Periodic solutions for the N-vortex problem via a superposition principle

We examine the $N$ -vortex problem on general domains $Ω\subset\mathbb{R}^2$ concerning the existence of nonstationary collision-free periodic solutions. The problem in question is a first order Hamiltonian system of the form

$Γ_k\dot{z}_k = J\nabla_{z_k}H(z_1,...,z_N),\ \ \ \ k = 1,...,N,$

where $Γ_k∈\mathbb{R}\setminus\{0\}$ is the strength of the $k$ th vortex at position $z_k(t)∈Ω$ , $J∈\mathbb{R}^{2× 2}$ is the standard symplectic matrix and

$H(z_1,...,z_N) = -\frac{1}{2π}\sum\limits_{\underset{k≠ j}{k,j = 1}}^NΓ_jΓ_k\log|z_k-z_j|-\sum\limits_{k,j = 1}^NΓ_jΓ_k g(z_k,z_j)$

with some regular and symmetric, but in general not explicitely known function $g:Ω×Ω \to \mathbb{R}$ . The investigation relies 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. We establish general conditions on both, the stationary solution and the configurations, under which multiple $T$ -periodic solutions are shown to exist for every $T>0$ small enough. The crucial condition holds in generic bounded domains and is explicitly verified for an example in the unit disc $Ω = B_1(0)$ . In particular we therefore obtain various examples of periodic solutions in $B_1(0)$ that are not rigidly rotating configurations.