Rotating vortex patches for the planar Euler equations in a disk

We construct a family of rotating vortex patches with fixed angular velocity for the two-dimensional Euler equations in a disk. As the vorticity strength goes to infinity, the limit of these rotating vortex patches is a rotating point vortex whose motion is described by the Kirchhoff-Routh equation. The construction is performed by solving a variational problem for the vorticity which is based on an adaption of Arnold's variational principle. We also prove nonlinear orbital stability of the set of maximizers in the variational problem under L-p perturbation when p is an element of [3/2, +infinity). (C) 2020 Elsevier Inc. All rights reserved.

Automated summary

The study presents a family of rotating vortex patches with fixed angular velocity in a disk, where the limit of these patches as the vorticity strength approaches infinity is a rotating point vortex. The construction is based on solving a variational problem for the vorticity using an adaption of Arnold's variational principle. Nonlinear orbital stability of the set of maximizers in the variational problem is then proven under L-p perturbation for p in the range of [3/2, +infinity).