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

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 and L∞ estimates. The L 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.