the δ 0 -partial compact subset of L 2 0 (T 2 ) by

The main results in this paper for KP-II equation are as follows.

Theorem 5.1 Let s > 0, R 0 > 0 be given. Then, there exist C > 0, γ > 0 and δ 0 > 0 such that the inequality u(t) H s (T 2 ) ≤ Ce -γt u 0 H s (T 2 ) , t > 0 holds for every solution u of damped KP-II equation (5. 1.6) with initial data in S δ 0 such that u 0 H s (T 2 ) ≤ R 0 .

From the usual arguments combining with the stabilization theorem and exact control result for small-data established in chapter 3, we have the exact control result for large data as follows:

Corollary 5.1 Let s > 0, R 0 > 0 be given. There exist T > 0, δ 0 > 0, such that if u 0 , u 1 ∈ H s 0 (T 2 ) ∩ S δ 0 satisfy u 0 H s (T 2 ) ≤ R 0 , u 1 H s (T 2 ) ≤ R 0 , then one can find a control input Gh with h ∈ L 2 ((0, T ); H s (T 2 )) such that the system (5.1.4) with l = 1 admits a solution u ∈ C([0, T ]; H s 0 (T 2 )) satisfying

For 5KP-II, less regularity of the initial data is needed for stabilization and exact controllability. More precisely, we have Theorem 5.2 Let s > 0, R 0 > 0 be given. Then, there exist C > 0, γ > 0 and δ 0 > 0 such that the inequality u(t) H s (T 2 ) ≤ Ce -γt u 0 H s (T 2 ) , t > 0 holds for every solution u of damped KP-II equation (5.1.7) with initial data in S δ 0 such that u 0 H s (T 2 ) ≤ R 0 .

Moreover,

for some ν > 0.All the constants C are independent of T .

Cauchy-Problem for nonlinear system with damping

First we prove global well-posedness of damped KP-II equation

Theorem 5.3 Fix any s ≥ 0. For any given T > 0 and any u 0 ∈ H s 0 (T 2 ), there exists a unique solution

. Moreover, we have the following estimate

The positive function α(T, R) depends polynomial in T .

Proof. Given u 0 ∈ H s 0 (T 2 ). For each v ∈ Z s, 1 2 (T ), we extend v to a function on R × T 2 and still denote by v. For any given u 0 ∈ H s 0 (T 2 ), define the map

For t ≤ T ≤ 1, we rewrite it as

From linear and bilinear estimates, we have

We estimate

provided that b 1 + b ≤ 0, thanks to Proposition 5.1. Applying the estimate above to b = -1 2 + and v = G * Gψ T u, we have

for some ν > 0, thanks to the condition b 1 < 3 8 . This implies that

Similar manipulation yields

Therefore, by taking r > 2C u 0 H s (T 2 ) and T = T ( u 0 ) > 0, the map Φ u 0 : B T (r) → B T (r) is well-defined and contraction. This implies that For any given u 0 ∈ H s 0 (T 2 ), there exists T 0 = T 0 ( u 0 H s (T 2 ) ) such that the solution u ∈ Z s, 1 2 (T 0 ) ∩ C([0, T 0 ]; H s 0 (T 2 )) exists and unique. Moreover, u

We next extend this local solution to global one. First we consider the case s = 0. From the mass dissipation

This implies the global existence for s = 0.

Now we want to prove the global existence for all s ≥ 0. Suppose that u 0 ∈ H 3 (T 2 ), let v = ∂ t u, w = ∂ y u, then v, w satisfy the same equation

Proof. The proof is similar as the proof of Theorem 5.3, with the minor difference of the choice of functional spaces. We just write down the contraction step here. Define the map

and the ball

(T ) ≤ r}. Applying Proposition 5.2 and Proposition 5.5, we have

where to the last inequality, we have used the estimate

(T ) . Similar manipulation yields

By choosing r > 0 large and T > 0 small enough, Φ u 0 is a contraction map on B T (r).

Long time a priori estimate

KP-II

Next we prove a lemma which ensures the smallness of low frequency portion as well as H -1,0 norm along KP-II flow.

Proposition 5.6

Assume R 0 > 0, T 0 > 0 are given. For any > 0, there exists a small number δ 0 > 0, such that for all solutions u of (5.1.6) with

Corollary 5.4

We have that for any u 1 , u 2 , any 0 < 1,

The proof of Proposition 5.7 follows the I-method firstly introduced by the authors of [START_REF] Colliander | Global well-posedness for schrödinger equations with derivative[END_REF]. In order to take advantage of several cancellation relations, we first introduce some notations briefly. For any two triples

iff there exists a permutation σ, such that

We denote by P all the equivalence classes and [N 1 , N 2 , N 3 ], the elements in P. P is the quotient set of (2 N ) 3 /S 3 and it gives a partition of (2 N ) 3 .

Proof of Proposition 5.7.

We split

where the dyadic summation is taken over all dyadic numbers N 1 , N 2 , N 3 ≥ 1. Moreover, since IP ≤N = 1, we can only take the summation above over all triples (N 1 , N 2 , N 3 ) such that

To estimate the increasing of Iu(t) 2 L 2 , we need integrate it over the time interval [0, T ] and we denote by I N 1 ,N 2 ,N 3 . Using the symmetric notation, we rewrite the summation we want to estimate as

To remove the annoyed non-smooth time localization 1 [0,T 0 ] , we perform as in [START_REF] Colliander | Global well-posedness for schrödinger equations with derivative[END_REF]. For any fixed term [N 1 , N 2 , N 3 ] ∈ P, we pick a smooth approximation to the identity G(t) of width

Observability for high frequencies

In this section, we will consider the inhomogeneous linearized KP-II type equations in a uniform way

where the system is KP-II if l = 1 and 5-KP-II if l = 2. Each Fourier mode in y, u m , satisfies

To capture the high frequency information |m| 1, we write |m| = 1 h l+1 and reduce the study of (5.5.2) to the semi-classical equation

(5.5.3)

Semi-classical estimate

To perform the one dimensional analysis and simplify the notation, we introduce semiclassical Bourgain space via the norm

We need a lemma. In what follows, we pick a family of parameter-dependent cutoff function

uniformly for any 0 < ≤ 1 and 0 < h 1 1.

Proof. The proof follows from direct calculation. We only prove (5.5.4) and (5.5.5) will follow in the same way. Take v = χ c 0 (h 1 D x )u, we have

Combining (5.5.7),(5.5.8), we have obtained that

for any function ϕ ∈ C ∞ (T). Fix any x 0 ∈ T and any smooth function θ, there exists ϕ ∈ C ∞ (T) such that ϕ (x) = θ(x) -θ(x -x 0 ). Now we take θ ≥ 0 such that supp(θ) ⊂ x ∈ T : g(x) > 1 4 and this implies that

Now we conclude by choosing a finite set S of x 0 such that

Lemma 5.5 Given T > 0. There exists 0 < 0 1 and 0 < h 0 1, h 0 0 , such that for any 1 0 < N ≤ 1 2 100 h and any solution w N of (5.5.3), we have

(5.5.10)

where the constant C 0 ,T depends only on 0 , T .

Proof. We first chose 0 > 0, small enough to fit the semi-classical pseudo-differential calculus, which will be made precise implicitly later. Define a new semi-classical parameter h 1 := h = 1 N ≤ 0 , we rewrite the equation (5.5.3) as

As in the proof of previous lemma, we compute the commutator

high-frequency estimate for inhomogeneous linear equation

Proposition 5.9

Given T > 0, there exist a large dyadic number N 0 > 0 and C = C(T, N 0 ) > 0 such that for any solutions u to (5.5.1) and any N 1 ≥ N 0 , we have

(5.5.21)

Moreover, the same inequality holds if the second term on the right hand side is replaced by

Proof. Take an integer m 0 > 0 such that m -1 0 ≤ h 0 , where h 0 > 0 is given in Proposition 5.8. For any integer m, |m| ≥ m 0 , we define a semi-classical parameter h = |m| -1 l+1 , and from Proposition 5.8, we have

Summing over all integers |m| ≥ m 0 , we obtain that

where we have used that fact that π m is interchangeable with the multiplication by g(x). Now we claim that the inequality (5.5.22) also holds true Indeed if we change π m u by π ≤m 0 u. This fact can be deduced easily by repeating the proof of Lemma 5.6.

Indeed, in the proof of Proposition 5.8, the semi-classical parameter is chosen so that h < h 0 ∼ 1 2 100 N 0 . This means that m 0 ∼ 2 100(l+1) N l+1 0 . The small number 0 > 0 can be firstly fixed to fit the h 1 pseudo-differential calculus (in x variable). The dyadic number N 0 1 0 has been fixed after fixing 0 . We can then choose

. Therefore, we can apply the proof of Lemma 5.6 to gain a factor N -l 1 in front of ψ(t)π m F X 0,-1 2 , uniformly in 0 < m ≤ m 0 (In the proof of Lemma 5.6, we do the semi-classical analysis with respect to h 1 ∼ h instead of h). Summing over all 0 < m ≤ m 0 yields

since g commutes with π ≤m 0 π =0 . To complete the proof of (5.5.21), it remains to treat the zero Fourier mode π 0 u. Notice that v = P ≥N 1 π =0 u satisfies the inhomogeneous KdV

equation and the argument in the proof of Lemma 5.6 still valid.

Finally, In order to replace ψ T (t)P ≥N 1 /2 (gu) L 2 (R×T 2 ) by ψ T (t)P ≥N 1 /2 (Gu) L 2 (R×T 2 ) , we write

Since g is smooth, we have

N 1 , and we finally obtain that We first prove the conditional observability for both general KP-II type equations

Stabilization

Proposition 5.10 Given R 0 > 0, T > 0, there exist 0 > 0 and C T > 0, such that for all u 0 ∈ L 2 0 (T 2 ) satisfying u 0 L 2 (T 2 ) ≤ R 0 , if the solution u to (5.6.1) with l = 1 or l = 2 satisfies

The constant C T depends only on T , R 0 and 0 .

Proof. We argue by contradiction. Suppose there exists a sequence u n,0 ∈ L 2 0 (T 2 ) with

and

Denote by α n := u n,0 L 2 (T 2 ) and β n := sup t∈[0,T ] u n (t) H -1,0 (T 2 ) . From Cauchy theory, we have u n Z 0, 1 2 (T ) ≤ α(T, R 0 ), if l = 1 and u n X 0,0;b,b 1 5

Case 1 : α n → α > 0.

After normalization, we may assume that u n,0 L 2 (T 2 ) = 1 for all n and

Thanks to Corollary 5.9, for any large number N 1 , we have

where the constant C 1 is independent of N 1 . From Cauchy-theory, commutator estimate and bilinear-estimate, we have

with δ n → 0 as n → ∞. Since

Thus we can apply Proposition 5.10 for time interval [nT, (n + 1)T ] with 0

This implies that

(5.6.8)

for n ≤ n 0 -1 and u(T 0 ) 2 L 2 (T 2 ) ≤ 2 0 . Since the L 2 norm is non increasing, (5.6.8) holds true for all n. This completes the proof.

The same argument yields the stabilization for 5KP-II.

Corollary 5.6

Let R 0 > 0 be given. There exist C > 0, γ > 0 and δ 0 > 0 such that the inequality

holds for every solution u of damped 5KP-II equation (5.1.7) with initial data in S δ 0 such that u 0 L 2 (T 2 ) ≤ R 0 .

Stabilization for in H s (T 2 )

Proof of Theorem 5.1. Now we turn to the proof of stabilization for general s ≥ 0. This is a consequence of L 2 -stabilization and a stability argument. We mainly follow the strategy in [START_REF] Laurent | Control and stabilization of the korteweg-de vries equation on a periodic domain[END_REF], in which the authors dealt with the stabilization for KdV equation.

Define the semi-group S G (t) be the solution map associated with the dissipative equation

We need the following lemma about the bilinear estimate for the semi-group S G (t).

Lemma 5.8

Let I ⊂ R be any finite time interval. Then the bilinear estimate

(5.6.10) The proof is similar as in [START_REF] Laurent | Control and stabilization of the korteweg-de vries equation on a periodic domain[END_REF]. For the sake of completeness, we briefly recall the proof here.

Proof. We may assume that I = [0, T ]. From Duhamel furlmula, we have S G (t)u 0 = S(t)u 0 -t 0 S(t -t )G * G(S G (t )u 0 )dt .

For the first inequality, we use the usual bilinear estimate and the fact(see the proof of Theorem 5.3)

(5.6.12)

to get S G (t)u 0 Z s, 1 2 (T ) ≤C u 0 H s (T 2 ) + CT ν S G (t)u 0 Z s, 1 2 (T ) .

Then (5.6.10) follows for all T < T 0 with T 0 ≤ (2C) -1 ν . For T ≥ T 0 , we can partition [0, T ] into finite many intervals of length at most T 0 and conclude by induction. Now we prove (5.6.11). Observe that .

Thus there exists T 0 > 0 such that CT ν 0 < 1 2 and (5.6.11) is true for all T ≤ T 0 . The case T > T 0 follows from (5.6.10) and induction. Now we prove the stabilization for any s ≥ 0. We first deal with the case s = 3. From L 2 -stabilization and Cauchy theory, for any T > 0, there exists a number C T = C(T, u 0 L 2 (T 2 ) ) such that for all t ≥ 0, u Z 0, 1 2 ([t,t+T ]) ≤ C T e -c 0 t u 0 L 2 (T 2 ) .