On the rigidity of uniformly rotating vortex patch near the Rankine vortex

Abstract

In this paper, we study the uniformly rotating vortex patch solutions for the 2D incompressible Euler equations. Specifically, we prove that if the patch solution is close to the Rankine vortex in a certain weak topology, it is either the Kirchhoff ellipses or the Rankine vortex.

In the current version, a much more direct proof of Theorem 1.2 is given without using Riemann mapping