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 and estimates. The 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.
@article{SLSEDP_2012-2013____A18_0, author = {Thomas Alazard}, title = {About global existence and asymptotic behavior for two dimensional gravity water waves}, journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications}, note = {talk:18}, pages = {1--16}, publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique}, year = {2012-2013}, doi = {10.5802/slsedp.44}, language = {en}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.44/} }
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%0 Journal Article %A Thomas Alazard %T About global existence and asymptotic behavior for two dimensional gravity water waves %J Séminaire Laurent Schwartz — EDP et applications %Z talk:18 %D 2012-2013 %P 1-16 %I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique %U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.44/ %R 10.5802/slsedp.44 %G en %F SLSEDP_2012-2013____A18_0
Thomas Alazard. About global existence and asymptotic behavior for two dimensional gravity water waves. Séminaire Laurent Schwartz — EDP et applications (2012-2013), Talk no. 18, 16 p. doi : 10.5802/slsedp.44. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.44/
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