The vanishing viscosity limit for the inhomogeneous incompressible Navier-Stokes equations around a small obstacle

Abstract

We investigate the interaction between a vanishing obstacle and the vanishing viscosity limit in the setting of the inhomogeneous incompressible Navier-Stokes equations with no-slip boundary conditions. We work in the space exterior to a small obstacle in three dimensions. We prove quantitative convergence rates, assuming that the obstacle is sufficiently small, namely smaller than a constant times the viscosity. The convergence holds as long as the solution to the limit problem is known to exist and stays sufficiently smooth. This extends the results of Iftimie, Lopes Filho and Nussenzveig Lopes to the inhomogeneous Navier-Stokes equations. Moreover, we can show improved convergence rates and convergence of the gradient in $\mathbb{R}^3$.