Abstract: In this talk, we investigate a steady planar flow of an ideal fluid in a bounded domain and focus on the vortex patch problem with prescribed vorticity strength. There are two methods to deal with the existence 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. 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.
