Derivation of Darcy’s law in randomly perforated domains

Abstract We consider the homogenization of a Poisson problem or Stokes system in randomly punctured domain with Dirichlet boundary conditions. assume that holes are spherical and have random centres radii. impose average distance between balls is size $$\varepsilon $$ ? their radius ^{\alpha }$$ ? , $$\alpha \in (1; 3)$$ ? ( 1 ? 3 ) . prove that, as periodic case (Allaire, G., Arch. Rational Mech. Anal. 113(113):261–298, 1991), solutions converge to solution Darcy’s law (or its scalar analogue Poisson). In same spirit (Giunti, A., Höfer, R., Ann. Inst. H. Poincare’- An. Nonl. 36(7):1829–1868, 2019; Giunti, Velàzquez, J.J.L., Comm. PDEs 43(9):1377–1412, 2018), we work under minimal conditions on integrability These ensure well-defined but do not rule out onset clusters holes.