Low regularity solutions for gravity water waves

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 $d\ge 2$, 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.

52 pages; reorganized and corrected appendix