Low regularity well-posedness for two-dimensional deep water waves

TL;DR

This paper proves local well-posedness for 2D gravity-capillary water waves in certain Sobolev spaces using a modified energy method, advancing understanding of their mathematical behavior.

Contribution

It introduces the cubic modified energy method to establish well-posedness for 2D gravity-capillary water waves with regularity s > 1.

Findings

Local well-posedness established for s > 1

Application of cubic modified energy method to water waves

Advances mathematical understanding of gravity-capillary wave dynamics

Abstract

The study of gravity-capillary water waves in two space dimensions has been an important question in mathematical fluid dynamics. By implementing the cubic modified energy method of Ifrim-Tataru in the context of gravity-capillary waves, we show that for $s> 1$, the two-dimensional gravity-capillary water wave system is locally well-posed in $\mathcal{H}^{s}$.