Improved low regularity theory for gravity-capillary waves

Abstract

This article concerns the Cauchy problem for the gravity-capillary water waves system in general dimensions. We establish local well-posedness for initial data in $H^s$, with $s > \frac{d}{2} + 2 - \mu$, with $\mu = \frac{3}{14}$ and $\mu = \frac37$ in the cases $d = 1$ and $d \geq 2$ respectively. This represents an improvement over the the state-of-the-art low regularity theory in $d \geq 2$ dimensions.