Existence of steady symmetric vortex patch in a disk

TL;DR

This paper constructs steady symmetric vortex patches in a disk by maximizing kinetic energy under vorticity constraints and shows they shrink to a minimum point of the Kirchhoff-Routh function as vorticity increases.

Contribution

It introduces a variational method to explicitly construct steady symmetric vortex patches in a disk and analyzes their asymptotic behavior as vorticity strength grows.

Findings

Vortex patches are constructed via energy maximization.

As vorticity strength increases, patches shrink to a minimum point of the Kirchhoff-Routh function.

The method provides explicit solutions for steady symmetric vortex configurations.

Abstract

In this paper we construct a family of steady symmetric vortex patches for the incompressible Euler equations in an open disk. The result is obtained by studying a variational problem in which the kinetic energy of the fluid is maximized subject to some appropriate constraints for the vorticity. Moreover, we show that these vortex patches shrink to a given minimum point of the corresponding Kirchhoff-Routh function as the vorticity strength parameter goes to infinity.