Existence of corotating and counter-rotating vortex pairs for active scalar equations

TL;DR

This paper proves the existence of corotating and counter-rotating vortex pairs for Euler and SQG equations, confirming numerical conjectures and extending results to fractional cases using contour dynamics and implicit function theorem.

Contribution

It provides direct analytical proofs of vortex pair existence for Euler and SQG equations, including cases with fractional b1, and offers insights into their structure beyond previous variational methods.

Findings

Confirmed existence of vortex pairs for Euler equations.

Extended existence results to b1 d; (0,1) for SQG.

Validated numerical conjectures with rigorous proofs.

Abstract

In this paper, we study the existence of corotating and counter-rotating pairs of simply connected patches for Euler equations and the $(\hbox{SQG})_\alpha$ equations with $\alpha\in (0,1).$ From the numerical experiments implemented for Euler equations in \cite{DZ, humbert, S-Z} it is conjectured the existence of a curve of steady vortex pairs passing through the point vortex pairs. There are some analytical proofs based on variational principle \cite{keady, Tur}, however they do not give enough information about the pairs such as the uniqueness or the topological structure of each single vortex. We intend in this paper to give direct proofs confirming the numerical experiments and extend these results for the $(\hbox{SQG})_\alpha$ equation when $\alpha\in (0,1)$. The proofs rely on the contour dynamics equations combined with a desingularization of the point vortex pairs and the application of the implicit function theorem.