Periodic solutions of N-vortex type Hamiltonian systems near the domain boundary

TL;DR

This paper proves the existence of near-boundary, collision-free periodic vortex solutions in a domain, with small periods and trajectories approaching the boundary, under general conditions on the Green function's behavior.

Contribution

It establishes new conditions for the existence of small-period, boundary-approaching vortex solutions in N-vortex Hamiltonian systems, including choreographies and curvature relations.

Findings

Existence of periodic solutions near boundary with small periods

Solutions form choreographies moving on the same trajectory

Vortex speed relates to boundary curvature

Abstract

The paper deals with the existence of nonstationary collision-free periodic solutions of singular first order Hamiltonian systems of $N$-vortex type in a domain $\Omega\subset\mathbb{C}$. These are solutions $z(t)=(z_1(t),\dots,z_N(t))$ of \[ \dot{z}_j(t)=-i\nabla_{z_j} H_\Omega\big(z(t)\big),\quad j=1,\dots,N, \tag{HS} \] where the Hamiltonian $H_\Omega$ has the form \[ H_\Omega(z_1,\dots,z_N) = -\sum_{{j,k=1}\over{j\ne k}}^N \frac{1}{2\pi}\log|z_j-z_k| -\sum_{j,k=1}^N g(z_j,z_k). \] The function $g:\Omega\times\Omega\to\mathbb{R}$ is required to be of class $C^3$ and symmetric, the regular part of a hydrodynamic Green function being our model. The Hamiltonian is unbounded from above and below, and the associated action integral is not defined on an open subset of the space of periodic $H^{1/2}$ functions. Given a closed connected component $\Gamma\subset\partial\Omega$ of class $C^3$ we are interested in periodic solutions of (HS) near $\Gamma$. We present quite general conditions on the behavior of $g$ near $\Gamma$ which imply that there exists a family of periodic solutions $z^{(r)}(t)$, $0<r<\overline{r}$, with arbitrarily small minimal period $T_r\to0$ as $r\to0$, and such that the "point vortices" $z_j^{(r)}(t)$ approach $\Gamma$ as $r\to0$. The solutions are choreographies, i.e.\ $z_j^{(r)}(t)$ moves on the same trajectory as $z_1^{(r)}(t)$ with a phase shift. We can also relate the speed of each vortex with the curvature of $\Gamma$.