Homogenization of Inhomogeneous Incompressible Navier-Stokes Equations in Domains with Very Tiny Holes

Abstract: In this paper, we study the homogenization problems of $3D$ inhomogeneous incompressible Navier-Stokes system perforated with very tiny holes whose diameters are much smaller than their mutual distances. The key is to establish the equations in the homogeneous domain without holes for the zero extensions of the weak solutions. This allows us to derive time derivative estimates and show the strong convergence of the density and the momentum by Aubin-Lions type argument. For the case of small holes, we finally show the limit equations remain unchanged in the homogenization limit.