Global solutions for the generalized SQG equation and rearrangements

TL;DR

This paper establishes the existence of rotating and traveling-wave solutions for the generalized SQG equation by energy maximization over rearrangements, including vortex configurations and desingularizations.

Contribution

It introduces a novel variational approach to construct explicit solutions for the gSQG equation, including vortex pairs and desingularized vortex configurations.

Findings

Existence of co-rotating vortex solutions with N-fold symmetry

Existence of translating vortex pair solutions

Solutions serve as desingularizations of point vortices

Abstract

In this paper, we study the existence of rotating and traveling-wave solutions for the generalized surface quasi-geostrophic (gSQG) equation. The solutions are obtained by maximization of the energy over the set of rearrangements of a fixed function. The rotating solutions take the form of co-rotating vortices with $N$-fold symmetry. The traveling-wave solutions take the form of translating vortex pairs. Moreover, these solutions constitute the desingularization of co-rotating $N$ point vortices and counter-rotating pairs. Some other quantitative properties are also established.