Low regularity solutions for gravity water waves

TL;DR

This paper establishes local well-posedness for gravity water waves without surface tension at lower regularity levels than previously known, by refining coordinate change analysis and optimizing Strichartz estimates.

Contribution

It extends prior results by lowering regularity requirements through improved coordinate change analysis and time-interval optimization of Strichartz estimates.

Findings

Proves local well-posedness for initial velocity in $H^s$, with $s$ below previous thresholds.

Improves analysis of the Eulerian to Lagrangian coordinate change.

Optimizes the use of localized Strichartz estimates over time intervals.

Abstract

We prove local well-posedness for the gravity water waves equations without surface tension, with initial velocity field in $H^s$, $s > \frac{d}{2} + 1 - \mu$, where $\mu = \frac{1}{10}$ in the case $d = 1$ and $\mu = \frac{1}{5}$ in the case $d \geq 2$, extending previous results of Alazard-Burq-Zuily. The improvement primarily arises in two areas. First, we perform an improved analysis of the regularity of the change of variables from Eulerian to Lagrangian coordinates. Second, we perform a time-interval length optimization of the localized Strichartz estimates.