Finite depth gravity water waves in holomorphic coordinates

TL;DR

This paper studies irrotational gravity water waves with finite bottom, reformulating the equations in holomorphic coordinates, proving local well-posedness, and establishing cubic lifespan bounds for small data solutions, consistent with infinite depth results.

Contribution

It introduces a holomorphic coordinate framework for finite depth water waves and proves lifespan bounds that align with known infinite depth results.

Findings

Holomorphic coordinate representation of water wave equations

Local well-posedness in the holomorphic setting

Cubic lifespan bounds for small data solutions

Abstract

In this article we consider irrotational gravity water waves with finite bottom. Our goal is two-fold. First, we represent the equations in holomorphic coordinates and discuss the local well-posedness of the problem in this context. Second, we consider the small data problem and establish cubic lifespan bounds for the solutions. Our results are uniform in the infinite depth limit, and match the earlier infinite depth result of Hunter-Ifrim-Tataru.