Two dimensional incompressible ideal flow around a small obstacle

Abstract

In this article we study the asymptotic behavior of incompressible, ideal, time-dependent two dimensional flow in the exterior of a single smooth obstacle when the size of the obstacle becomes very small. Our main purpose is to identify the equation satisfied by the limit flow. We will see that the asymptotic behavior depends on $γ$, the circulation around the obstacle. For smooth flow around a single obstacle, $γ$ is a conserved quantity which is determined by the initial data. We will show that if $γ= 0$, the limit flow satisfies the standard incompressible Euler equations in the full plane but, if $γ\neq 0$, the limit equation acquires an additional forcing term. We treat this problem by first constructing a sequence of approximate solutions to the incompressible 2D Euler equation in the full plane from the exact solutions obtained when solving the equation on the exterior of each obstacle and then passing to the limit on the weak formulation of the equation. We use an explicit treatment of the Green's function of the exterior domain based on conformal maps, {\it a priori} estimates obtained by carefully examining the limiting process and the Div-Curl Lemma, together with a standard weak convergence treatment of the nonlinearity for the passage to the limit.