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

This article is concerned with the study of existence of stationary solutions for the dynamics of N point vortices in an idealised fluid constrained to a bounded two-dimensional domain Ω, which is governed by a Hamiltonian system{Γidxidt=∂HΩ∂yi(z1,…,zN)Γidyidt=−∂HΩ∂xi(z1,…,zN)wherezi=(xi,yi),i=1,…,N,under selected conditions on the “vorticities” ΓiΓi and various topological and geometrical assumptions on Ω. Here HΩ(z):=∑j=1NΓj2h(zj)+∑i,j=1,i≠jNΓiΓjG(zi,zj) is the so-called Kirchhoff–Routh-path function. In particular, we will prove that HΩHΩ has a critical point, if it is possible to align the vortices along a line, such that the signs of the ΓiΓi are alternating and |Γi||Γi| is non-increasing. We are also able to derive a critical point of HΩHΩ, if ∑j∈JΓj2>∑i,j∈Ji≠j|ΓiΓj| for all J⊂{1,…,N}J⊂{1,…,N}, |J|≥2|J|≥2 and Ω is not simply connected.