Existence and Regularity for Vortex Patch Solutions of the 2D Euler Equations

TL;DR

This paper proves the regularity and existence of vortex patch solutions to the 2D Euler equations using a level-set framework, extending previous a priori estimates and demonstrating higher regularity propagation.

Contribution

It introduces a method to establish vortex patch regularity and existence solely within the level-set framework, building on prior a priori estimates.

Findings

Constructed vortex patch solutions with higher Hölder regularity

Proved propagation of regularity over time

Extended tools for a priori estimates in the level-set framework

Abstract

In "Global regularity for vortex patches" (Commun. Math. Phys. 1993), Bertozzi and Constantin formulate the vortex patch problem in the level-set framework and prove a priori estimates for this active scalar equation. By extending the tools used to prove these estimates, we construct solutions and show propagation of higher H\"older regularity. This constitutes a proof of the regularity of vortex patches, carried out solely in the level-set framework.