Reiterated homogenization of parabolic systems with several spatial and temporal scales

Abstract

We consider quantitative estimates in the homogenization of second-order parabolic systems with periodic coefficients that oscillate on multiple spatial and temporal scales,∂t−div(A(x,t,x/ε1,…,x/εn,t/ε1′,…,t/εm′)∇), where εℓ=εαℓ,εk′=εβk,ℓ=1,...,n,k=1,...,m, with 0<α1<...<αn<∞ and 0<β1<...<βm<∞. The convergence rate in the homogenization is derived in the L2 space, and the large-scale interior and boundary Lipschitz estimates are also established. In the case n=m=1, such issues have been addressed by Geng and Shen (2020) [12] based on an interesting scale reduction technique developed therein. Our investigation relies on a quantitative reiterated homogenization theory.