Viscous flow passing a small obstacle with non-zero velocity at infinity

In this paper, we consider the incompressible Navier-Stokes equations in the exterior of a rigid obstacle in $ \mathbb{R}^d $(d = 2, 3). For the case that the velocity vanishes at infinity, it is well-known that the Navier-Stokes equations have a Leray-Hopf weak solution with initial data in $ L^2 $. Moreover, Iftimie et al. [8] proved that, as the viscosity and the size of the obstacle both tend to zero, the limit flow satisfies the Euler equations in the whole space. The purpose here is to extend the analysis to the case that the velocity does not vanish at infinity. We prove that the system in this case has a global weak solution, and the convergence to the Euler equations happens likewise when the viscosity and the size of the obstacle both vanish.