Doubly connected V-states for the generalized surface quasi-geostrophic equations

TL;DR

This paper proves the existence of doubly connected V-states for generalized SQG equations, revealing bifurcation from annuli and including both positive and negative rotation speeds, with numerical insights into their limits.

Contribution

It establishes the existence of doubly connected V-states for generalized SQG equations with explicit bifurcation analysis, extending prior results for Euler equations.

Findings

Doubly connected V-states bifurcate from annuli at explicit eigenvalues.

V-states can rotate with both positive and negative angular velocities.

Numerical experiments explore the limiting shapes of V-states.

Abstract

In this paper, we prove the existence of doubly connected V-states for the generalized SQG equations with $\alpha\in ]0,1[.$ They can be described by countable branches bifurcating from the annulus at some explicit "eigenvalues" related to Bessel functions of the first kind. Contrary to Euler equations \cite{H-F-M-V}, we find V-states rotating with positive and negative angular velocities. At the end of the paper we discuss some numerical experiments concerning the limiting V-states.