On radial symmetry of rotating vortex patches in the disk

摘要

In this note, we consider the radial symmetry property of rotating vortex patches for the 2D incompressible Euler equations in the unit disk. 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 >= max{1/2, (2l(2))/(1 - l(2))(2)} or Omega <= -(2l(2))/(1 - l(2))(2), where l = sup(x is an element of D) vertical bar x vertical bar, must be a disk. The main idea of the proof, which has a variational flavor, comes from a recent paper of Gomez-Serrano-Park-Shi-Yao, arXiv :1908 .01722, where radial symmetry of rotating vortex patches in the whole plane was studied.

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