Periodic solutions of the N-vortex Hamiltonian system in planar domains

Abstract

We investigate the existence of collision-free nonconstant periodic solutions of the $N$-vortex problem in domains $\Omega\subset\mathbb{C}$. These are solutions $z(t)=(z_1(t),\dots,z_N(t))$ of the first order Hamiltonian system \[ \dot{z}_k(t)=-i\nabla_{z_k} H_\Omega\big(z(t)\big),\quad k=1,\dots,N, \] where the Hamiltonian $H_\Omega$ has the form \[ H_\Omega(z_1,\dots,z_N) = \frac1{2\pi}\sum_{{j,k=1}\atop{j\ne k}}^N \log\frac1{|z_j-z_k|} - F(z). \] The function $F:\Omega^N\to\mathbb{R}$ depends on the regular part of the hydrodynamic Green's function and is unbounded from above. The Hamiltonian is unbounded from above and below, it is singular, not integrable, energy surfaces are not compact and not known to be of contact type. We prove the existence of a family of periodic solutions $z^r(t)$, $0<r<r_0$, with arbitrarily small minimal period $T_r\to0$ as $r\to0$. The solutions are close to the singular set of $H_\Omega$. Our result applies in particular to generic bounded domains, which may be simply or multiply connected. It also applies to certain unbounded domains. Depending on the domain there are multiple such families.