Two dimensional water waves in holomorphic coordinates
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.
Keywords
Cite
@article{arxiv.1401.1252,
title = {Two dimensional water waves in holomorphic coordinates},
author = {John Hunter and Mihaela Ifrim and Daniel Tataru},
journal= {arXiv preprint arXiv:1401.1252},
year = {2014}
}
Comments
72 pages. We completely removed subsection 2.4. As a consequence, the proofs in Section 6 are much simpler and straightforward. Minor typos have been fixed