Abstract
We consider the homogenization of the linear parabolic problem which exhibits a mismatch between the spatial scales in the sense that the coefficient of the elliptic part has one frequency of fast spatial oscillations, whereas the coefficient of the time derivative contains a faster spatial scale. It is shown that the faster spatial microscale does not give rise to any corrector term and that there is only one local problem needed to characterize the homogenized problem. Hence, the problem is not of a reiterated type even though two rapid scales of spatial oscillation appear.
Highlights
The field of homogenization has its main source of inspiration in the problem of finding the macroscopic properties of strongly heterogeneous materials
A two-scale convergence was invented by Nguetseng [15] as a new approach for the homogenization of problems with fast oscillations in one scale in space
The concept in the following definition is used as an assumption in the proofs of the compactness results in Theorems 3 and 7
Summary
We consider the homogenization of the exhibits a mismatch between the spatial linear scales parabolic problem ρ(x/ε2)∂tuε(x, t) − in the sense that the coefficient a(x/ε1. ∇t/⋅ε(12a)(oxf/tεh1,ete/lεl12ip)∇tiucεp(xar, tt)h)a=s ofn(exf,rte)qwuehnicchy of fast spatial oscillations, whereas the coefficient ρ(x/ε2) of the time derivative contains a faster spatial scale. It is shown that the faster spatial microscale does not give rise to any corrector term and that there is only one local problem needed to characterize the homogenized problem. The problem is not of a reiterated type even though two rapid scales of spatial oscillation appear
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