Local uniqueness of vortices for 2D steady Euler flow

We study the steady planar Euler flow in a bounded simply connected domain, where the profile function of vortices is $f=t_+^p$ with $p>0$ and the vorticity strength is prescribed. By studying the local uniqueness, we prove that the vorticity method and the stream function method actrally give the same solution. In this paper, we also show that if the domain is convex, the vortex problem with this kind of profile function has a unique solution.