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

Abstract

We consider Hamiltonian systems with two degrees of freedom of point vortex type \[ κ_j \dot{z}_j = J \nabla_{z_j} H_Ω(z_1,z_2), \quad j=1,2, \] 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 \[ H_Ω(z_1,z_2) = -\frac{κ_1 κ_2}π \log |z_1-z_2| - 2κ_1 κ_2g(z_1,z_2) - κ_1^2 h(z_1) - κ_2^2 h(z_2), \] 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 Poincaré-Birkhoff theorem due to Fonda and Ureña.

17 pages

References in corpus (2)

• Global continua of periodic solutions of singular first-order Hamiltonian systems of N-vortex type
• Periodic solutions of N-vortex type Hamiltonian systems near the domain boundary