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

TL;DR

This paper proves the existence of nonstationary, collision-free periodic solutions for the N-vortex problem in general domains by superposing stationary solutions and rotating vortex clusters, extending known results to more complex configurations.

Contribution

It introduces a superposition principle approach to construct periodic solutions for the N-vortex problem in arbitrary domains, including non-rotating configurations, which was not previously established.

Findings

Established conditions for existence of periodic solutions in general domains.

Verified explicit conditions for the unit disc domain.

Constructed examples of non-rigidly rotating vortex configurations.

Abstract

We examine the $N$-vortex problem on general domains $\Omega\subset\mathbb{R}^2$ concerning the existence of nonstationary collision-free periodic solutions. The problem in question is a first order Hamiltonian system of the form $$ \Gamma_k\dot{z}_k=J\nabla_{z_k}H(z_1,\ldots,z_N),\quad k=1,\ldots,N, $$ where $\Gamma_k\in\mathbb{R}\setminus\{0\}$ is the strength of the $k$th vortex at position $z_k(t)\in\Omega$, $J\in\mathbb{R}^{2\times 2}$ is the standard symplectic matrix and $$ H(z_1,\ldots,z_N)=-\frac{1}{2\pi}\sum_{\underset{k\neq j}{k,j=1}}^N\Gamma_j\Gamma_k\log|z_k-z_j|-\sum_{k,j=1}^N\Gamma_j\Gamma_k g(z_k,z_j) $$ with some regular and symmetric, but in general not explicitely known function $g:\Omega\times\Omega\rightarrow \mathbb{R}$. 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 explicitely verified for an example in the unit disc $\Omega=B_1(0)$. In particular we therefore obtain various examples of periodic solutions in $B_1(0)$ that are not rigidly rotating configurations.