Periodic solutions with prescribed minimal period of vortex type problem in domains

TL;DR

This paper proves the existence of infinitely many periodic vortex solutions with prescribed minimal periods in bounded domains, combining slow center motion near level lines of the Robin function and fast vortex rotation.

Contribution

It introduces a novel approach to find periodic vortex solutions with specific periods using a higher-dimensional Poincaré-Birkhoff theorem.

Findings

Existence of infinitely many periodic solutions with prescribed periods.

Solutions involve a superposition of slow and fast vortex motions.

Application of advanced topological methods to vortex dynamics.

Abstract

We consider Hamiltonian systems with two degrees of freedom of point vortex type \[ \kappa_j \dot{z}_j = J \nabla_{z_j} H_\Omega(z_1,z_2), \quad j=1,2, \] for $z_1,z_2$ in a domain $\Omega\subset\mathbb{R}^2$. In the classical point vortex context the Hamiltonian $H_\Omega$ is of the form \[ H_\Omega(z_1,z_2) = -\frac{\kappa_1 \kappa_2}{\pi} \log |z_1-z_2| - 2\kappa_1 \kappa_2g(z_1,z_2) - \kappa_1^2 h(z_1) - \kappa_2^2 h(z_2), \] where $g:\Omega\times\Omega\to\mathbb{R}$ is the regular part of a hydrodynamic Green function in $\Omega$, $h:\Omega\to\mathbb{R}$ is the Robin function: $h(z)=g(z,z)$, and $\kappa_1$, $\kappa_2$ are the vortex strengths. We prove the existence of infinitely many periodic solutions with prescribed minimal period that are superpositions of a slow motion of the center of vorticity close to a star-shaped level line of $h$ and of a fast rotation of the two vortices around their center of vorticity. The proofs are based on a recent higher dimensional version of the Poincar\'e-Birkhoff theorem due to Fonda and Ure\~na.