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About global existence and asymptotic behavior for two dimensional gravity water waves

Thomas Alazard

Séminaire Laurent Schwartz — EDP et applications

The main result of this talk is a global existence theorem 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 estimates. The L 2 bounds are proved in the paper [5]. They rely on a normal forms paradifferential method allowing one to obtain energy estimates on the...

Alentours de la limite incompressible

Thomas Alazard

Séminaire Équations aux dérivées partielles

Le résultat principal de cet exposé énonce que le problème de Cauchy pour les équations adimensionnées d’un fluide général est bien posé sur un intervalle de temps indépendant des nombres de Mach, Reynolds et Péclet.

Strichartz estimates for water waves

Thomas AlazardNicolas BurqClaude Zuily — 2011

Annales scientifiques de l'École Normale Supérieure

In this paper we investigate the dispersive properties of the solutions of the two dimensional water-waves system with surface tension. First we prove Strichartz type estimates with loss of derivatives at the same low level of regularity we were able to construct the solutions in [3]. On the other hand, for smoother initial data, we prove that the solutions enjoy the optimal Strichartz estimates (i.e, without loss of regularity compared to the system linearized at ( η = 0 , ψ = 0 )).

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