Regularization of point vortices for the Euler equation in dimension two, part II

TL;DR

This paper develops a method to construct stationary solutions to the 2D Euler equations by regularizing point vortices, using elliptic PDEs, and relates solutions to critical points of the Kirchhoff-Routh function.

Contribution

It introduces a novel elliptic PDE approach to approximate singular vortex solutions, extending the regularization of point vortices in 2D Euler equations.

Findings

Constructed stationary solutions approximating vortex configurations.

Linked solutions to critical points of Kirchhoff-Routh function.

Validated the approach for simply-connected smooth domains.

Abstract

In this paper, we continue to construct stationary classical solutions of the incompressible Euler equation approximating singular stationary solutions of this equation. This procedure now is carried out by constructing solutions to the following elliptic problem \[ {cases} -\ep^2 \Delta u=(u-q-\frac{\kappa}{2\pi}\ln\frac{1}{\ep})_+^p-(q-\frac{\kappa}{2\pi}\ln\frac{1}{\ep}-u)_+^p, \quad & x\in\Omega, u=0, \quad & x\in\partial\Omega, {cases} \] where $p>1$, $\Omega\subset\mathbb{R}^2$ is a bounded domain, $q$ is a harmonic function. We showed that if $\Omega$ is a simply-connected smooth domain, then for any given non-degenerate critical point of Kirchhoff-Routh function $\mathcal{W}(x_1^+,...,x_m^+,x_1^-,...,x_n^-)$ with $\kappa^+_i=\kappa>0\,(i=1,...,m)$ and $\kappa^-_j=-\kappa\,(j=1,...,n)$, there is a stationary classical solution approximating stationary $m+n$ points vortex solution of incompressible Euler equations with total vorticity $(m-n)\kappa$.