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

Abstract

The main contribution of this paper is the homogenization of the linear parabolic equation∂tuε(x,t)-∇·(a(x/εq1,...,x/εqn,t/εr1,...,t/εrm)∇uε(x,t))=f(x,t)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 obtainnlocal 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 withq1=1,q2=2, and0<r1<r2.

Homogenization of the Heat Equation with a Vanishing Volumetric Heat Capacity

“…From (9) we know that {u ε } is bounded in L 2 (0, T ; H 1 0 (Ω)) and therefore {∇u ε } is bounded in L 2 (Ω T ) N and we have…”

“…In [16] (see also the appendix of [9]), compactness results were given for an arbitrary number of scales in both space and time, extending the concept of multiscale convergence to an analogous evolution setting.…”

“…The present form of the concept was given for an arbitrary number of spatial scales in [6], where also the name "very weak multiscale convergence" was established. Following [17] and [9], we give the evolution version of very weak multiscale convergence including arbitrarily many spatial and temporal scales.…”

“…, can be found in e.g. [9]. Here, we use a different approach where the boundedness of the time derivative is replaced by a certain condition, see (2) in Theorem 5.…”

Homogenization of Linear Parabolic Equations with a Certain Resonant Matching Between Rapid Spatial and Temporal Oscillations in Periodically Perforated Domains

“…Also [37], by Persson, deals with monotone parabolic problems, but with an arbitrary number of temporal microscales, where none of them has to be identical with the rapid spatial scale or even has to be a power of ε. In [21] we return to the case of linear parabolic homogenization for arbitrary numbers of spatial and temporal scales benefitting from the concept of jointly separated scales introduced in [35].…”