A sharp Cauchy theory for the 2D gravity-capillary waves

TL;DR

This paper establishes a well-posedness theory for 2D gravity-capillary water waves with less regular initial data than previously known, leveraging advanced Strichartz estimates and paracomposition techniques.

Contribution

It introduces a sharp Cauchy theory for 2D gravity-capillary waves, reducing the regularity requirements for initial data and employing novel semiclassical Strichartz estimates.

Findings

Unique solvability for initial data with 1/4 derivatives less regular than previous thresholds

Global quantitative results for paracomposition theory

Gain of Holder regularity in the solution

Abstract

This article is devoted to the Cauchy problem for the 2D gravity-capillary water waves in fluid domains with general bottoms. We prove that the Cauchy problem in Sobolev spaces is uniquely solvable for data $\frac{1}{4}$ derivatives less regular than the energy threshold (obtained by Alazard-Burq-Zuily), which corresponds to the gain of Holder regularity of the semiclassical Strichartz estimate for the fully nonlinear system. To obtain this result, we establish global, quantitative results for the paracomposition theory of Alinhac.