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

$$\Gamma_k\dot{z}_k(t)=J\nabla_{z_k} H \left(z(t)\right),\quad k=1,\dots,N,$$

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:

$$H(z_1,\dots,z_N) = -\frac{1}{2\pi} \sum\limits_{\mathop {j,k=1}\limits_{j \neq k}}^N \Gamma_j\Gamma_k\log\mid{z_j-z_k}\mid - \sum_{j,k=1}^N\Gamma_j\Gamma_kg(z_j,z_k).$$

Here ${g:\Omega\times\Omega\to\mathbb{R}}$ is an arbitrary symmetric function of class ${\mathcal{C}^2}$, e.g., the regular part of a hydrodynamic Green function. Given a non-degenerate critical point ${a_0\in\Omega}$ of ${h(z)=g(z,z)}$ and a non-degenerate 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 ${z^{(r)}(t)=(z^{(r)}_1(t),\dots,z^{(r)}_N(t))\in\Omega^N}$, ${0<r<r_0}$, 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 ${Z(t)}$.