Global solutions and asymptotic behavior  for two dimensional gravity water waves

Alazard, Thomas; Delort, Jean-Marc

doi:10.24033/asens.2268

Global solutions and asymptotic behavior for two dimensional gravity water waves
[Solutions globales et comportement asymptotique pour l'équation des ondes de gravité en dimension deux]

Alazard, Thomas ; Delort, Jean-Marc

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 5, pp. 1149-1238.

Résumé
Abstract

Cet article est consacré à une preuve d'un résultat d'existence globale pour l'équation des ondes de gravité à données de Cauchy régulières, petites et décroissantes à l'infini. On obtient de plus une description asymptotique de la solution dans les coordonnées physiques, qui montre qu'il y a diffusion modifiée.

La démonstration est basée sur un argument inductif faisant intervenir des estimations a priori dans $L^{2}$ et $L^{\infty}$ . Les bornes $L^{2}$ sont prouvées dans [5], texte complémentaire au présent article. Elles reposent sur une méthode de formes normales paradifférentielles permettant d'obtenir des estimations d'énergie sur la formulation eulérienne de l'équation des ondes de gravité. Nous donnons ici une démonstration des bornes uniformes, en interprétant l'équation de manière semi-classique, et en combinant la méthode des champs de vecteurs de Klainerman avec la description de la solution en termes de distributions lagrangiennes semi-classiques. Cela nous permet, compte tenu des estimations $L^{2}$ de [5], d'en déduire notre principal résultat d'existence globale.

This paper is devoted to the proof of a global existence result for the water waves equation with smooth, small, and decaying at infinity Cauchy data. We obtain moreover an asymptotic description in physical coordinates of the solution, which shows that modified scattering holds.

The proof is based on a bootstrap argument involving $L^{2}$ and $L^{\infty}$ estimates. The $L^{2}$ bounds are proved in the companion paper [5] of this article. They rely on a normal forms paradifferential method allowing one to obtain energy estimates on the Eulerian formulation of the water waves equation. We give here the proof of the uniform bounds, interpreting the equation in a semi-classical way, and combining Klainerman vector fields with the description of the solution in terms of semi-classical Lagrangian distributions. This, together with the $L^{2}$ estimates of [5], allows us to deduce our main global existence result.

Publié le : 2015-09-24

MR | 1 citation dans Numdam

DOI : 10.24033/asens.2268

Classification : 76B15, 35L60.
Keywords: Water waves equations, global solutions, Klainerman vector fields, Lagrangian distributions, semiclassical analysis.
Mot clés : Équation des ondes de gravité, solutions globales, champs de vecteurs de Klainerman, distributions lagrangiennes, analyse semi-classique.

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Alazard, Thomas; Delort, Jean-Marc. Global solutions and asymptotic behavior  for two dimensional gravity water waves. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 5, pp. 1149-1238. doi : 10.24033/asens.2268. http://www.numdam.org/articles/10.24033/asens.2268/

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