Traveling water waves with compactly supported vorticity

TL;DR

This paper proves the existence of two-dimensional traveling water waves with localized vorticity, including point vortices and vortex patches, using bifurcation theory to construct solutions for both periodic and non-periodic cases.

Contribution

It introduces a global bifurcation approach to establish the existence of finite-amplitude solutions with compactly supported vorticity in water waves.

Findings

Existence of solutions with point vortex vorticity

Existence of solutions with vortex patches

Construction of solution continua for periodic and non-periodic cases

Abstract

In this paper, we prove the existence of two-dimensional, traveling, capillary-gravity, water waves with compactly supported vorticity. Specifically, we consider the cases where the vorticity is a $\delta$-function (a point vortex), or has small compact support (a vortex patch). Using a global bifurcation theoretic argument, we construct a continuum of finite-amplitude, finite-vorticity solutions for the periodic point vortex problem. For the non-periodic case, with either a vortex point or patch, we prove the existence of a continuum of small-amplitude, small-vorticity solutions.