Homogenization of the Stokes System in a Non-Periodically Perforated Domain

Abstract

In our recent work [8], we have studied the homogenization of the Poisson equation in a class of non periodically perforated domains. In this paper, we examine the case of the Stokes system. We consider a porous medium in which the characteristic distance between two holes, denoted by $\epsilon$, is proportional to the characteristic size of the holes. It is well known (see [1],[17] and [19]) that, when the holes are periodically distributed in space, the velocity converges to a limit given by the Darcy's law when the size of the holes tends to zero. We generalize these results to the setting of [8]. The non-periodic domains are defined as a local perturbation of a periodic distribution of holes. We obtain classical results of the homogenization theory in perforated domains (existence of correctors and regularity estimates uniform in $\epsilon$) and we prove $H^2$--convergence estimates for particular force fields.