Liselott Flodén

Very weak multiscale convergence

We briefly recall the concept of multiscale convergence, which is a generalization of two-scale convergence. Then we investigate a related concept, called very weak multiscale convergence, and prove a compactness result with respect to this type of convergence. Finally we illustrate how this result can be used to study homogenization problems with several scales of oscillations.

Homogenization of Parabolic Equations with an Arbitrary Number of Scales in Both Space and Time

The main contribution of this paper is the homogenization of the linear parabolic equation exhibiting an arbitrary finite number of both spatial and temporal scales. We briefly recall some fundamentals of multiscale convergence and provide a characterization of multiscale limits for gradients, in an evolution setting adapted to a quite general class of well-separated scales, which we name by jointly well-separated scales (see appendix for the proof). We proceed with a weaker version of this concept called very weak multiscale convergence. We prove a compactness result with respect to this latter type for jointly well-separated scales. This is a key result for performing the homogenization of parabolic problems combining rapid spatial and temporal oscillations such as the problem above. Applying this compactness result together with a characterization of multiscale limits of sequences of gradients we carry out the homogenization procedure, where we together with the homogenized problem obtain local problems, that is, one for each spatial microscale. To illustrate the use of the obtained result, we apply it to a case with three spatial and three temporal scales with , , and .

Some Remarks on Homogenization in Perforated Domains

Mathematical homogenization theory deals with the question of finding effective properties and microvariations in heterogeneous materials. Usually, the difficulty consists in handling the rapid periodic oscillations of coefficients governing some partial differential equation. Sometimes, though, there are also many small periodically arranged holes in the material, i.e., the domain of the equation. In this latter case we have to distinguish between the situations where the holes have Neumann (e.g., isolating holes) and Dirichlet (e.g., constant temperature) boundary conditions. The aim of this chapter is to investigate an intermediate case, where holes with constant zero temperature are coated with a thin layer of a material with low heat-conduction number.

A Strange Term in the Homogenization of Parabolic Equations with Two Spatial and Two Temporal Scales

We study the homogenization of a parabolic equation with oscillations in both space and time in the coefficient a(x/e,t/e2) in the elliptic part and spatial oscillations in the coefficient ρ(x/e) that is multiplied with the time derivative ∂tue. We obtain a strange term in the local problem. This phenomenon appears as a consequence of the combination of the spatial oscillation in ρ(x/e) and the temporal oscillation in a(x/e,t/e2) and disappears if either of these oscillations is removed.

A discussion of a homogenization procedure for a degenerate linear hyperbolic-parabolic problem

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 e→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 e−1∂tue 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.

Homogenization of a Hyperbolic-Parabolic Problem with Three Spatial Scales

We study the homogenization of a certain linear hyperbolic-parabolic problem exhibiting two rapid spatial scales {e, e2}. 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.

A Note on Parabolic Homogenization with a Mismatch between the Spatial Scales

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.