On the rapidly rotating vorticity in the unit disk

Abstract

In this paper, we obtain families of uniformly rotating vorticity in the unit disk with a sufficiently large angular velocity. The solution is either a small nearly-ellipse vortex patch highly concentrated near the origin or a 2+1 vorticity configuration where another two vortical domains are sufficiently close to the boundary of fluid domain. The vorticity components are not necessarily symmetric nor small in the same order. These solutions are constructed via a perturbational approach and exhibit an interesting multi-scale phenomenon in the planar vortex flows.