UNIFIED QUANTITATIVE ANALYSIS OF THE STOKES EQUATIONS IN DILUTE PERFORATED DOMAINS VIA LAYER POTENTIALS

We develop a unified method to obtain the quantitative homogenization of Stokes systems in periodically perforated domains with no-slip boundary conditions on the perforating holes. The main novelty of our paper is a quantitative analysis of the asymptotic behavior of the two-scale cell correctors via periodic Stokes layer potentials. The two-scale cell correctors were introduced and analyzed qualitatively by Allaire in the early 90's [G. Allaire, Ann. Sc. Norm. Super. Pisa Cl. Sci., 18 (1991), pp. 475-499]. Thanks to our layer potential approach, we also provide a novel explanation of the conductivity matrix in Darcy's model, of the Brinkman term in Brinkman's model, and explain the special behavior for d = 2. Finally, we also prove quantitative homogenization error estimates in various regimes of ratios between the size of the perforating holes and the typical distance between holes. In particular we handle a subtle issue in the dilute Darcy regime related to the nonvanishing of the Darcy velocity on the boundary.