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

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