Existence of nested polygonal vortex patches for the generalized SQG equation

TL;DR

This paper constructs families of nested polygonal vortex patches for the generalized SQG equation, providing detailed geometric descriptions of their evolution and expanding understanding of coherent structures in active scalar equations.

Contribution

It introduces a novel implicit function approach to prove the existence of co-rotating nested polygonal vortex patches in the gSQG equation under nondegeneracy conditions.

Findings

Existence of co-rotating nested polygonal vortex patches.

Precise asymptotic descriptions of patch boundary geometry.

Extension of vortex patch solutions to more singular regimes.

Abstract

This paper investigates time-periodic solutions of both the surface quasi-geostrophic (SQG) equation and its generalized form (gSQG) within the more singular regime, focusing on the evolution of patch-type structures. Assuming the underlying point vortex equilibrium satisfies a natural nondegeneracy condition, we employ an implicit function argument to construct families of co-rotating nested polygonal vortex patch solutions. These configurations provide precise asymptotic descriptions of the geometry of the evolving patch boundaries. Our results contribute to the broader understanding of coherent rotating structures arising in active scalar equations with singular velocity coupling.