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

Abstract

We consider the homogenization of the linear parabolic problem 𝜌(𝑥/𝜀₂)𝜕_𝑡𝑢^𝜀(𝑥,𝑡)−∇⋅(𝑎(𝑥/𝜀₁,𝑡/𝜀²₁)∇𝑢^𝜀(𝑥,𝑡))=𝑓(𝑥,𝑡) which exhibits a mismatch between the spatial scales in the sense that the coefficient 𝑎(𝑥/𝜀₁,𝑡/𝜀²₁) of the elliptic part has one frequency of fast spatial oscillations, whereas the coefficient 𝜌(𝑥/𝜀₂) 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.

1. Introduction

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 𝜀 associated with the length scales of the heterogeneities tends to zero. The sequence of solutions 𝑢^𝜀 converges to the solution 𝑢 to a so-called homogenized problem governed by a coefficient 𝑏, where 𝑏 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 𝜌𝑥𝜀₂𝜕_𝑡𝑢^𝜀()𝑎𝑥𝑥,𝑡−∇⋅𝜀₁,𝑡𝜀²₁∇𝑢^𝜀()()𝑥,𝑡=𝑓𝑥,𝑡inΩ_𝑇,𝑢^𝜀()(),𝑢𝑥,𝑡=0on𝜕Ω×0,𝑇_𝜀(𝑥,0)=𝑢⁰(𝑥)inΩ, (1) where Ω_𝑇=Ω×(0,𝑇), Ω is an open, bounded set in ℝ^𝑁 with locally Lipschitz boundary, where both 𝑎 and 𝜌 possess unit periodicity in their respective arguments and the scales 𝜀₁, 𝜀₂, and 𝜀²₁ fulfill a certain separatedness assumption.

The problem exhibits rapid spatial oscillations in 𝜌 and spatial as well as temporal oscillations in 𝑎. Furthermore, there is a “mismatch” between the spatial scales in the sense that the frequency of the spatial oscillations in 𝜌(𝑥/𝜀₂) is higher than that of 𝑎(𝑥/𝜀₁,𝑡/𝜀²₁). Since there are two spatial microscales represented in (1), one might expect two local problems with respect to one corrector each, see, for example, [1]. However, it is shown that no corrector corresponding to the scale emanating from 𝜌(𝑥/𝜀₂) 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 [2] that the corrector 𝑢₂ associated with the gradient for the second rapid spatial scale 𝑦₂ 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 𝑢^𝜀.

Thinking in terms of heat conduction, our result means that the heat capacity 𝜌 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 𝜌≡1 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 [5], 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 𝜌≡1, Nandakumaran and Rajesh [12] 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 𝑌_𝑘=(0,1)^𝑁 for 𝑘=1,…,𝑛, 𝑌^𝑛=𝑌₁×⋯×𝑌_𝑛, 𝑦^𝑛=(𝑦₁,…,𝑦_𝑛), 𝑑𝑦^𝑛=𝑑𝑦₁⋯𝑑𝑦_𝑛, 𝑆_𝑗=𝑆=(0,1) for 𝑗=1,…,𝑚, 𝑆^𝑚=𝑆₁×⋯×𝑆_𝑚, 𝑠^𝑚=(𝑠₁,…,𝑠_𝑚), and 𝑑𝑠^𝑚=𝑑𝑠₁⋯𝑑𝑠_𝑚. Let 𝜀_𝑘(𝜀), 𝑘=1,…,𝑛, and 𝜀^′_𝑗(𝜀), 𝑗=1,…,𝑚, be positive and go to zero when 𝜀 does. Furthermore, let 𝐹_♯((0,1)^𝑀) be the space of all functions in 𝐹_loc(ℝ^𝑀) that are (0,1)^𝑀-periodic repetitions of some function in 𝐹((0,1)^𝑀).

2. Multiscale Convergence

A two-scale convergence was invented by Nguetseng [15] as a new approach for the homogenization of problems with fast oscillations in one scale in space. The method was further developed by Allaire [16] and generalized to multiple scales by Allaire and Briane [1]. To homogenize problem (1), we use the further generalization in the definition below, adapted to evolution settings, see, for example, [8].

Definition 1. A sequence {𝑢^𝜀} in 𝐿²(Ω_𝑇) is said to (𝑛+1,𝑚+1)-scale converge to 𝑢₀∈𝐿²(Ω_𝑇×𝑌^𝑛×𝑆^𝑚) if _Ω𝑇𝑢^𝜀()𝑣𝑥𝑥,𝑡𝑥,𝑡,𝜀₁𝑥,…,𝜀_𝑛,𝑡𝜀^′₁𝑡,…,𝜀^′_𝑚⟶𝑑𝑥𝑑𝑡_{Ω𝑇𝑌𝑛𝑆𝑚}𝑢₀𝑥,𝑡,𝑦^𝑛,𝑠^𝑚×𝑣𝑥,𝑡,𝑦^𝑛,𝑠^𝑚𝑑𝑦^𝑛𝑑𝑠^𝑚𝑑𝑥𝑑𝑡, (2) for any 𝑣∈𝐿²(Ω_𝑇;𝐶_♯(𝑌^𝑛×𝑆^𝑚)). We write 𝑢^𝜀(𝑥,𝑡)⇀^{𝑛+1,𝑚+1}𝑢₀𝑥,𝑡,𝑦^𝑛,𝑠^𝑚. (3)

Usually, some assumptions are made about how the scales are related to each other. We say that the scales in a list {𝜀₁,…,𝜀_𝑛} are separated if lim_𝜀→0𝜀_𝑘+1𝜀_𝑘=0, (4) for 𝑘=1,…,𝑛−1 and that the scales are well-separated if there exists a positive integer 𝑙 such that lim_𝜀→01𝜀_𝑘𝜀_𝑘+1𝜀_𝑘^𝑙=0, (5) for 𝑘=1,…,𝑛−1.

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 {𝜀₁,…,𝜀_𝑛} and {𝜀^′₁,…,𝜀^′_𝑚} 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 {𝜀₁,…,𝜀_𝑛} and {𝜀^′₁,…,𝜀^′_𝑚} are said to be jointly well separated.

In the theorem below, which will be used in the homogenization procedure in Section 3, 𝑊¹₂(0,𝑇;𝐻¹₀(Ω),𝐿²(Ω)) denotes all functions 𝑢∈𝐿²(0,𝑇;𝐻¹₀(Ω)) such that 𝜕_𝑡𝑢∈𝐿²(0,𝑇;𝐻⁻¹(Ω)), see, for example, [18, Chapter  23].

Theorem 3. Let {𝑢^𝜀} be a bounded sequence in 𝑊¹₂(0,𝑇;𝐻¹₀(Ω),𝐿²(Ω)), and suppose that the lists {𝜀₁,…,𝜀_𝑛} and {𝜀^′₁,…,𝜀^′_𝑚} are jointly well separated. Then there exists a subsequence such that 𝑢^𝜀(𝑥,𝑡)⟶𝑢(𝑥,𝑡)𝑖𝑛𝐿²Ω_𝑇,𝑢^𝜀(𝑥,𝑡)⇀𝑢(𝑥,𝑡)𝑖𝑛𝐿²0,𝑇;𝐻¹₀(Ω),∇𝑢^𝜀(𝑥,𝑡)⇀^{𝑛+1,𝑚+1}∇𝑢(𝑥,𝑡)+^𝑛_𝑗=1∇_𝑦𝑗𝑢_𝑗𝑥,𝑡,𝑦^𝑗,𝑠^𝑚, (6)(7) where 𝑢∈𝑊¹₂(0,𝑇;𝐻¹₀(Ω),𝐿²(Ω)), 𝑢₁∈𝐿²(Ω_𝑇×𝑆^𝑚;𝐻¹_♯(𝑌₁)/ℝ), and 𝑢_𝑗∈𝐿²(Ω_𝑇×𝑌^𝑗−1×𝑆^𝑚;𝐻¹_♯(𝑌_𝑗)/ℝ) for 𝑗=2,…,𝑛.

Proof. See [17, Theorem 2.74].

To treat evolution problems with fast time oscillations, such as (1), we also need the concept of very weak multiscale convergence, see, for example, [2, 5].

Definition 4. A sequence {𝑔^𝜀} in 𝐿¹(Ω_𝑇) is said to (𝑛+1,𝑚+1)-scale converge very weakly to 𝑔₀∈𝐿¹(Ω_𝑇×𝑌^𝑛×𝑆^𝑚) if _Ω𝑇𝑔^𝜀()𝑣𝑥𝑥,𝑡𝑥,𝜀₁𝑥,…,𝜀_𝑛−1𝑡×𝑐𝑡,𝜀^′₁𝑡,…,𝜀^′_𝑚𝜑𝑥𝜀_𝑛⟶𝑑𝑥𝑑𝑡_{Ω𝑇𝑌𝑛𝑆𝑚}𝑔₀𝑥,𝑡,𝑦^𝑛,𝑠^𝑚𝑣𝑥,𝑦^𝑛−1×𝑐𝑡,𝑠^𝑚𝜑𝑦_𝑛𝑑𝑦^𝑛𝑑𝑠^𝑚𝑑𝑥𝑑𝑡, (8) for any 𝑣∈𝐷(Ω,𝐶^∞_♯(𝑌^𝑛−1)), 𝑐∈𝐷(0,𝑇;𝐶^∞_♯(𝑆^𝑚)), and 𝜑∈𝐶^∞_♯(𝑌_𝑛)/ℝ, where _𝑌𝑛𝑔₀𝑥,𝑡,𝑦^𝑛,𝑠^𝑚𝑑𝑦_𝑛=0. (9) We write 𝑔^𝜀(𝑥,𝑡)⇀^{𝑛+1,𝑚+1}_𝑣𝑤𝑔₀𝑥,𝑡,𝑦^𝑛,𝑠^𝑚. (10)

Remark 5. The requirement (9) is imposed in order to ensure the uniqueness of the limit. For details, see [17, Proposition  2.26].

Remark 6. The convergence in Definition 1 may take place only if {𝑢^𝜀} is bounded in 𝐿²(Ω_𝑇) and hence also is a weakly convergent in 𝐿²(Ω_𝑇), 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 {𝑢^𝜀/𝜀_𝑛}, which are in general not bounded in 𝐿²(Ω_𝑇). 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 {𝑢^𝜀} be a bounded sequence in 𝑊¹₂(0,𝑇;𝐻¹₀(Ω),𝐿²(Ω)), and assume that the lists {𝜀₁,…,𝜀_𝑛} and {𝜀^′₁,…,𝜀^′_𝑚} are jointly well separated. Then there exists a subsequence such that 𝑢^𝜀(𝑥,𝑡)𝜀_𝑛^{𝑛+1,𝑚+1}⇀_𝑣𝑤𝑢_𝑛𝑥,𝑡,𝑦^𝑛,𝑠^𝑚, (11) where, for 𝑛=1, 𝑢₁∈𝐿²(Ω_𝑇×𝑆^𝑚;𝐻¹_♯(𝑌₁)/ℝ) and, for 𝑛=2,3,…, 𝑢_𝑛∈𝐿²(Ω_𝑇×𝑌^𝑛−1×𝑆^𝑚;𝐻¹_♯(𝑌_𝑛)/ℝ) are the same as those in Theorem 3.

Proof. See [17, Theorem 2.54].

Remark 8. For a sequence of solutions {𝑢^𝜀} to (1), we may replace the requirement that {𝜕_𝑡𝑢^𝜀} should be bounded in 𝐿²(0,𝑇;𝐻⁻¹(Ω)) by the assumption that {𝑢^𝜀} is bounded in 𝐿^∞(Ω_𝑇) and still obtain (6), see [12, Lemmas 3.3 and (4.1)] and thereby also (7) and (11). The only difference is that 𝑢 will belong to 𝐿²(0,𝑇;𝐻¹₀(Ω)) instead of the space 𝑊¹₂(0,𝑇;𝐻¹₀(Ω),𝐿²(Ω)). See also [13].

3. Homogenization

Let us now investigate the heat conduction problem 𝜌𝑥𝜀₂𝜕_𝑡𝑢^𝜀()𝑎𝑥𝑥,𝑡−∇⋅𝜀₁,𝑡𝜀²₁∇𝑢^𝜀()()𝑥,𝑡=𝑓𝑥,𝑡inΩ_𝑇,𝑢^𝜀()(),𝑢𝑥,𝑡=0on𝜕Ω×0,𝑇_𝜀(𝑥,0)=𝑢⁰(𝑥)inΩ, (12) which takes into consideration heat capacity oscillations. We assume that 𝜌∈𝐶^∞_♯(𝑌₂), is positive, 𝑓∈𝐿²(Ω_𝑇), 𝑢⁰∈𝐿²(Ω), and 𝑎𝑦₁,𝑠𝜉⋅𝜉≥𝛼𝜉² (13) for some 𝛼>0, all (𝑦₁,𝑠)∈ℝ^𝑁+1, and all 𝜉∈ℝ^𝑁, where 𝑎∈𝐶_♯(𝑌₁×𝑆)^𝑁×𝑁. Moreover, we assume that {𝑢^𝜀} is bounded in 𝐿^∞(Ω_𝑇), see Remark 8, and that the lists {𝜀₁,𝜀₂} and {𝜀²₁} are jointly well separated. Note that this separatedness assumption implies, for example, that 𝜀₂ tends to zero faster than 𝜀₁, 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 𝑢₂ actually vanishes.

Theorem 9. Let {𝑢^𝜀} be a sequence of solutions to (12). Then 𝑢^𝜀(𝑥,𝑡)⇀𝑢(𝑥,𝑡)𝑖𝑛𝐿²0,𝑇;𝐻¹₀(Ω),∇𝑢^𝜀(𝑥,𝑡)⇀^3,2∇𝑢(𝑥,𝑡)+∇_𝑦1𝑢₁𝑥,𝑡,𝑦₁,𝑠+∇_𝑦2𝑢₂𝑥,𝑡,𝑦²,𝑠, (14)(15) where 𝑢 is the unique solution to _𝑌2𝜌𝑦₂𝑑𝑦₂𝜕_𝑡𝑢()((()𝑥,𝑡−∇⋅𝑏∇𝑢𝑥,𝑡))=𝑓𝑥,𝑡𝑖𝑛Ω_𝑇,𝑢()(),𝑢()𝑥,𝑡=0𝑜𝑛𝜕Ω×0,𝑇𝑥,0=𝑢⁰(𝑥)𝑖𝑛Ω, (16) with ()=𝑏∇𝑢𝑥,𝑡_𝑆𝑌1𝑎𝑦₁,𝑠∇𝑢(𝑥,𝑡)+∇_𝑦1𝑢₁𝑥,𝑡,𝑦₁,𝑠𝑑𝑦₁𝑑𝑠. (17) Here, 𝑢₁∈𝐿²(Ω_𝑇×𝑆;𝐻¹_♯(𝑌₁)/ℝ) uniquely solves _𝑌2𝜌𝑦₂𝑑𝑦₂𝜕_𝑠𝑢₁𝑥,𝑡,𝑦₁,𝑠−∇_𝑦1⋅𝑎𝑦₁,𝑠∇𝑢(𝑥,𝑡)+∇_𝑦1𝑢₁𝑥,𝑡,𝑦₁,𝑠=0. (18) Furthermore, the corrector 𝑢₂ vanishes.

Remark 10. After a separation of variables, we can write the local problem as _𝑌2𝜌𝑦₂𝑑𝑦₂𝜕_𝑠𝑧_𝑘𝑦₁−,𝑠_{𝑁𝑖,𝑗=1}𝜕_𝑦𝑖𝑎_𝑖𝑗𝑦₁,𝑠𝛿_𝑗𝑘+𝜕_𝑦𝑗𝑧_𝑘𝑦₁,𝑠=0 (19) and the homogenized coefficient as 𝑏_𝑖𝑘=_𝑆𝑌1^𝑁_𝑗=1𝑎_𝑖𝑗𝑦₁,𝑠𝛿_𝑗𝑘+𝜕_𝑦𝑗𝑧_𝑘𝑦₁,𝑠𝑑𝑦₁𝑑𝑠, (20) where 𝑘=1,…,𝑁 and 𝑢₁𝑥,𝑡,𝑦₁,𝑠=^𝑁_𝑘=1𝜕_𝑥𝑘𝑢(𝑥,𝑡)⋅𝑧_𝑘𝑦₁,𝑠. (21)

Remark 11. Periodic homogenization problems of, for example, elliptic or parabolic type may be seen as special cases of the more general concepts of 𝐺-convergence, which gives a characterization of the limit problem but no suggestion of how to compute the homogenized matrix. Essential features of 𝐺-convergence for parabolic problems are that boundary conditions, and initial conditions are preserved in the limit. 𝐺-convergence for linear parabolic problems were studied already in [19] by Spagnolo and extended to the monotone case by Svanstedt in [20]. A treatment of this problem in a quite general setting is found in the recent work [21] by Paronetto.

Proof of Theorem 9. Following the procedure in Section  23.9 in [18], we obtain that {𝑢^𝜀} is bounded in 𝐿²(0,𝑇;𝐻¹₀(Ω)), see also [22]. Hence, (14) holds up to a subsequence. We proceed by studying the weak form of (12); that is, _Ω𝑇𝑥−𝜌𝜀₂𝑢^𝜀(𝑥,𝑡)𝑣(𝑥)𝜕_𝑡𝑐(𝑡)𝑥+𝑎𝜀₁,𝑡𝜀²₁∇𝑢^𝜀()(𝑥)𝑐(𝑡)=𝑥,𝑡∇𝑣𝑑𝑥𝑑𝑡_Ω𝑇𝑓(𝑥,𝑡)𝑣(𝑥)𝑐(𝑡)𝑑𝑥𝑑𝑡, (22) for all 𝑣∈𝐻¹₀(Ω) and 𝑐∈𝐷(0,𝑇). We pass to the limit by applying (6), taking into consideration Remark 8, and (7) with 𝑛=1 and 𝑚=1 and arrive at the homogenized problem _{Ω𝑇𝑆𝑌1}−_𝑌2𝜌𝑦₂𝑑𝑦₂𝑢(𝑥,𝑡)𝑣(𝑥)𝜕_𝑡𝑐(𝑡)+𝑎𝑦₁,𝑠∇𝑢(𝑥,𝑡)+∇_𝑦1𝑢₁𝑥,𝑡,𝑦₁(𝑥)𝑐(𝑡),𝑠×∇𝑣𝑑𝑦₁=𝑑𝑠𝑑𝑥𝑑𝑡_Ω𝑇𝑓(𝑥,𝑡)𝑣(𝑥)𝑐(𝑡)𝑑𝑥𝑑𝑡. (23) To find the local problem associated with 𝑢₁, let us again consider (22) in which we choose 𝑣(𝑥)=𝜀₁𝑣₁(𝑥)𝑣₂𝑥𝜀₁;𝑣₁∈𝐷(Ω),𝑣₂∈𝐶^∞_♯𝑌₁ℝ,𝑐(𝑡)=𝑐₁(𝑡)𝑐₂𝑡𝜀²₁;𝑐₁∈𝐷(0,𝑇);𝑐₂∈𝐶^∞_♯(𝑆); (24)(25) that is, we study _Ω𝑇𝑥−𝜌𝜀₂𝑢^𝜀(𝑥,𝑡)𝑣₁(𝑥)𝑣₂𝑥𝜀₁×𝜀₁𝜕_𝑡𝑐₁(𝑡)𝑐₂𝑡𝜀²₁+𝜀₁⁻¹𝑐₁(𝑡)𝜕_𝑠𝑐₂𝑡𝜀²₁𝑥+𝑎𝜀₁,𝑡𝜀²₁∇𝑢^𝜀()⋅𝜀𝑥,𝑡₁∇𝑣₁(𝑥)𝑣₂𝑥𝜀₁+𝑣₁(𝑥)∇_𝑦1𝑣₂𝑥𝜀₁×𝑐₁(𝑡)𝑐₂𝑡𝜀²₁=𝑑𝑥𝑑𝑡_Ω𝑇𝑓(𝑥,𝑡)𝜀₁𝑣₁(𝑥)𝑣₂𝑥𝜀₁𝑐₁(𝑡)𝑐₂𝑡𝜀²₁𝑑𝑥𝑑𝑡. (26) We first investigate the second term of the part of the expression containing time derivatives. We have _Ω𝑇𝑥−𝜌𝜀₂𝑢^𝜀(𝑥,𝑡)𝑣₁(𝑥)𝑣₂×𝑥𝜀₁𝜀₁⁻¹𝑐₁(𝑡)𝜕_𝑠𝑐₂𝑡𝜀²₁=𝑑𝑥𝑑𝑡_Ω𝑇−𝜀₁⁻¹𝑢^𝜀(𝑥,𝑡)𝑣₁(𝑥)𝑣₂×𝑥𝜀₁𝜌𝑥𝜀₂−_𝑌2𝜌𝑦₂𝑑𝑦₂×𝑐₁(𝑡)𝜕_𝑠𝑐₂𝑡𝜀²₁𝑑𝑥𝑑𝑡+_Ω𝑇−𝜀₁⁻¹𝑢^𝜀(𝑥,𝑡)𝑣₁(𝑥)𝑣₂𝑥𝜀₁×_𝑌2𝜌𝑦₂𝑑𝑦₂𝑐₁(𝑡)𝜕_𝑠𝑐₂𝑡𝜀²₁⟶𝑑𝑥𝑑𝑡_{Ω𝑇𝑆𝑌1}−_𝑌2𝜌𝑦₂𝑑𝑦₂𝑢₁𝑥,𝑡,𝑦₁,𝑠×𝑣₁(𝑥)𝑣₂𝑦₁𝑐₁(𝑡)×𝜕_𝑠𝑐₂(𝑠)𝑑𝑦₁𝑑𝑠𝑑𝑥𝑑𝑡, (27) where we have applied (11) with 𝑛=2 and 𝑚=1 and with 𝑛=1 and 𝑚=1, respectively, in the last step. The passage to the limit in the remaining part of (26) is a simple application of (7) with 𝑛=1 and 𝑚=1. This provides us with the weak form, _{Ω𝑇𝑆𝑌1}−_𝑌2𝜌𝑦₂𝑑𝑦₂𝑢₁𝑥,𝑡,𝑦₁,𝑠×𝑣₁(𝑥)𝑣₂𝑦₁𝑐₁(𝑡)𝜕_𝑠𝑐₂(𝑠)+𝑎𝑦₁,𝑠∇𝑢(𝑥,𝑡)+∇_𝑦1𝑢₁𝑥,𝑡,𝑦₁,𝑠⋅𝑣₁(𝑥)∇_𝑦1𝑣₂𝑦₁𝑐₁(𝑡)𝑐₂(𝑠)𝑑𝑦₁𝑑𝑠𝑑𝑥𝑑𝑡=0, (28) of the local problem (18). This means that 𝑢₁, and thus also 𝑢, is uniquely determined and hence the entire sequence {𝑢^𝜀} converges and not just the extracted subsequence.
This far, we have only used test functions oscillating with a period 𝜀₁, and hence we have not given the coefficient 𝜌(𝑥/𝜀₂) a fair chance to produce a second corrector 𝑢₂. In order to do so, we use a slightly different set of test functions in (22). Again, we let 𝑐 be as in (25), whereas 𝑣 is chosen according to 𝑣(𝑥)=𝜀₂𝑣₁(𝑥)𝑣₂𝑥𝜀₁𝑣̃𝑥𝜀₂;𝑣₁∈𝐷(Ω),𝑣₂∈𝐶^∞_♯𝑌₁, (29) where 𝑣̃𝑦₂=𝑣₃𝑦₂−𝐾𝜌𝑦₂;𝑣₃∈𝐶^∞_♯𝑌₂, (30) with 𝐾=_𝑌2𝜌𝑦₂𝑣₃𝑦₂𝑑𝑦₂. (31) Note that _𝑌2𝜌𝑦₂𝑣̃𝑦₂𝑑𝑦₂=0, (32) which means that 𝜌𝑣̃∈𝐶^∞_♯(𝑌₂)/ℝ. We get _Ω𝑇𝑥−𝜌𝜀₂𝑢^𝜀(𝑥,𝑡)𝑣₁(𝑥)𝑣₂𝑥𝜀𝑣̃𝑥𝜀₂×𝜀₂𝜕_𝑡𝑐₁(𝑡)𝑐₂𝑡𝜀²₁+𝜀₂𝜀²₁𝑐₁(𝑡)𝜕_𝑠𝑐₂𝑡𝜀²₁𝑥+𝑎𝜀₁,𝑡𝜀²₁∇𝑢^𝜀()⋅𝜀𝑥,𝑡₂∇𝑣₁(𝑥)𝑣₂𝑥𝜀₁𝑣̃𝑥𝜀₂+𝜀₂𝜀₁𝑣₁(𝑥)∇_𝑦1𝑣₂𝑥𝜀₁𝑣̃𝑥𝜀₂+𝑣₁(𝑥)𝑣₂𝑥𝜀₁∇_𝑦2𝑣̃𝑥𝜀₂×𝑐₁(𝑡)𝑐₂𝑡𝜀²₁=𝑑𝑥𝑑𝑡_Ω𝑇𝑓(𝑥,𝑡)𝜀₂𝑣₁(𝑥)𝑣₂𝑥𝜀₁×𝑣̃𝑥𝜀₂𝑐₁(𝑡)𝑐₂𝑡𝜀²₁𝑑𝑥𝑑𝑡, (33) and applying (11) with 𝑛=2 and 𝑚=1 together with (15), that is, (7) with 𝑛=2 and 𝑚=1, we achieve _{Ω𝑇𝑆𝑌2}𝑎𝑦₁×(),𝑠∇𝑢𝑥,𝑡+∇_𝑦1𝑢₁𝑥,𝑡,𝑦₁,𝑠+∇_𝑦2𝑢₂𝑥,𝑡,𝑦²,𝑠⋅𝑣₁(𝑥)𝑣₂𝑦₁∇_𝑦2𝑣̃𝑦₂𝑐₁(𝑡)𝑐₂(𝑠)𝑑𝑦²𝑑𝑠𝑑𝑥𝑑𝑡=0. (34) Noting that 𝑎, 𝑢, and 𝑢₁ are all independent of 𝑦₂, (34) reduces to _{Ω𝑇𝑆𝑌2}𝑎𝑦₁,𝑠∇_𝑦2𝑢₂𝑥,𝑡,𝑦²,𝑠⋅𝑣₁(𝑥)𝑣₂𝑦₁∇_𝑦2𝑣̃𝑦₂𝑐₁(𝑡)𝑐₂(𝑠)𝑑𝑦²𝑑𝑠𝑑𝑥𝑑𝑡=0. (35) Recalling (30), we have _{Ω𝑇𝑆𝑌2}𝑎𝑦₁,𝑠∇_𝑦2𝑢₂𝑥,𝑡,𝑦²,𝑠⋅𝑣₁(𝑥)𝑣₂𝑦₁∇_𝑦2𝑣₃𝑦₂−𝐾𝜌𝑦₂×𝑐₁(𝑡)𝑐₂(𝑠)𝑑𝑦²𝑑𝑠𝑑𝑥𝑑𝑡=0, (36) which after rearranging can be written as _{Ω𝑇𝑆𝑌2}𝑎𝑦₁,𝑠∇_𝑦2𝑢₂𝑥,𝑡,𝑦²,𝑠⋅𝑣₁(𝑥)𝑣₂𝑦₁∇_𝑦2𝑣₃𝑦₂𝑐₁(𝑡)𝑐₂(𝑠)𝑑𝑦²𝑑𝑠𝑑𝑥𝑑𝑡=𝐾_{Ω𝑇𝑆𝑌1𝑌2}𝑎𝑦₁,𝑠∇_𝑦2𝑢₂𝑥,𝑡,𝑦²,𝑠⋅∇_𝑦21𝜌𝑦₂𝑑𝑦₂×𝑣₁(𝑥)𝑣₂𝑦₁𝑐₁(𝑡)𝑐₂(𝑠)𝑑𝑦₁𝑑𝑠𝑑𝑥𝑑𝑡. (37) If we replace 𝑣̃ with 1/𝜌 in (33), let 𝜀→0, and use (6) and (7) with 𝑛=2 and 𝑚=1, we find that _{Ω𝑇𝑆𝑌2}𝑎𝑦₁,𝑠∇_𝑦2𝑢₂𝑥,𝑡,𝑦²,𝑠⋅𝑣₁(𝑥)𝑣₂𝑦₁∇_𝑦21𝜌𝑦₂𝑐₁(𝑡)×𝑐₂(𝑠)𝑑𝑦²𝑑𝑠𝑑𝑥𝑑𝑡=0. (38) This means that the right-hand side in (37) is zero. Applying several times the variational lemma on the remaining part, we obtain _𝑌2𝑎𝑦₁,𝑠∇_𝑦2𝑢₂𝑥,𝑡,𝑦²,𝑠⋅∇_𝑦2𝑣₃𝑦₂𝑑𝑦₂=0, (39) and hence the corrector 𝑢₂ is zero.

Remark 12. That 𝑢₂ vanishes means that 𝑢^𝜀/𝜀₂ tends to zero in the sense of very weak (3,2)-scale convergence. However, there might still be oscillations originating from the oscillations of 𝜌(𝑥/𝜀₂) that have an impact on 𝑢^𝜀. The possibility is that their amplitude is so small that the magnification by 1/𝜀₂ 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.