On radial symmetry of rotating vortex patches in the disc

TL;DR

This paper proves that under certain angular velocity conditions, simply-connected rotating vortex patches in a unit disc must be circular, extending symmetry results from the plane to the disc using a variational approach.

Contribution

It establishes radial symmetry of rotating vortex patches in the disc for specific angular velocities, generalizing previous plane results.

Findings

Rotating vortex patches with certain angular velocities are necessarily discs.

The proof employs a variational method inspired by recent plane symmetry results.

The result applies to simply-connected patches in the unit disc.

Abstract

In this note, we consider the radial symmetry property of rotating vortex patches for the 2D incompressible Euler equations in the unit disc. By choosing a suitable vector field to deform the patch, we show that each simply-connected rotating vortex patch $D$ with angular velocity $\Omega$, $\Omega\geq \max\{{1}/{2},({2 l^2})/{(1-l^2)^2}\}$ or $\Omega\leq -({2 l^2})/{(1-l^2)^2}$, where $l=\sup_{x\in D}|x|$, must be a disc. The main idea of the proof, which has a variational flavor, comes from a very recent paper of G\'omez-Serrano--Park--Shi--Yao, arXiv:1908.01722, where radial symmetry of rotating vortex patches in the whole plane was studied.