Global solutions and asymptotic behavior for two dimensional gravity water waves

TL;DR

This paper proves global existence and describes the asymptotic behavior of solutions to the 2D gravity water waves equations with small, smooth, decaying initial data, demonstrating modified scattering phenomena.

Contribution

It establishes a comprehensive proof of global solutions and asymptotic analysis for 2D gravity water waves, combining normal forms, paradifferential calculus, and semi-classical analysis.

Findings

Global existence of solutions for small initial data

Modified scattering behavior of water waves

Uniform bounds achieved through combined analytical methods

Abstract

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 a companion paper 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 the companion paper, allows us to deduce our main global existence result.