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

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 $${\Gamma }_k{\dot{z}}_k=J{\nabla }_{z_k}H(z_1,\dots ,z_N),k=1,\dots ,N,$$ 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 $$H(z_1,\dots ,z_N)=-\frac{1}{2\pi }\sum _{\underset{k\ne j}{k,j=1}}^N{\Gamma }_j{\Gamma }_k\log |z_k-z_j|-\sum _{k,j=1}^N{\Gamma }_j{\Gamma }_{k}g(z_k,z_j)$$ with some regular and symmetric, but in general not explicitely known function $g:\Omega \times \Omega \to \mathbb{R}$. 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 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.

27 pages, 1 figure

Cited by in corpus (3)

• Burst of Point Vortices and Non-Uniqueness of 2D Euler Equations
• The N-vortex Problem on a Riemann Sphere
• Stability of periodic solutions of the N-vortex problem in general domains