Vortex dynamics in a two-dimensional domain with holes and the small obstacle limit

In this work we examine the asymptotic behavior of solutions of the incompressible two-dimensional Euler equations on a domain with several holes, when one of the holes becomes small. We show that the limit flow satisfies a modified Euler system in the domain with the small hole removed. In vorticity form, the limit system is the usual equation for transport of vorticity, coupled with a modified Biot-Savart law which includes a point vortex at the point where the small hole disappears, together with the appropriate correction for the harmonic part of the flow. This work extends results by Iftimie, Lopes Filho, and Nussenzveig Lopes, obtained in the context of the exterior of a single small obstacle in the plane; see [ Comm. Partial Differential Equations, 28 ( 2003), pp. 349 - 379]. The main difficulty in the present situation lies in controlling the behavior of the harmonic part of the flow, which is not an exact conserved quantity. As part of our analysis we develop a new description of two-dimensional vortex dynamics in a general domain with holes.