Vanishing Viscosity Limit for Incompressible Flow around a Sufficiently Small Obstacle

In this article we consider viscous flow in the exterior of an obstacle satisfying the standard no-slip boundary condition at the surface of the obstacle. We look for conditions under which solutions of the NavierStokes system in the exterior domain converge to solutions of the Euler system in the full space when both viscosity and the size of the obstacle vanish. We prove that this convergence is true assuming two hypothesis: first, that the initial exterior domain velocity converges strongly (locally) in L to the full-space initial velocity and second, that the diameter of the obstacle is smaller than a suitable constant times viscosity, or, in other words, that the obstacle is sufficiently small. The convergence holds as long as the solution to the limit problem is known to exist and stays sufficiently smooth. To fix the O(1) spatial scale, we consider flows with an initial vorticity which is compacly supported, vanishes near obstacle, and does not depend on viscosity and the on size of the obstacle. In [3], Iftimie proved that any such vorticity gives rise to a family of exterior flows which converges in L to the corresponding full-space flow. For exterior two dimensional flow, topology implies that the initial velocity is not determined by vorticity alone, but also by its harmonic part. In the case of two dimensional flow, we prove strong convergence of initial data, as required by our main result, if the harmonic part of the family of initial velocities is chosen so that the circulation of the initial flow around the small obstacle vanishes. This work complements the study of incompressible flow around small obstacles, which has been carried out in [3, 4, 5]