Existence of Vortex Patch Equilibria for Active Scalars Equations

TL;DR

This paper proves the existence of vortex patch equilibria for generalized surface quasi-geostrophic equations with specific configurations, using desingularization and the implicit function theorem.

Contribution

It introduces a novel approach to construct time-periodic vortex patch solutions for gSQG equations near point vortex equilibria.

Findings

Existence of vortex patch equilibria for gSQG equations with $ ext{α} ext{ in } [1,2)$.

Construction of time-periodic solutions near nondegenerate point vortex equilibria.

Asymptotic descriptions of vortex patch boundaries provided.

Abstract

In this paper, we investigate the existence of a finite number of vortex patches for the generalized surface quasi-geostrophic (gSQG) equations with $\alpha \in [1,2)$, focusing on configurations that may rotate uniformly, translate, or remain stationary. Using a desingularization technique, we reformulate the problem to resolve singularities arising in the point vortex limit. Assuming a nondegenerate equilibrium of the point vortices, we apply the implicit function theorem to construct time-periodic solutions to the gSQG equations, offering asymptotic descriptions of the vortex patch boundaries.