Title: Uniqueness of planar vortex patch in incompressible steady flow,
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.