Qualitative/quantitative homogenization of some non-Newtonian flows in perforated domains

In this paper, we consider the homogenization of stationary and evolutionary incompressible viscous non-Newtonian flows of Carreau–Yasuda type in domains perforated with a large number of periodically distributed small holes in R³ R 3 , where the mutual distance between the holes is measured by a small parameter ε >0 ε > 0 and the size of the holes is ε α ε α with α ∈ (1, 3) α ∈ ( 1 , 3 ) . The Darcy’s law is recovered in the limit, thus generalizing the results from Bourgeat and Mikelić (Nonlinear Anal Theory Methods 26(7):1221–1253, 1996. https://doi.org/10.1016/0362-546X(94)00285-P ) and Lu and Qian (J Differ Equ 411: 619–639, 2024. https://doi.org/10.1016/j.jde.2024.08.021 ) for α =1 α = 1 . Instead of using their restriction operator to derive the estimates of the pressure extension by duality, we use the Bogovskiĭ type operator in perforated domains [constructed in Diening et al. (ESAIM Control Optim Calc Var 23:851–868, 2017. https://doi.org/10.1051/cocv/2016016 )] to deduce the uniform estimates of the pressure directly. Moreover, quantitative convergence rates are given.