Symmetric equilibria for the N-vortex problem

Abstract

This paper is concerned with the study of existence and properties of symmetric stationary solutions for the dynamics of N point vortices in an ideal fluid constrained to a dihedrally symmetric two-dimensional domain ${\Omega}$, which is governed by a Hamiltonian system

$$\Gamma_{i}\frac{{\rm d}x_{i}}{{\rm d}t} = \frac{\partial H_{\Omega}}{\partial y_{i}} (z_{1}, ... , z_{N}), \quad \Gamma_{i}\frac{{\rm d}y_{i}}{{\rm d}t} = -\frac{\partial H_{\Omega}}{\partial x_{i}} (z_{1}, ... , z_{N})$$

, where ${z_i = (x_i, y_i), i = 1, . . . ,N}$, and where

$$H_{\Omega} (z) := \sum_{j = 1}^{N} {\Gamma_{j}^{2}h(z_{j})} + \sum_{i, j = 1, i \neq j}^{N}{\Gamma_{i}\Gamma_{j}G(z_{i}, z_{j})}$$

is the so-called Kirchhoff–Routh path function under some conditions on the vorticities ${\Gamma_{i}}$.