Periodic solutions of the N-vortex Hamiltonian system in planar domains

Abstract

We investigate the existence of collision-free nonconstant periodic solutions of the N-vortex problem in domains Ω⊂C. These are solutions z(t)=(z1(t),…,zN(t))∈ΩN of the first order Hamiltonian systemz˙k(t)=−i∇zkHΩ(z(t)),k=1,…,N, where the Hamiltonian HΩ has the formHΩ(z1,…,zN)=12π∑j,k=1j≠kNlog⁡1|zj−zk|−F(z). The function F:ΩN→R depends on the regular part of the hydrodynamic Green's function and is unbounded from above: F(z)→∞ if zk→∂Ω for some k. 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. Using singular perturbation techniques and Conley index theory we prove the existence of a family of periodic solutions zr(t), 0<r<r0, with arbitrarily small minimal period Tr→0 as r→0. The solutions are close to the singular set of HΩ. 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.