On 3D water waves system above a flat bottom

TL;DR

This paper advances understanding of the Dirichlet-Neumann operator in 3D water waves over a flat bottom and derives new energy estimates, laying groundwork for proving global regularity of gravity waves.

Contribution

It improves the analysis of the Dirichlet-Neumann operator and introduces a novel $L^2-L^ Infty$ energy estimate for 3D gravity waves.

Findings

Enhanced understanding of the Dirichlet-Neumann operator in 3D water waves.

Derived a new $L^2-L^ Infty$ energy estimate with good structure.

Foundation for proving global regularity of 3D gravity waves.

Abstract

As a starting point of studying the long time behavior of the $3D$ water waves system in the flat bottom setting, in this paper, we try to improve the understanding of the Dirichlet-Neumann operator in this setting. As an application, we study the $3D$ gravity waves system and derive a new $L^2-L^\infty$ type energy estimate, which has a good structure in the $L^\infty-$type space. In our second paper, base on the results we obtained in this paper, we prove the global regularity of the $3D$ gravity waves system for suitably small initial data.