Existence of stationary vortex patches for the gSQG in bounded domains

TL;DR

This paper proves the existence of time-periodic vortex patches in bounded domains for the generalized surface quasi-geostrophic equation with specific parameters, using linearization and the implicit function theorem.

Contribution

It establishes the existence of stationary vortex patches for gSQG in bounded domains for b3 in (1,2), a novel result in this setting.

Findings

Vortex patches exist near non-degenerate critical points of the Kirchhoff--Routh equation.

Construction applies for b3 in (1,2).

Method combines linearization analysis and the implicit function theorem.

Abstract

In this paper we show the existence of time-periodic vortex patches for the generalized surface quasi-geostrophic equation within a bounded domain. This construction is carried out for values of $\gamma$ in the range of $(1,2)$. The resulting vortex patches possess a fixed vorticity and total flux, and they are located in the neighborhood of critical points that are non-degenerate for the Kirchhoff--Routh equation. The proof is accomplished through a combination of analyzing the linearization of the contour dynamics equation and employing the implicit function theorem as well as carefully selected function spaces.