Equilibria for the $N$-vortex-problem in a general bounded domain

TL;DR

This paper investigates the existence of stationary solutions for the N-vortex problem in bounded domains, establishing conditions under which critical points of the Kirchhoff-Routh function exist, depending on vortex configurations and domain topology.

Contribution

It provides new existence results for stationary vortex configurations in bounded domains, including line alignments and non-simply connected cases, under specific vorticity conditions.

Findings

Critical points exist when vortices are aligned with alternating signs and increasing magnitudes.

Existence of critical points in non-simply connected domains under certain vorticity sum conditions.

Conditions depend on domain topology and vortex strength arrangements.

Abstract

This article is concerned with the study of existence and properties of stationary solutions for the dynamics of $N$ point vortices in an idealised fluid constrained to a bounded two--dimen\-sional domain $\Omega$, which is governed by a Hamiltonian system \[ \left\{\begin{aligned} \Gamma_i\frac{d x_i}{d t} &=\frac{\partial H_\Omega}{\partial y_i}(z_1,\dots,z_N)\\ \Gamma_i\frac{d y_i}{d t} &=-\frac{\partial H_\Omega}{\partial x_i}(z_1,\dots,z_N) \end{aligned} \hspace{2cm}\text{where}\ z_i=(x_i,y_i),\ i=1,\dots,N, \right. \] where $H_\Omega(z):=\sum_{j=1}^N\Gamma_j^2h(z_j)+\sum_{i,j=1, i\not=j}^N\Gamma_i\Gamma_jG(z_i,z_j)$ is the so--called Kirchhoff--Routh--path function under various conditions on the "vorticities" $\Gamma_i$ and various topological and geometrical assumptions on $\Omega$. In particular, we will prove that (under an additional technical assumption) if it is possible to align the vortices along a line, such that the signs of the $\Gamma_i$ are alternating and $|\Gamma_i|$ is increasing, $H_\Omega$ has a critical point. If $\Omega$ is not simply connected, we are able to derive a critical point of $H_\Omega$, if $\sum_{j\in J}\Gamma_j^2>\sum_{\substack{i,j\in J\\ i\not=j}}|\Gamma_i\Gamma_j|$ for all $J\subset\{1,\dots,N\}$, $|J|\ge 2$.