Regularization of point vortices for the Euler equation in dimension two

TL;DR

This paper develops a method to construct stationary solutions to the 2D Euler equations that approximate singular vortex solutions by solving an elliptic PDE with regularization, linking vortex points to critical points of a specific function.

Contribution

It introduces a regularization technique for point vortices in the Euler equations, establishing existence of solutions near critical points of the Kirchhoff-Routh function.

Findings

Constructed stationary solutions approximating multiple vortices.

Linked vortex positions to critical points of Kirchhoff-Routh function.

Extended previous results on single vortex solutions.

Abstract

In this paper, we construct stationary classical solutions of the incompressible Euler equation approximating singular stationary solutions of this equation. This procedure is carried out by constructing solutions to the following elliptic problem [ -\ep^2 \Delta u=(u-q-\frac{\kappa}{2\pi}\ln\frac{1}{\ep})_+^p, \quad & x\in\Omega, u=0, \quad & x\in\partial\Omega, ] where $p>1$, $\Omega\subset\mathbb{R}^2$ is a bounded domain, $q$ is a harmonic function. We showed that if $\Omega$ is simply-connected smooth domain, then for any given non-degenerate critical point of Kirchhoff-Routh function $\mathcal{W}(x_1,...,x_m)$ with the same strength $\kappa>0$, there is a stationary classical solution approximating stationary $m$ points vortex solution of incompressible Euler equations with vorticity $m\kappa$. Existence and asymptotic behavior of single point non-vanishing vortex solutions were studied by D. Smets and J. Van Schaftingen (2010).