Low Regularity Solutions for Gravity Water Waves II: The 2D Case

Abstract

The gravity water waves equations describe the evolution of the surface of an incompressible, irrotational fluid in the presence of gravity. The classical regularity threshold for the well-posedness of this system requires initial velocity field in $$H^s$$, with $$s > \frac{1}{2}+ 1$$, and can be obtained by proving standard energy estimates. This threshold was improved by additionally proving Strichartz estimates with loss. In this article, we establish the well-posedness result for $$s > \frac{1}{2}+ 1 - \frac{1}{8}$$, corresponding to proving lossless Strichartz estimates. This provides the sharp regularity threshold with respect to the approach of combining Strichartz estimates with energy estimates.