On the capillary water waves with constant vorticity

TL;DR

This paper proves local well-posedness for deep water gravity-capillary waves with constant vorticity in two dimensions, using paralinearization, energy estimates, and Strichartz estimates for Sobolev spaces.

Contribution

It establishes the local well-posedness of the water wave system with constant vorticity in Sobolev spaces for the first time using dispersive and energy methods.

Findings

Well-posedness holds for s > 5/4 in Sobolev space H^s.

The water wave system can be paralinearized into a quasilinear dispersive form.

Energy and Strichartz estimates are effective for analyzing the system.

Abstract

This article is devoted to the study of local well-posedness for deep water waves with constant vorticity in two space dimensions on the real line. The water waves can be paralinearized and written as a quasilinear dispersive system of equations. By using the energy estimate and the Strichartz estimate, we show that for $s> \frac{5}{4}$, the gravity-capillary water wave system with constant vorticity is locally well-posed in $\mathcal{H}^{s}(\mathbb{R})$.