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

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 $J\subset\{1,\dots,N\}$, $|J|\ge 2$.