Abstract
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 .
1. Introduction
In this paper, we study the homogenization of where and . Here , where is an open bounded subset of with smooth boundary and is periodic with respect to the unit cube in in the first variables and with respect to the unit interval in the remaining variables. The homogenization of (1) consists in studying the asymptotic behavior of the solutions as tends to zero and finding the limit equation which admits the limit of this sequence as its unique solution. The main contribution of this paper is the proof of a homogenization result for (1), that is, for parabolic problems with an arbitrary finite number of scales in both space and time.
Parabolic problems with rapid oscillations in one spatial and one temporal scale were investigated already in [1] using asymptotic expansions. Techniques of two-scale convergence type, see, for example, [2–4], for this kind of problems were first introduced in [5]. One of the main contributions in [5] is a compactness result for a more restricted class of test functions compared with usual two-scale convergence, which has a key role in the homogenization procedure. In [6], a similar result for an arbitrary number of well-separated spatial scales is proven and the type of convergence in question is formalized under the name of very weak multiscale convergence.
A number of recent papers address various kinds of parabolic homogenization problems applying techniques related to those introduced in [5]. [7] treats a monotone parabolic problem with the same choices of scales as in [5] in the more general setting of -convergence. In [8], the case with two fast temporal scales is treated with one of them identical to a single fast spatial scale. These results with the same choice of scales are extended to a more general class of differential operators in [9] and in [10], the two fast spatial scales are fixed to be , , while only one fast temporal scale appears. Significant progress was made in [11], where the case with an arbitrary number of temporal scales is treated and none of them has to coincide with the single fast spatial scale. A first study of parabolic problems where the number of fast spatial and temporal scales both exceeds one is found in [12], where the fast spatial scales are , and the rapid temporal scales are chosen as , , and . Similar techniques have also been recently applied to hyperbolic problems. In [13] the two fast spatial scales are well separated and the fast temporal scale coincides with the slower of the fast spatial scales and in [14] the set of scales is the same as in [8, 9]. Clearly all of these previous results include strong restrictions on the choices of scales. Our aim here is to provide a unified approach with the choices of scales in the examples above as special cases. The homogenization procedure for (1) covers arbitrary numbers of spatial and temporal scales and any reasonable choice of the exponents and defining the fast spatial and temporal scales, respectively. The key to this is the result on very weak multiscale convergence proved in Theorem 7 which adapts the original concept in [6] to the appropriate evolution setting. Let us note that techniques used for the proof of the special case with , in [10] do not apply to the case with arbitrary numbers of scales studied here.
The present paper is organized as follows. In Section 2 we briefly recall the concepts of multiscale convergence and evolution multiscale convergence and give a characterization of gradients with respect to this latter type of convergence under a certain well-separatedness assumption. In Section 3 we consider very weak multiscale convergence in the evolution setting and give the key compactness result employed in the homogenization of (1), which is carried out in Section 4. In this final section, we also illustrate how this general homogenization result can be used by applying it to the particular case governed by where .
Notation. is the space of all functions in that are -periodic repetitions of some function in . We denote for , , , , for , , , , and . Moreover, we let , , and , , be strictly positive functions such that and go to zero when does. More explanations of standard notations for homogenization theory are found in [15].
2. Multiscale Convergence
Our approach for the homogenization procedure in Section 4 is based on the two-scale convergence method, first introduced in [2] and generalized to include several scales in [16]. Following [16], we say that a sequence in ()-scale converges to if for any and we write This type of convergence can be adapted to the evolution setting; see, for example, [12]. We give the following definition of evolution multiscale convergence.
Definition 1. A sequence in is said to ()-scale converge to if for any . We write
Normally, some assumptions are made on the relation between the scales. 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 .
We also need the concept in the following definition.
Definition 2. Let and be lists of well-separated scales. Collect all elements from both lists in one common list. If from possible duplicates, where by duplicates we mean scales which tend to zero equally fast, one member of each such 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.
In the remark below, we give some further comments on the concept introduced in Definition 2.
Remark 3. To include also the temporal scales alongside with the spatial scales allows us to study a much richer class of homogenization problems such as all the cases included in (1). For a more technically formulated definition and some examples, see Section 2.4 in [17]. Note that the lists and of spatial and temporal scales, respectively, in (1) are jointly well-separated for any choice of and .
Below we provide a characterization of evolution multiscale limits for gradients, which will be used in the proof of the homogenization result in Section 4. Here is the space of all functions in such that the time derivative belongs to ; see, for example, Chapter 23 in [18].
Theorem 4. 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 Theorem 2.74 in [17] and the appendix of this paper.
3. Very Weak Multiscale Convergence
A first compactness result of very weak convergence type was presented in [5] for the purpose of homogenizing linear parabolic equations with fast oscillations in one spatial scale and one temporal scale. A compactness result for the case with oscillations in well-separated spatial scales was proven in [6], where the notion of very weak convergence was introduced. It states that for any bounded sequence in and the scales in the list well-separated it holds up to subsequence that for any and , where is the same as in the right-hand side of the original time independent version of the gradient characterization in Theorem 4, that is found in [16]. In Theorem 7 below we present a generalized result including oscillations in time with a view to homogenizing (1). First we define very weak evolution multiscale convergence.
Definition 5. We say that a sequence in -scale converges very weakly to if for any , and . A unique limit is provided by requiring that We write
The following proposition (see Theorem 3.3 in [16]) is needed for the proof of Theorem 7.
Proposition 6. Let be a function such that and assume that the scales in the list are well-separated. Then is bounded in .
We are now ready to state the following theorem which is essential for the homogenization of (1); see also Theorem 7 in [19] and Theorem 2.78 in [17].
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 in Theorem 4.
Proof. We want to prove that for any , and ,
for some suitable subsequence. First we note that any can be expressed as
for some (see, e.g., Remark 3.2 in [7]). Furthermore, let
and observe that
because of the -periodicity of . By (18), the left-hand side of (17) can be expressed as
Integrating by parts with respect to , we obtain
To begin with, we consider the first term. Passing to the multiscale limit using Theorem 4, we arrive up to subsequence at
and due to (20) all but the last term vanish. We have
Moreover, (8) means that the second term of (22) up to a subsequence approaches
where the last equality is a result of (20).
It remains to investigate the last term of (22). We write
Clearly, is bounded in for by Proposition 6. Observing that is assumed to be bounded in , this means that, for any integer , there are constants such that
Hence, all the terms in the sum (26) vanish as as a result of the separatedness of the scales. Then (24) is all that remains after passing to the limit in (22). Finally, integrating (24) by parts, we obtain
which is the right-hand side of (17).
Remark 8. The notion of very weak multiscale convergence is an alternative type of multiscale convergence. It is remarkable in the sense that it enables us to provide a compactness result of multiscale convergence type for sequences that are not bounded in any Lebesgue space. In fact, it deals with the normally forbidden situation of finding a limit for a quotient, where the denominator goes to zero while the numerator does not. The price to pay for this is that we have to use much smaller class of admissible testfunctions. In the set of modes of multiscale convergence usually applied in homogenization that we find in Definition 1 and Theorem 4, very weak multiscale convergence provides us with the missing link. As we will see in the homogenization procedure in the next section Theorems 4 and 7 give us the cornerstones for the homogenization procedure that allows us to tackle all appearing passages to limits in a unified way by means of two distinct theorems and without ad hoc constructions. Moreover, Theorem 7 provides us with appropriate upscaling to detect microoscillations in solutions of typical homogenization problems, which are usually of vanishing amplitude, while the global tendency is filtered away as a result of the choice of test functions. See [12].
4. Homogenization
We are now ready to give the main contribution of this paper, the homogenization of the linear parabolic problem (1). The gradient characterization in Theorem 4 and the very weak compactness result from Theorem 7 are crucial for proving the homogenization result, which is presented in Section 4.1. An illustration of how this result can be used in practice is given in Section 4.2.
4.1. The General Case
We study the homogenization of the problem where , , , and where we assume that(A1).(A2) for all , all and some .Under these conditions, (29) allows a unique solution and for some positive constant ,
Given the scale exponents and , we may define some numbers in order to formulate the theorem below in a convenient way. We define (the number of temporal scales faster than the square of the spatial scale in question) and (indicates whether there is nonresonance or resonance), , as follows.(i)If ,then , if for some , then , and if , then .(ii)If for some , that is we have resonance, we let ; otherwise, .Note that from the definition of we have in fact in the definition of that in the case of resonance.
Finally, we recall that the lists and are jointly well-separated.
Theorem 9. Let be a sequence of solutions in to (29). Then it holds that where is the unique solution to with Here and , , are the unique solutions to the system of local problems for , where is independent of .
Remark 10. In the case , we naturally interpret the integration in (34) as if there is no local temporal integration involved and that there is no independence of any local temporal variable.
Remark 11. Note that if,for example, is independent of the function space that belongs to simplifies to and when is also independent of , we have that and so on.
Proof of Theorem 9. Since is bounded in and the lists of scales are jointly well-separated, we can apply Theorem 4 and obtain that, up to a subsequence,
where , , and , .
To obtain the homogenized problem, we introduce the weak form
of (29) where and , and letting , we get using Theorem 4
We proceed by deriving the system of local problems (34) and the independencies of the local temporal variables. Fix and choose
with , for , , and for . Here and will be fixed later. Using this choice of test functions in (36), we have
where, for and , the interpretation should be that the partial derivative acts on and , respectively, and where the and terms are defined analogously. We let and using Theorem 4, we obtain
and extracting a factor in the first term, we get
Suppose that and (which also guarantees that as required above); then, by Theorems 7 and 4, we have left
which is the point of departure for deriving the local problems and the independency.
We distinguish four different cases where is either zero (nonresonance) or one (resonance) and is either zero or positive.
Case 1. Consider and . We choose and . This means that since and . This implies that (42) is valid. We get
where we let and obtain by means of Theorems 7 and 4
By the Variational Lemma, we have
a.e. in for all and by density for all . This is the weak form of the local problem in this case. In what follows Theorems 7 and 4, the variational lemma and the density argument are used in a corresponding way.
Case 2. Consider and . We again choose and . We then have since and and which implies that we may again use (42). We get
and, passing to the limit,
By the variational lemma
a.e. for all and , which is the weak form of the local problem in this second case.
Case 3. Consider and . Let be fixed and successively be . Choose which immediately yields that . Furthermore, by the restriction of and the definition of .
Thus we have from (42)
We let tend to zero and obtain
and we have left
a.e. for all . This means that is independent of ; thus, does not depend on . Next we choose and . We have and and we may again use (42). We have
where a passage to the limit yields
and finally
a.e. for all , which is the weak form of the local problem.
Case 4. Consider and . Let be fixed and successively be . Choose directly implying that . Moreover, by the restriction of and the definition of and . Hence using (42), we obtain
Passing to the limit, we get
That is,
a.e. for all , and hence is independent of . Next we choose and in (42). Thus we have and and we get
We let go to zero obtaining
and finally we arrive at
a.e. for all and , the weak form of the local problem.
Remark 12. The result above can be extended to any meaningful choice of jointly well-separated scales by means of the general compactness results in Theorems 4 and 7 and are hence not restricted to scales that are powers of ; see, for example, [11] for the case with an arbitrary number of temporal scales but only one spatial micro scale. To make the exposition clear, we have assumed linearity, but the result can be extended to monotone, not necessarily linear, problems using standard methods.
Remark 13. The wellposedness of the homogenized problem follows from -convergence; see, for example, Sections 3 and 4 in [20]. See also Theorem 4.19 in [17] for an easily accessible description of the regularity of the -limit . The existence of solutions to the local problems follows from the fact that they appear as limits in appropriate convergence processes. Concerning uniqueness, the coercivity of the elliptic part follows along the lines of the proof of Theorem 2.11 in [16] and for those containing a derivative with respect to some local time scale general theory for linear parabolic equations apply, see, for example, Section 23 in [18]. Normally multiscale homogenization results are formulated as in Theorem 9 without separation of variables and if we study slightly more general problems, for example, those with monotone operators where the linearity has been relaxed, such separation is not possible. However, in Corollary 2.12 in [16], a technique similar to separation of variables of the type sometimes used for conventional homogenization problems is developed. Here one scale at the time is removed in an inductive process and the homogenized coefficient is computed. We believe that a similar procedure could be successful also for the type of problem studied here but would be quite technical.
4.2. Illustration of Theorem 9
To illustrate the use of Theorem 9, we apply it to the -scaled parabolic homogenization problem where , ,, and the structure conditions(B1)(B2) for all , all and some are satisfied.
We note that the assumptions of Theorem 9 are satisfied in this case. Hence the convergence results in (31) hold and, for the homogenized matrix, Furthermore, and are the unique solutions to the system of local problems for , where is independent of .
To find the local problems and the independencies explicitly, we need to identify which values of , and to use. To find , we simply count the number of temporal scales faster than the square of the th spatial scale for different choices of and . Moreover, resonance () occurs when the square of the th spatial scale coincides with one of the temporal scales.
First we consider the slowest spatial scale; that is, we let . Note that . If , then , if then and if , then . Regarding resonance, if or ; then ; otherwise, . For lucidity, we present which values of and that give the different values of and in Table 1.
In a similar way as above, we get for Table 2.
We start by sorting out the independencies of the local temporal variables. As noted, for , is independent of , which means that if , then is independent of and if , then is independent of both and . In terms of and , we have that for , is independent of and for also independent of , for , is independent of and moreover, for it holds that is also independent of .
To find the local problems, we examine all possible combinations of and , where 13 are realizable depending on which values and may assume. Each row in the tables gives rise to a local problem via (63). This means that each combination gives two local problems. If a row occurs in several combinations, the same local problem reappears. If we start by choosing the first row in the second table, that is , this can be combined with all five rows from the first table, which means that the local problem descending from is common to these combinations. By (63), this common local problem is If we combine with we have in terms of and that . The other local problem in this case is In combination with , that is, , we obtain instead and for , which means that , we have The fourth possible combination, that is, with , that is , gives and finally for , that is , the second local problem is
Next we consider in Table 2, which corresponds to and gives the local problem Here we have three possible combinations, namely with , , and . We note that we have already derived the local problems corresponding to these rows. Thus, the second local problem for and is given by (67) for and by (68) and for by (69).
We proceed by choosing in Table 2, yielding The choice can be combined with three different rows from Table 1, , , and . In combination with , which means that and , we have which is essentially the same as (67) but with the integration over directly on since both and are independent of . For , that is, and , we have which is the same as (68), but where we may integrate directly on in the same manner as above. For the third possibility, , , we get the same as (69), except for the position of the integration over .
The next row in Table 2 to consider is , which can be combined only with . This combination corresponds to and gives and again (74).
Finally, for the row together with , that is, , we get where the latter is essentially the same as (69) and (74).
Thus, having considered all possible combinations of and , we have obtained 13 different cases, A–M in Figure 1, governed by two local problems each.
In the figure, cases B, D, F, H, J, and L (straight line segments) correspond to single resonance, whereas in the case G (a single point), there is double resonance. In the remaining cases (open two-dimensional regions), there is no resonance.
Remark 14. Note that for a problem with fixed scales the finding of the local problems is very straightforward. For example, if we study (61) with and , we have , , , , , and . We obtain that both and are independent of . Inserting , in (34) immediately gives the problem (73) and , results in (71). The example chosen above with variable time scale exponents reveals more of the applicability and comprehensiveness of the theorem.
Remark 15. The problem (61) was studied already in [17, 19], but using Theorem 9, the process is considerably shortened.
Appendix
Proof of Theorem 4
This section is devoted to the proof of Theorem 4. The theorem was first formulated and proven in a detailed preprint version from 2010 of [11]. It was also given as Theorem 2.74 in [17] together with the proof. We first need the following fundamental compactness result; see also, for example, Theorem 2.66 in [17]. Observe that the concept of jointly separated scales amounts to the obvious modification of jointly well-separated scales.
Theorem A.1. Let be a bounded sequence in and suppose that the lists and are jointly separated. Then there exists in such that, up to a subsequence,
Proof. Introduce the spatiotemporal variable in and let the corresponding local variable , where is the number of pairs of duplicates, that is, scales which tend to zero equally fast (see Definition 2), be defined in the following manner. Suppose that the resulting combined spatiotemporal list generated from the lists and is . Fix ; then, we have three mutually exclusive possibilities for the spatiotemporal scale . Firstly, if tends to zero equally fast as for some but not equally fast as for any , then where is a temporal “ghost” variable. Secondly, if tends to zero equally fast as for some but not equally fast as for any , then where is a spatial “ghost” variable. Finally, if tends to zero equally fast as both and for some and , then . We collect the introduced “ghost” variables in the total “ghost” variable where is a Cartesian product of copies of and copies of .
Within the framework of spatiotemporal quantities as introduced above, let
for any . Note that the sequence is bounded in and that is independent of the local “ghost” variables.
We have by definition
Since is separated, Theorem 2.4 in [16] ensures that, up to a subsequence the sequence does -converge to ; that is,
as tends to zero where
belongs to due to the fact that
by Jensen’s inequality. To conclude, we have shown (A.1) and we are done.
Remark A.2. In the proof above in the case when all spatial and temporal scales can be matched into pairs, we naturally interpret the formal instances of integration over the empty set as if there is no local spatiotemporal “ghost” integration involved.
We are now prepared to give the main proof of the appendix.
Proof of Theorem 4. Since is bounded in ,
for some unique and, by Lemmas 8.2 and 8.4 in [21],
From Theorem A.1 and again using the boundedness of in we have, up to a subsequence,
for some in .
We will now characterize in terms of gradients. Let and where is the subspace of generalized divergence-free functions in defined according to
Using as a test function in (A.9) we get, up to a subsequence,
Using partial integration on , the fact that and vanish on (we only need that one of them does, though) and that , the left-hand side of (A.11) may be written
We claim now that where
. Indeed, for any , we have and
where we have simply employed the definition of being in making the multiple integrals to vanish, so . Thus, by Corollary 3.4 in [16], we have that is bounded in for all . This boundedness yields an estimation
where we in the first inequality have utilized the Hölder inequality and in the last step have used that the scales are separated. We thus conclude that the left-hand side of (A.9) converges to
for all and all . Hence, from the right-hand side of (A.11), we obtain
By the Variational Lemma and utilizing the density property (i) of Lemma 3.7 in [16] it holds for every that
that is is in the orthogonal of almost everywhere in . By property (ii) of Lemma 3.7 in [16], We conclude that
where and for almost everywhere in .
What remains is to prove that and for . We will perform a proof by induction accomplished in two steps: the Base Case followed by the Inductive Step.
Base Case. We show that . We have, almost everywhere in ,
where the second equality follows from the fact that is -periodic. Thus,
Clearly, , and since , we have that
belongs to by the Hölder inequality. We get
and the Base Case is verified.
Inductive Step. Fix where . Assume that and, provided that , that for all . We must show that this assumption implies that . We have, almost everywhere in ,
where the second equality follows from the fact that is -periodic. We get the estimation
Using the same arguments as in the Base Case, and
belongs to . By the inductive assumption, we have that and for all . Thus
and the Inductive Step is complete and we are done.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.