Lars-Erik Persson and John Fabricius: Homogenization of evolution Stokes equation with two small parameters via two-scale convergence

First we will present how homogenization theory was introduced in Sweden at Luleå University of Technology (LTU) around 1988. We will shortly describe how this group at LTU has developed, including some of its most important results. Finally, we will present a concrete problem we are currently working on and the results obtained so far. It concerns the asymptotic behav- ior of solutions of the evolution Stokes equation in a thin domain with moving “rough” boundaries. The problem involves two small parameters ε > 0 and μ > 0 which describe domain thickness and roughness wavelength respectively. Depending on λ = lim ε/μ, three different flow regimes are obtained as both ε and μ tend to zero: Reynolds roughness (λ = 0), Stokes roughness (0 < λ < ∞) and the high frequency regime (λ = ∞). In all three cases the limit flow is governed by a time-dependent equation of Reynolds type. The analysis is based on two-scale convergence.