The incompressible $\alpha$--Euler equations in the exterior of a vanishing disk

TL;DR

This paper studies the behavior of the $ ext{alpha}$--Euler equations outside a small disk and proves convergence to a modified equation with a point vortex as the disk shrinks to zero.

Contribution

It demonstrates the convergence of solutions in the exterior domain to a modified full-plane equation with a point vortex as the disk radius approaches zero.

Findings

Solution converges to a modified $ ext{alpha}$--Euler equation with a Dirac delta at the origin.

The circulation remains independent of the disk size during the limit process.

The initial potential vorticity's support and independence from $ ext{epsilon}$ are crucial for the convergence.

Abstract

In this article we consider the $\alpha$--Euler equations in the exterior of a small fixed disk of radius $\epsilon$. We assume that the initial potential vorticity is compactly supported and independent of $\epsilon$, and that the circulation of the unfiltered velocity on the boundary of the disk does not depend on $\epsilon$. We prove that the solution of this problem converges, as $\epsilon \to 0$, to the solution of a modified $\alpha$--Euler equation in the full plane where an additional Dirac located at the center of the disk is imposed in the potential vorticity.