We investigate the existence of collision-free nonconstant periodic solutions of the Nvortex problem in domains Ω ⊂ C. These are solutions z(t) = (z 1 (t), . . . , z N (t)) of the first order Hamiltonian systeṁwhere the Hamiltonian H Ω has the formThe function F : Ω N → R depends on the regular part of the hydrodynamic Green's function and is unbounded from above. The Hamiltonian is unbounded from above and below, it is singular, not integrable, energy surfaces are not compact and not known to be of contact type. We prove the existence of a family of periodic solutions z r (t), 0 < r < r 0 , with arbitrarily small minimal period T r → 0 as r → 0. The solutions are close to the singular set of H Ω . Our result applies in particular to generic bounded domains, which may be simply or multiply connected. It also applies to certain unbounded domains. Depending on the domain there are multiple such families.MSC 2010: Primary: 37J45; Secondary: 37N10, 76B47

No abstract