Sobolev estimates for two dimensional gravity water waves

TL;DR

This paper develops Sobolev and $L^2$ estimates for 2D gravity water waves using a normal forms approach and paradifferential calculus, enabling further results on global existence and scattering.

Contribution

It introduces a purely Eulerian paradifferential change of variables that eliminates quadratic terms affecting Sobolev energy estimates for water waves.

Findings

Sobolev estimates for water wave solutions are established.

$L^2$-estimates for derivatives involving Klainerman vector fields are proved.

The results support global existence and scattering analysis in a subsequent paper.

Abstract

Our goal in this paper is to apply a normal forms method to estimate the Sobolev norms of the solutions of the water waves equation. We construct a paradifferential change of unknown, without derivatives losses, which eliminates the part of the quadratic terms that bring non zero contributions in a Sobolev energy inequality. Our approach is purely Eulerian: we work on the Craig-Sulem-Zakharov formulation of the water waves equation. In addition to these Sobolev estimates, we also prove $L^2$-estimates for the $\partial_x^\alpha Z^\beta$-derivatives of the solutions of the water waves equation, where $Z$ is the Klainerman vector field $t\partial_t +2x\partial_x$. These estimates are used in another paper where we prove a global existence result for the water waves equation with smooth, small, and decaying at infinity Cauchy data, and we obtain an asymptotic description in physical coordinates of the solution, which shows that modified scattering holds. The proof of this global in time existence result relies on the simultaneous bootstrap of some H\"older and Sobolev a priori estimates for the action of iterated Klainerman vector fields on the solutions of the water waves equation. The present paper contains the proof of the Sobolev part of that bootstrap.