Critical points of the N-vortex Hamiltonian in bounded planar domains and steady state solutions of the incompressible Euler equations

TL;DR

This paper proves the existence of critical points for the N-vortex Hamiltonian in bounded planar domains, leading to new vortex equilibria and smooth steady solutions of the Euler equations, including counter-rotating vortices.

Contribution

It introduces new critical points of the N-vortex Hamiltonian in bounded domains, extending the understanding of vortex equilibria and their relation to steady Euler flows.

Findings

Existence of critical points for N=3 or 4 vortices under certain conditions.

Critical points correspond to vortex equilibria in Euler equations.

Desingularization yields smooth steady-state solutions with localized vorticity.

Abstract

We prove the existence of critical points of the $N$-vortex Hamiltonian $H_\Omega (x_1,\ldots, x_N) =\sum\limits^N_{i=1}\Gamma^2_i h(x_i) + \sum\limits_{i,j=1\atop j\not= k}^N \Gamma_i\Gamma_jG(x_i,x_j)+2\sum\limits_{i=1}^N\Gamma_i\psi_0(x_i)$ in a bounded domain $\Omega\subset{\mathbb R}^2$ which may be simply or multiply connected. Here $G$ denotes the Green function for the Dirichlet Laplace operator in $\Omega$, more generally a hydrodynamic Green function, and $h$ the Robin function. Moreover $\psi_0\in C^1(\overline\Omega)$ is a harmonic function on $\Omega$. We obtain new critical points $x=(x_1,\dots,x_N)$ for $N=3$ or $N=4$ under conditions on the vorticities $\Gamma_i\in{\mathbb R}\setminus\{0\}$. These critical points correspond to point vortex equilibria of the Euler equation in vorticity form. The case $\Gamma_i=(-1)^i$ of counter-rotating vortices with identical vortex strength is included. The point vortex equilibria can be desingularized to obtain smooth steady state solutions of the Euler equations for an ideal fluid. The velocity of these steady states will be irrotational except for $N$ vorticFity blobs near $x_1,\dots,x_N$.