Global solution for the 3D gravity water waves system above a flat bottom

TL;DR

This paper proves global regularity for the 3D gravity water wave system with small, localized initial data, and shows the non-existence of small traveling waves in 3D, contrasting with 2D cases.

Contribution

It establishes the first global regularity result for 3D gravity water waves with finite depth and rules out small traveling waves in 3D.

Findings

Global regularity for 3D water waves with small initial data

Non-existence of small traveling waves in 3D

Contrast with 2D water wave behavior

Abstract

Given any suitably small, localized, and smooth initial data, in this paper, we prove global regularity for the $3D$ finite depth gravity water wave system. As a byproduct, we rule out the small, localized traveling waves in $3D$, which do exist for the same system in $2D$.