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

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 $\mathrm{\Omega }$, which is governed by a Hamiltonian system $$\{\begin{array}{rl}{\displaystyle {\mathrm{\Gamma }}_i\frac{d{x}_i}{dt}}& {\displaystyle =\frac{\mathrm{\partial }{H}_{\mathrm{\Omega }}}{\mathrm{\partial }{y}_i}(z_1,\dots ,z_N)}\\ {\displaystyle {\mathrm{\Gamma }}_i\frac{d{y}_i}{dt}}& {\displaystyle =-\frac{\mathrm{\partial }{H}_{\mathrm{\Omega }}}{\mathrm{\partial }{x}_i}(z_1,\dots ,z_N)}\end{array}\text{where }{z}_i=(x_i,y_i),i=1,\dots ,N,$$ 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” ${\mathrm{\Gamma }}_i$ and various topological and geometrical assumptions on $\mathrm{\Omega }$. 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 ${\mathrm{\Gamma }}_i$ are alternating and $\mathrm{\mid }{\mathrm{\Gamma }}_i\mathrm{\mid }$ is increasing, $H_{\mathrm{\Omega }}$ has a critical point. If $\mathrm{\Omega }$ is not simply connected, we are able to derive a critical point of $H_{\mathrm{\Omega }}$, if $\sum_{j\in J}\Gamma_j^2>\sum_{\substack{i,j\in J\ i\not=j}}|\Gamma_i\Gamma_j|$forall$J\subset{1,\dots,N}$,$|J|\ge 2$.