Stability of periodic solutions of the N-vortex problem in general domains

TL;DR

This paper studies the stability of specific periodic vortex solutions in general planar domains, showing how the stability depends on the nature of a critical point of the Robin function, with results verified for up to four vortices.

Contribution

It establishes a link between the stability of vortex solutions and the type of critical point of the Robin function, extending stability analysis to general domains beyond the plane.

Findings

Saddle points lead to unstable solutions.

Non-saddle critical points yield stable solutions.

Existence of orbitally stable solutions for two vortices.

Abstract

We investigate stability properties of a type of periodic solutions of the $N$-vortex problem on general domains $\Omega\subset \mathbb{R}^2$. The solutions in question bifurcate from rigidly rotating configurations of the whole-plane vortex system and a critical point $a_0\in\Omega$ of the Robin function associated to the Dirichlet Laplacian of $\Omega$. Under a linear stability condition on the initial rotating configuration, which can be verified for examples consisting of up to 4 vortices, we show that the linear stability of the induced solutions is solely determined by the type of the critical point $a_0$. If $a_0$ is a saddle, they are unstable. Otherwise they are stable in a certain linear sense. The proof uses a criterion for the bifurcation of multiple eigenvalues, which is applied to suitable Poincar\'e sections. Beyond linear stability, Herman's last geometric theorem allows us to prove the existence of isoenergetically orbitally stable solutions in the case of $N=2$ vortices.