Quantitative homogenization of the compressible Navier-Stokes equations towards Darcy's law

Abstract

We consider the solutions ${\rho }_{\epsilon },{\mathbf{u}}_{\epsilon }$ to the compressible Navier-Stokes equations (NSE) in a domain periodically perforated by holes of diameter $\epsilon >0$. We focus on the case where the diameter of the holes is of the same order as the distance between neighboring holes. This is the same setting investigated in the paper by Masmoudi [\url{http://www.numdam.org/article/COCV_2002__8__885_0.pdf}], where convergence ${\rho }_{\epsilon },{\mathbf{u}}_{\epsilon }$ of the system to the porous medium equation has been shown. We prove a quantitative version of this convergence result provided that the solution of the limiting system is sufficiently regular. The proof builds on the relative energy inequality satisfied by the compressible NSE.