Long-term regularity of 3D gravity water waves

TL;DR

This paper proves long-term regularity and well-posedness of 3D gravity water wave solutions with small initial data, including unweighted and weighted cases, and analyzes lifespan in periodic settings.

Contribution

It establishes almost global and global well-posedness results for the 3D gravity water wave equation with small initial data, extending understanding of solution longevity.

Findings

Almost global well-posedness for unweighted Sobolev initial data.

Global well-posedness for weighted Sobolev initial data with any positive weight.

Lifespan estimate in periodic case proportional to R divided by epsilon squared and logarithmic factors.

Abstract

We study a fundamental model in fluid mechanics--the 3D gravity water wave equation, in which an incompressible fluid occupying half the 3D space flows under its own gravity. In this paper we show long-term regularity of solutions whose initial data is small but not localized. Our results include: almost global wellposedness for unweighted Sobolev initial data and global wellposedness for weighted Sobolev initial data with weight $|x|^\alpha$, for any $\alpha > 0$. In the periodic case, if the initial data lives on an $R$ by $R$ torus, and $\epsilon$ close to the constant solution, then the life span of the solution is at least $R/(\epsilon^2(\log R)^2)$.