Steady vortex patch solutions to the vortex-wave system

TL;DR

This paper proves the existence of steady vortex patch solutions in a 2D ideal fluid system with a single point vortex, showing their supports shrink to a minimum point as background vorticity increases.

Contribution

It establishes the existence of steady vortex patch solutions with prescribed background vorticity in the vortex-wave system and analyzes their asymptotic behavior.

Findings

Existence of steady vortex patch solutions with prescribed background vorticity.

Supports of solutions shrink to a minimum point as background vorticity strength increases.

Solutions' supports tend to concentrate at critical points of the Kirchhoff-Routh function.

Abstract

The vortex-wave system describes the motion of a two-dimensional ideal fluid in which the vorticity includes continuously distributed vorticity, which is called the background vorticity, and a finite number of concentrated vortices. In this paper we restrict ourselves to the case of a single point vortex in bounded domains. We prove the existence of steady vortex patch solutions to this system with prescribed distribution for the background vorticity. Moreover, we show that the supports of these solutions "shrink" to a minimum point of the Kirchhoff-Routh function as the strength parameter of the background vorticity goes to infinity.