In this work we study the asymptotic behavior of solutions of the incompressible two-dimensional Euler equations in the exterior of a single smooth obstacle when the obstacle becomes very thin tending to a curve. We extend results by Iftimie, Lopes Filho and Nussenzveig Lopes, obtained in the context of an obstacle tending to a point, see [Comm. PDE, 28 (2003), 349-379].

Abstract. We prove uniqueness for the vortex-wave system with a single point vortex introduced by Marchioro and Pulvirenti [7] in the case where the vorticity is initially constant near the point vortex. Our method relies on the Eulerian approach for this problem and in particular on the formulation in terms of the velocity.

In this paper we prove that the motion of a solid body in a two dimensional incompressible perfect fluid converges, when the body shrinks to a point with fixed mass and circulation, to a variant of the vortex-wave system where the vortex, placed in the point occupied by the shrunk body, is accelerated by a lift force similar to the Kutta-Joukowski force of the irrotational theory.