Desingularization of nondegenerate rotating vortex patches

TL;DR

This paper develops a desingularization method for steady rotating vortex patches in 2D Euler equations, showing they can be approximated by smooth solutions with symmetry and nondegeneracy conditions.

Contribution

It introduces a novel desingularization technique for vortex patches, applicable to general steady rotating solutions satisfying a natural nondegeneracy condition.

Findings

Vortex patches satisfying the nondegeneracy condition are limits of smooth rotating solutions.

Kirchhoff ellipses satisfy the nondegeneracy condition and are included.

The method constructs exotic vortex solutions near a given nondegenerate state.

Abstract

This paper analyzes the space of steady rotating solutions to the two-dimensional incompressible Euler equations nearby vortex patch solutions satisfying a natural nondegeneracy condition. We address the question of desingularization and prove that such vortex patch states are the limit of rotating Euler solutions that are smooth to infinite order, have compact vorticity support, and respect dihedral symmetry. Our nondegeneracy condition is proved to be satisfied by Kirchhoff ellipses and along the local bifurcation curves emanating from the Rankine vortex. The construction, which is based on a local stream function formulation in a tubular neighborhood of the patch boundary, is a synthesis of delicate analysis on thin domains, nonlinear a priori estimates, and a custom version of Newton's method. Our techniques are robust enough to additionally allow us to construct exotic families of singular rotating vortex patch-like solutions nearby a given nondegenerate state. To the best of the authors' knowledge, this work constitutes the first desingularization procedure applicable to general families of steady rotating vortex patches.