Uniqueness of planar vortex patch in incompressible steady flow

TL;DR

This paper proves the local uniqueness of solutions for steady planar vortex patches in bounded domains and confirms that different solution methods yield the same result, especially in convex domains.

Contribution

It establishes the equivalence of vorticity and stream function methods and proves uniqueness of vortex patch solutions in convex domains.

Findings

Vorticity and stream function methods produce the same solutions.

Solutions are locally unique for the vortex patch problem.

Uniqueness holds in convex domains.

Abstract

We investigate a steady planar flow of an ideal fluid in a bounded simple connected domain and focus on the vortex patch problem with prescribed vorticity strength. There are two methods to deal with the existence of solutions for this problem: the vorticity method and the stream function method. A long standing open problem is whether these two entirely different methods result in the same solution. In this paper, we will give a positive answer to this problem by studying the local uniqueness of the solutions. Another result obtained in this paper is that if the domain is convex, then the vortex patch problem has a unique solution.