On radial symmetry of rotating vortex patches in the disc

2019/09/01 by Wang, Guodong, Zuo, Bijun
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1909.00359

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 Ω, Ω≥ max\1/2,(2 l²)/(1-l²)²\ or Ω≤ -(2 l²)/(1-l²)², where l=sup_{x∈ D}|x|, must be a disc. The main idea of the proof, which has a variational flavor, comes from a very recent paper of Gómez-Serrano--Park--Shi--Yao, arXiv:1908.01722, where radial symmetry of rotating vortex patches in the whole plane was studied.

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