Periodic solutions of singular first-order Hamiltonian systems of N-vortex type

Abstract

We are concerned with the dynamics of $N$ point vortices $z_1,\dots,z_N\in\Omega\subset\mathbb{R}^2$ in a planar domain. This is described by a Hamiltonian system \[ \Gamma_k\dot{z}_k(t)=J\nabla_{z_k} H\big(z(t)\big),\quad k=1,\dots,N, \] where $\Gamma_1,\dots,\Gamma_N\in\mathbb{R}\setminus\{0\}$ are the vorticities, $J\in\mathbb{R}^{2\times2}$ is the standard symplectic $2\times2$ matrix, and the Hamiltonian $H$ is of $N$-vortex type: \[ H(z_1,\dots,z_N) = -\frac1{2\pi}\sum_{j\ne k}^N \Gamma_j\Gamma_k\log|z_j-z_k| - \sum_{j,k=1}^N\Gamma_j\Gamma_kg(z_j,z_k). \] Here $g:\Omega\times\Omega\to\mathbb{R}$ is an arbitrary symmetric function of class $C^2$, e.g.\ the regular part of a hydrodynamic Green function. Given a nondegenerate critical point $a_0\in\Omega$ of $h(z)=g(z,z)$ and a nondegenerate relative equilibrium $Z(t)\in\mathbb{R}^{2N}$ of the Hamiltonian system in the plane with $g=0$, we prove the existence of a smooth path of periodic solutions $z^{(r)}(t)=\big(z^{(r)}_1(t),\dots,z^{(r)}_N(t)\big)\in\Omega^N$, $0<r<r_0$, with $z^{(r)}_k(t)\to a_0$ as $r\to0$. In the limit $r\to0$, and after a suitable rescaling, the solutions look like $Z(t)$.