Global Solutions for the Gravity Water Waves Equation in Dimension 3

TL;DR

This paper proves the existence of global solutions for the 3D gravity water waves equation with small initial data by combining energy and dispersive estimates, emphasizing the role of resonance analysis in Fourier space.

Contribution

It introduces a novel approach that combines energy and dispersive estimates with resonance analysis to establish global solutions for the 3D gravity water waves equation.

Findings

Global solutions exist for small data in 3D water waves

Dispersive estimates are obtained via Fourier space analysis

Resonance analysis is crucial for decay estimates

Abstract

We show existence of global solutions for the gravity water waves equation in dimension 3, in the case of small data. The proof combines energy estimates, which yield control of L^2 related norms, with dispersive estimates, which give decay in L^\infty. To obtain these dispersive estimates, we use an analysis in Fourier space; the study of space and time resonances is then the crucial point.