Two-dimensional Incompressible Viscous Flow around a Small Obstacle

In this work we study the asymptotic behavior of viscous incompressible 2D flow in the exterior of a small material obstacle. We fix the initial vorticity ω0 and the circulation γ of the initial flow around the obstacle. We prove that, if γ is sufficiently small, the limit flow satisfies the full-plane Navier-Stokes system, with initial vorticity ω0 + γδ, where δ is the standard Dirac measure. The result should be contrasted with the corresponding inviscid result obtained by the authors in [15], where the effect of the small obstacle appears in the coefficients of the PDE and not only on the initial data. The main ingredients of the proof are L −L estimates for the Stokes operator in an exterior domain, a priori estimates inspired on Kato’s fixed point method, energy estimates, renormalization and interpolation.