Rotating vortex patches for the planar Euler equations in a disk

TL;DR

This paper constructs rotating vortex patches in a disk for the 2D Euler equations, showing their limit as vorticity increases and proving their nonlinear orbital stability under certain perturbations.

Contribution

It introduces a variational construction of rotating vortex patches and establishes their stability, extending understanding of vortex dynamics in bounded domains.

Findings

Limit of vortex patches as vorticity increases is a rotating point vortex.

Constructed vortex patches are orbitally stable under $L^p$ perturbations.

Variational approach based on Arnold's principle successfully models vortex behavior.

Abstract

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 $p\in[{3}/{2},+\infty)$.