Two dimensional water waves in holomorphic coordinates

TL;DR

This paper studies the two-dimensional infinite depth water wave equation in holomorphic coordinates, providing improved, unified results on local well-posedness and almost global solutions with simpler proofs.

Contribution

It offers a unified framework for well-posedness and long-time solutions, with sharper results and simplified proofs compared to previous work.

Findings

Established local well-posedness in Sobolev spaces.

Proved existence of almost global solutions for small data.

Provided simpler, sharper proofs for these results.

Abstract

This article is concerned with the infinite depth water wave equation in two space dimensions. We consider this problem expressed in position-velocity potential holomorphic coordinates. Viewing this problem as a quasilinear dispersive equation, we establish two results: (i) local well-posedness in Sobolev spaces, and (ii) almost global solutions for small localized data. Neither of these results are new; they have been recently obtained by Alazard-Burq-Zuily \cite{abz}, respectively by Wu \cite{wu} using different coordinates and methods. Instead our goal is improve the understanding of this problem by providing a single setting for both problems, by proving sharper versions of the above results, as well as presenting new, simpler proofs. This article is self contained.