Mid Sweden University

In this paper we prove a general homogenization result for monotone parabolic problems with an arbitrary number of microscopic scales in space as well as in time, where the scale functions are not necessarily powers of epsilon. The main tools for the homogenization procedure are multiscale convergence and very weak multiscale convergence, both adapted to evolution problems. At the end of the paper an example is given to concretize the use of the main result.

In this paper we establish compactness results of multiscale and very weak multiscale type for sequences bounded in L2 (0, T; H10 (Ω)), fulfilling a certain condition. We apply the results in the homogenization of the parabolic partial differential equation εp ∂t uε(x, t) − ∇ · (a(xε−1, xε−2, tε−q, tε−r)∇uε(x, t)) = f(x, t), where 0 < p < q < r. The homogenization result reveals two special phenomena, namely that the homogenized problem is elliptic and that the matching for which the local problem is parabolic is shifted by p, compared to the standard matching that gives rise to local parabolic problems.

The focus of this thesis is the theory of periodic homogenization of partial differential equations and some applicable concepts of convergence. More precisely, we study parabolic problems exhibiting both spatial and temporal microscopic oscillations and a vanishing volumetric heat capacity type of coefficient. We also consider a hyperbolic-parabolic problem with two spatial microscopic scales. The tools used are evolution settings of multiscale and very weak multiscale convergence, which are extensions of, or closely related to, the classical method of two-scale convergence. The novelty of the research in the thesis is the homogenization results and, for the studied parabolic problems, adapted compactness results of multiscale convergence type.

This paper is devoted to the study of the linear parabolic problem  by means of periodic homogenization. Two interesting phenomena arise as a result of the appearance of the coefficient  in front of the timederivative. First, we have an elliptic homogenized problem although the problem studiedis parabolic. Secondly, we get a parabolic local problem even though the problem has adifferent relation between the spatial and temporal scales than those normally giving rise to parabolic local problems. To be able to establish the homogenization result, adapting to the problem we state and prove compactness results for the evolution setting of multiscale and very weak multiscale convergence. In particular, assumptions on the sequence  different from the standard setting are used, which means that these results are also of independent interest.

We study the homogenization of a hyperbolic-parabolic PDE with oscillations in one fast spatial scale. Moreover, the first order time derivative has a degenerate coefficient passing to infinity when ϵ→0. We obtain a local problem which is of elliptic type, while the homogenized problem is also in some sense an elliptic problem but with the limit for ϵ-1∂tuϵ as an undetermined extra source term in the right-hand side. The results are somewhat surprising and work remains to obtain a fully rigorous treatment. Hence the last section is devoted to a discussion of the reasonability of our conjecture including numerical experiments.

We study the homogenization of a certain linear hyperbolic-parabolic problem exhibiting two rapid spatial scales {ε; ε²}. The homogenization is performed by means of evolution multiscale convergence, a generalization of the concept of two-scale convergence to include any number of scales in both space and time. In particular we apply a compactness result for gradients. The outcome of the homogenization procedure is that we obtain a homogenized problem of hyperbolic-parabolic type together with two elliptic local problems, one for each rapid scale, for the correctors.