Two Dimensional Water Waves in Holomorphic Coordinates

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 (Invent Math 198:71–163, 2014 ), respectively by Wu (Invent Math 177:45–135, 2009 ) using different coordinates and methods. Instead our goal is to 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.