Vortex patches choreography for active scalar equations

TL;DR

This paper demonstrates the existence of rotating vortex patch configurations arranged at the vertices of regular polygons for Euler and SQG$_eta$ equations, extending classical point vortex solutions to continuous vorticity distributions.

Contribution

It introduces a method to desingularize polygonal point vortex configurations into vortex patches for Euler and SQG$_eta$ equations using contour dynamics and the Implicit Function theorem.

Findings

Existence of rotating vortex patches at polygon vertices.

Extension of point vortex solutions to vortex patches.

Method applicable to Euler and SQG$_eta$ equations.

Abstract

This paper deals with the existence of $N$ vortex patches located at the vertex of a regular polygon with $N$ sides that rotate around the center of the polygon at a constant angular velocity. That is done for Euler and (SQG)$_\beta$ equations, with $\beta\in(0,1)$, but may be also extended to more general models. The idea is the desingularization of the Thomsom polygon for the $N$ point vortex system, that is, $N$ point vortices located at the vertex of a regular polygon with $N$ sides. The proof is based on the study of the contour dynamics equation combined with the application of the infinite dimensional Implicit Function theorem and the well--chosen of the function spaces.