Improved low regularity theory for gravity-capillary waves

TL;DR

This paper advances the mathematical understanding of gravity-capillary water waves by establishing local well-posedness for initial data with lower regularity than previously known, in multiple dimensions.

Contribution

It improves the low regularity well-posedness results for the gravity-capillary water waves system in general dimensions, reducing the regularity threshold needed.

Findings

Established local well-posedness for initial data in $H^s$ with lower $s$ thresholds.

Extended low regularity theory to higher dimensions ($d eq 1$).

Improved upon previous state-of-the-art results in the field.

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.