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

Thomas Alazard[1]

  • [1] Département de Mathématiques et Applications École normale supérieure et CNRS UMR 8553 45, rue d’Ulm F-75230 Paris, France

Séminaire Laurent Schwartz — EDP et applications (2012-2013)

  • Volume: 2012-2013, page 1-16
  • ISSN: 2266-0607

Abstract

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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 Eulerian formulation of the water waves equation. The uniform bounds, and the proof of the global existence result, are presented in [4]. These uniform bounds are proved 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.

How to cite

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Alazard, Thomas. "About global existence and asymptotic behavior for two dimensional gravity water waves." Séminaire Laurent Schwartz — EDP et applications 2012-2013 (2012-2013): 1-16. <http://eudml.org/doc/275706>.

@article{Alazard2012-2013,
abstract = {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^\infty $ 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 Eulerian formulation of the water waves equation. The uniform bounds, and the proof of the global existence result, are presented in [4]. These uniform bounds are proved 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.},
affiliation = {Département de Mathématiques et Applications École normale supérieure et CNRS UMR 8553 45, rue d’Ulm F-75230 Paris, France},
author = {Alazard, Thomas},
journal = {Séminaire Laurent Schwartz — EDP et applications},
keywords = {incompressible Euler equations; global existence theorem; water waves equation; asymptotic description},
language = {eng},
pages = {1-16},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {About global existence and asymptotic behavior for two dimensional gravity water waves},
url = {http://eudml.org/doc/275706},
volume = {2012-2013},
year = {2012-2013},
}

TY - JOUR
AU - Alazard, Thomas
TI - About global existence and asymptotic behavior for two dimensional gravity water waves
JO - Séminaire Laurent Schwartz — EDP et applications
PY - 2012-2013
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2012-2013
SP - 1
EP - 16
AB - 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^\infty $ 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 Eulerian formulation of the water waves equation. The uniform bounds, and the proof of the global existence result, are presented in [4]. These uniform bounds are proved 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.
LA - eng
KW - incompressible Euler equations; global existence theorem; water waves equation; asymptotic description
UR - http://eudml.org/doc/275706
ER -

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