An equivariant degree and periodic solutions of the N-vortex problem

Abstract

We examine the N-vortex problem on general two-dimensional domains concerning the existence of nonstationary collision-free periodic solutions. The problem in question is a first order Hamiltonian system arising as a singular limit (not only) in two-dimensional fluid dynamics. The Hamiltonian contains logarithmic singularities and is except for some special domains not explicitely known.We present two types of periodic solutions that can be found in general domains. The first one is based on the idea to superpose a stationary solution of a system of less than N vortices and several clusters of vortices that are close to rigidly rotating configurations of the whole-plane system. The second type consists of choreographic solutions following approximately a boundary component of the domain.The proofs in both cases rely on a suitable rescaling of the problem, investigation of the limiting system and implicit-function-like methods for a local continuation of existing solutions. Moreover, the modification of a S^1-equivariant degree theory allows us to prove that the continuation occurs globally.