Qualitative derivation of a density dependent incompressible Darcy law

Abstract: This paper provides the first study of the homogenization of the 3D non-homogeneous incompressible Navier--Stokes system in perforated domains with holes of supercritical size. The diameter of the holes is of order $\varepsilon^{\alpha} \ (1<\alpha<3)$, where $\varepsilon > 0$ is a small parameter measuring the mutual distance between the holes. We show that as $\varepsilon\to 0$, the asymptotic limit behavior of velocity and density is governed by Darcy's law under the assumption of a strong solution of the limiting system. Moreover, convergence rates are obtained. Finally, we show the existence of strong solutions to the inhomogeneous incompressible Darcy law, which might be of independent interest.