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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.
The field of homogenization has its main source of inspiration in the problem of finding the macroscopic properties of strongly heterogeneous materials. Mathematically, the approach is to study a sequence of partial differential equations where a parameter associated with the length scales of the heterogeneities tends to zero. The sequence of solutions converges to the solution to a so-called homogenized problem governed by a coefficient , where gives the effective property of the material and can be characterized by certain auxiliary problems called the local problems.
In this paper, we study the homogenization of the linear parabolic problem where , is an open, bounded set in with locally Lipschitz boundary, where both and possess unit periodicity in their respective arguments and the scales , , and fulfill a certain separatedness assumption.
The problem exhibits rapid spatial oscillations in and spatial as well as temporal oscillations in . Furthermore, there is a “mismatch” between the spatial scales in the sense that the frequency of the spatial oscillations in is higher than that of . Since there are two spatial microscales represented in (1), one might expect two local problems with respect to one corrector each, see, for example, . However, it is shown that no corrector corresponding to the scale emanating from appears in the local and homogenized problem and accordingly there is only one local problem appearing in the formulated theorem. Hence, the problem is not of a reiterated type. We prove by means of very weak multiscale convergence  that the corrector associated with the gradient for the second rapid spatial scale actually vanishes. Already, in [3, 4], it was observed that having more than one rapid temporal scale in parabolic problems does not generate a reiterated problem and in this paper we can see that nor does the addition of a spatial scale if it is contained in a coefficient that is multiplied with the time derivative of .
Thinking in terms of heat conduction, our result means that the heat capacity may oscillate with completely different periodic patterns without making any difference for the homogenized coefficient as long as the arithmetic mean over one period is the same.
Parabolic homogenization problems for have been studied for different combinations of spatial and temporal scales in several papers by means of techniques of two-scale convergence type with approaches related to the one first introduced in , see, for example, [2, 3, 6–8], and in, for example, [9–11], techniques not of two-scale convergence type are applied. Concerning cases where, as in (1) above, we do not have , Nandakumaran and Rajesh  studied a nonlinear parabolic problem with the same frequency of oscillation in time and space, respectively, in the elliptic part of the equation and an operator oscillating in space with the same frequency appearing in the temporal differentiation term. Recently, a number of papers have addressed various kinds of related problems where the temporal scale is not assumed to be identical with the spatial scale, see for example, [13, 14]. Up to the authors’ knowledge, this is the first study of this type of problems where the oscillations of the coefficient of the term including the time derivative do not match the spatial oscillations of the elliptic part.
Notation. We denote for , , , , for , , , and . Let , , and , , be positive and go to zero when does. Furthermore, let be the space of all functions in that are -periodic repetitions of some function in .
2. Multiscale Convergence
A two-scale convergence was invented by Nguetseng  as a new approach for the homogenization of problems with fast oscillations in one scale in space. The method was further developed by Allaire  and generalized to multiple scales by Allaire and Briane . To homogenize problem (1), we use the further generalization in the definition below, adapted to evolution settings, see, for example, .
Definition 1. A sequence in is said to -scale converge to if for any . We write
Usually, some assumptions are made about how the scales are related to each other. We say that the scales in a list are separated if for and that the scales are well-separated if there exists a positive integer such that for .
The concept in the following definition is used as an assumption in the proofs of the compactness results in Theorems 3 and 7. For a more technically formulated definition and some examples, see [17, Section 2.4].
Definition 2. Let and be lists of well-separated scales. Consider all elements from both lists. If from possible duplicates, where by duplicates we mean scales which tend to zero equally fast, one member of each pair is removed and the list in order of magnitude of all the remaining elements is well separated, the lists and are said to be jointly well separated.
Theorem 3. Let be a bounded sequence in , and suppose that the lists and are jointly well separated. Then there exists a subsequence such that where , , and for .
Proof. See [17, Theorem 2.74].
Definition 4. A sequence in is said to -scale converge very weakly to if for any , , and , where We write
Remark 6. The convergence in Definition 1 may take place only if is bounded in and hence also is a weakly convergent in , at least up to suitable subsequences. For very weak multiscale convergence, this is not so. The main intention with the concept is to study sequences of the type , which are in general not bounded in . This requires a more restrictive class of test functions.
The theorem below is a key result for the homogenization procedure in Section 3.
Theorem 7. Let be a bounded sequence in , and assume that the lists and are jointly well separated. Then there exists a subsequence such that where, for , and, for , are the same as those in Theorem 3.
Proof. See [17, Theorem 2.54].
Remark 8. For a sequence of solutions to (1), we may replace the requirement that should be bounded in by the assumption that is bounded in and still obtain (6), see [12, Lemmas 3.3 and (4.1)] and thereby also (7) and (11). The only difference is that will belong to instead of the space . See also .
Let us now investigate the heat conduction problem which takes into consideration heat capacity oscillations. We assume that , is positive, , , and for some , all , and all , where . Moreover, we assume that is bounded in , see Remark 8, and that the lists and are jointly well separated. Note that this separatedness assumption implies, for example, that tends to zero faster than , which means that we have a mismatch between the spatial scales in (12).
We give a homogenization result for this problem in the theorem below. In the proof, it is shown that the local problem associated with the slower spatial microscale is enough to characterize the homogenized problem; that is, the fastest spatial scale does not give rise to any corrector involved in the homogenization. We also prove that the second corrector actually vanishes.
Theorem 9. Let be a sequence of solutions to (12). Then where is the unique solution to with Here, uniquely solves Furthermore, the corrector vanishes.
Remark 10. After a separation of variables, we can write the local problem as and the homogenized coefficient as where and
Remark 11. Periodic homogenization problems of, for example, elliptic or parabolic type may be seen as special cases of the more general concepts of -convergence, which gives a characterization of the limit problem but no suggestion of how to compute the homogenized matrix. Essential features of -convergence for parabolic problems are that boundary conditions, and initial conditions are preserved in the limit. -convergence for linear parabolic problems were studied already in  by Spagnolo and extended to the monotone case by Svanstedt in . A treatment of this problem in a quite general setting is found in the recent work  by Paronetto.
Proof of Theorem 9. Following the procedure in Section 23.9 in , we obtain that is bounded in , see also . Hence, (14) holds up to a subsequence. We proceed by studying the weak form of (12); that is,
for all and . We pass to the limit by applying (6), taking into consideration Remark 8, and (7) with and and arrive at the homogenized problem
To find the local problem associated with , let us again consider (22) in which we choose
that is, we study
We first investigate the second term of the part of the expression containing time derivatives. We have
where we have applied (11) with and and with and , respectively, in the last step. The passage to the limit in the remaining part of (26) is a simple application of (7) with and . This provides us with the weak form,
of the local problem (18). This means that , and thus also , is uniquely determined and hence the entire sequence converges and not just the extracted subsequence.
This far, we have only used test functions oscillating with a period , and hence we have not given the coefficient a fair chance to produce a second corrector . In order to do so, we use a slightly different set of test functions in (22). Again, we let be as in (25), whereas is chosen according to where with Note that which means that . We get and applying (11) with and together with (15), that is, (7) with and , we achieve Noting that , , and are all independent of , (34) reduces to Recalling (30), we have which after rearranging can be written as If we replace with in (33), let , and use (6) and (7) with and , we find that This means that the right-hand side in (37) is zero. Applying several times the variational lemma on the remaining part, we obtain and hence the corrector is zero.
Remark 12. That vanishes means that tends to zero in the sense of very weak -scale convergence. However, there might still be oscillations originating from the oscillations of that have an impact on . The possibility is that their amplitude is so small that the magnification by is not enough for the oscillations to be recognized in the limit. In this sense, the concept of very weak multiscale convergence gives us a more precise idea of what a corrector equals zero means.
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