On the rapidly rotating vorticity in the unit disk

Abstract: In this paper, we obtain uniformly rotating vorticity with sufficiently large angular velocity in the unit disk. The solution consists of either a small nearly-ellipse vortex patch which is highly concentrated near the origin or a $2+1$ configuration in which the another two vortical components are very close to the boundary of fluid domain. The total vorticity are prescribed and the sizes of vortical domains are not necessarily small in the same order. The construction, which is based on a perturbation argument, exhibits a subtle multiscale phenomenon in this highly rotating fluid.