On the Cauchy problem for gravity water waves

TL;DR

This paper investigates the initial regularity thresholds for gravity water waves without surface tension, focusing on low-regularity initial conditions with unbounded curvature and Lipschitz velocities, using paradifferential methods.

Contribution

It establishes optimal regularity thresholds for initial data in gravity water waves, particularly for surfaces of class $C^{3/2+ ext{epsilon}}$ and Lipschitz velocities, advancing understanding of low-regularity solutions.

Findings

Initial surfaces can be only $C^{3/2+\epsilon}$ with unbounded curvature.

Initial velocities can be only Lipschitz.

Paradifferential approach effectively reduces the system for analysis.

Abstract

We are interested in the system of gravity water waves equations without surface tension. Our purpose is to study the optimal regularity thresholds for the initial conditions. In terms of Sobolev embeddings, the initial surfaces we consider turn out to be only of~$C^{3/2+\epsilon}$-class for some $\epsilon>0$ and consequently have unbounded curvature, while the initial velocities are only Lipschitz. We reduce the system using a paradifferential approach.