On the homogenization of multicomponent transport

This paper is devoted to the homogenization of weakly coupled cooperative parabolic systems in strong convection regime with purely periodic coefficients. Our approach is to factor out oscillations from the solution via principal eigenfunctions of an associated spectral problem and to cancel any exponential decay in time of the solution using the principal eigenvalue of the same spectral problem. We employ the notion of two-scale convergence with drift in the asymptotic analysis of the factorized model as the lengthscale of the oscillations tends to zero. This combination of the factorization method and the method of two-scale convergence is applied to upscale an adsorption model for multicomponent flow in an heterogeneous porous medium.


Introduction
Upscaling reactive transport models in porous media is a problem of great practical importance and homogenization theory is a method of choice for achieving this goal (see [15] and references therein). In this paper we focus on a model problem of reactive multicomponent transport for N diluted chemical species in a saturated periodically varying media. The fluid velocity is assumed to be known. On top of usual convective and diffusive effects we consider linear reaction terms which satisfy a specific condition, namely that the reaction matrix is cooperative (see the precise definition in Section 2). This assumption is quite natural for a linear system, as we consider here, since it ensures a maximum (or positivity) principle for solutions which, being concentrations, should indeed be non-negative for obvious physical reasons. As usual the ratio between the period of the coefficients and a characteristic lengthscale of the porous domain is denoted by a small parameter 0 < ε ≪ 1. Denoting the unknown concentrations by u ε α , for 1 ≤ α ≤ N , we study in the entire space R d the following weakly coupled (i.e., no coupling in the derivatives) system of N parabolic equations with periodic bounded coefficients: Date: November 20, 2014. 1 for 1 ≤ α ≤ N , where b α are velocity fields, D α are symmetric and coercive diffusion tensors and Π is the reaction (or coupling) matrix, assumed to be cooperative (see (2.7) for a precise definition). All coefficients are Y -periodic, where Y :=]0, 1[ d is the unit cell in R d . Our main result, Theorem 18, states that a solution to the Cauchy problem for (1.1) admits the following asymptotic representation (for every 1 ≤ α ≤ N ): where {λ, (ϕ α ) 1≤α≤N } is the first eigenpair for a periodic system posed in the unit cell Y :=]0, 1[ d , b * is a constant drift vector and v(t, x) solves a scalar parabolic homogenized problem with constant coefficients. Our result generalizes the works [7] and [14], which were restricted to a single (scalar) parabolic equation. In [9], [10] a similar result was obtained for a cooperative elliptic system without convective terms. Our present work is thus the first to combine large convective terms and multiple equations. Let us explain the specific ε-scaling of the coefficients in (1.1), which is not new and is well explained, e.g., in [6]. Before adimensionalization, the physical system of equations, in original time-space coordinates (τ, y), is, for 1 ≤ α ≤ N , Interested by a macroscopic view and long time behaviour of this parabolic system, we perform a "parabolic" scaling of the time-space variables, i.e., (τ, y) → (ε −2 t, ε −1 x), which yields the scaled model (1.1).

Remark 1.
Another scaling that one could consider is the "hyperbolic" scaling, i.e., (τ, y) → (ε −1 t, ε −1 x). This has been addressed in [21] (for N = 2) where the scaled system is: The main result of [21] is that the solution to the Cauchy problem for the above system admits the asymptotic representation: where φ α is the first eigenfunction and there is no time exponential because λ = 0 happens to be the first eigenvalue for the specific choice of cooperative matrix Π αβ made in [21]. In the above equation δ is the Dirac mass which appears because of a concentration assumption on the initial data. The main difference with the parabolic scaling in our work is that there is no diffusion homogenized problem. The drift velocity can be interpreted as b * = ∇H(0) with H being some effective Hamiltonian.
The organization of this paper is as follows. In Section 2, we describe the mathematical model of cooperative parabolic systems and the precise hypotheses made on the coefficients. Section 3 briefly recalls the existence and uniqueness theory for system (1.1). Since no uniform a priori estimates can be obtained for (1.1), a factorization principle (or change of unknowns) is performed in Section 4. Then, uniform a priori bounds are deduced for the solution of this factorized problem. The definition of two-scale convergence with drift is recalled in Section 5. Then, based on the uniform a priori estimates of Section 4, we obtain a two-scale compactness result for the sequence of solutions (see Theorem 15). Our main homogenization result is Theorem 18 which is proved in Section 6. Eventually, Section 7 generalizes our analysis to a similar, but more involved, system which is meaningful from a physical point of view. The differences are that (i) the convection-diffusion takes place in a perforated porous medium and (ii) the chemical reactions are localized on the holes' boundaries rather than in the fluid bulk. This is a frequent case for adsorption or deposition of the chemical on the solid surface (cf. the discussion and references in [6]).

The model
Before we present our model, let us introduce the following shorthands: where the small positive parameter ε ≪ 1 represents the lengthscale of oscillations. We consider the following Cauchy problem: For a normed vector space H, we use the following standard notation for Yperiodic function spaces: The assumptions made on the coefficients of (2.1) are the following: and ∃c α > 0 s.t. c α |ξ| 2 ≤ D α (y)ξ · ξ for all ξ ∈ R d and for almost every y ∈ R d (where (D α ) * is the adjoint or transposed matrix of D α ), we also assume that the coupling matrix Π is irreducible, i.e., there exists no partition B = ∅,B ′ = ∅ of {1, · · · , N } such that This irreducibility assumption ensures that the system (2.1) cannot be decoupled in two disjoint subsystems (see Remark 19 below).
The only assumption made on the convective fields b α in (2.4) is that they are bounded as well as their divergences. No divergence-free assumption is made on these vector fields. The hypotheses (2.6)-(2.7) are borrowed from [24,22,3,10]. A matrix satisfying (2.6) is sometimes referred to as "cooperative matrix" (up to the addition of a multiple of the identity it is also an M -matrix).
Hence the system (2.1) gets the name "cooperative parabolic system".
Finally, we assume that the initial data in (2.2) has following regularity: u in α ∈ L 2 (R d ) for each 1 ≤ α ≤ N .

Qualitative Analysis
Results of existence and uniqueness of solutions to (2.1) are classical. The "cooperative" hypothesis (2.6) is actually not necessary to obtain well-posedness. Standard approach is to derive a priori estimates on the solution. Classical technique is to multiply (2.1) by u ε α and integrate over R d : Since the divergences of the convective fields are bounded, summing the above expression over 1 ≤ α ≤ N followed by the application of Cauchy-Schwarz inequality, Young's inequality, Gronwall's lemma and an integration over (0, T ) leads to the following a priori estimates: where the constant C ε depends on the small parameter ε. For any fixed 0 < ε, we can use the a priori estimates (3.1) and Galerkin method to establish existence and uniqueness of the solution u ε α ∈ L 2 ((0, T ); Maximum principles are a different story altogether. In general we have no maximum principles for systems. However, the hypotheses (2.6)-(2.7) guarantee a maximum principle. In [24,22], weakly coupled cooperative elliptic systems with coupling matrices satisfying (2.6)-(2.7) are studied with emphasis on maximum principles and on the well-posedness of associated spectral problems. The results of [24] on the cooperative elliptic systems have their parabolic counterpart. We state the result from [24] adapted to cooperative parabolic systems: Lemma 3 (see [24,22] for a proof). Let the conditions (2.3)-(2.7) on the coefficients of (2.1) be satisfied. Then, for any fixed ε > 0, the following holds: (i) There is a unique solution u ε α ∈ L 2 ((0, T ); In order to make an asymptotic analysis on (2.1), as ε → 0, one demands uniform (with respect to ε) estimates on the solution u ε α . But the estimates in (3.1) are not uniform in ε. This renders the application of standard compactness theorems from homogenization theory useless for (2.1).

Factorization Principle
The difficulty with the derivation of a priori estimates in presence of large lower order terms has long been recognized [25,17,18,3,4,14]. The idea is to use information from an associated spectral cell problem. The basic principle is to factor out principal eigenfunction from the solution to arrive at a new "factorized system", amenable to uniform a priori estimates. This idea of factoring our oscillations from the solution was first introduced in [25] in the context of elliptic eigenvalue problems. In case of scalar parabolic equations it is shown in [17,18,14,7] that the factorized equations have no zero order terms and that the first order terms are divergence free. In case of cooperative elliptic systems with large lower order terms studied in [3,10], however, it is shown that the factorized systems still have zero order terms and are transformed as "difference terms". We adopt the "factorization principle", extensively used in the above mentioned references, to remedy the difficulty we have with the derivation of uniform a priori estimates for (2.1). We first define the following spectral problem associated with (2.1) and posed in the unit cell with periodic boundary conditions: The above spectral cell problem is not self-adjoint. The associated adjoint problem is: where Π * is the transpose of Π. The well-posedness of the above spectral problems is a delicate issue which is addressed in [24,22]. The following proposition is an adaptation to our periodic setting of the main result of [24,22].
Proposition 5 (see [24] for a proof). Under the assumptions (2.3)-(2.7) on the coefficients, the spectral problems (4.1) and (4.2) admit a common first eigenvalue (i.e., smallest in modulus) which satisfies: (i) the first eigenvalue λ is real and simple, (ii) the corresponding first eigenfunctions (ϕ α ) 1≤α≤N ∈ (H 1 # (Y )) N for (4.1), 2) are positive, ϕ α , ϕ * α > 0 for 1 ≤ α ≤ N , and unique up to normalization. Remark 6. The first eigenvalue λ in Proposition 5 measures the balance between convection-diffusion and reaction. Also, the uniqueness of first eigenfunctions in Proposition 5 is up to a chosen normalization. The normalization that we consider is the following: In the proof of our a priori estimates it will be convenient to scale the spectral problems (4.1)-(4.2) to the entire domain R d via the change of variables y → ε −1 x. More precisely, (4.1)-(4.2) are equivalent to Now, we get down to the task of reducing (2.1) to a "factorized system". As explained in [3,4,14,7] the first eigenvalue λ governs the time decay or growth of the solution u ε α . So, as is done in the references cited, we perform time renormalization in the spirit of the factorization principle. Also the first eigenfunction ϕ ε α is factored out of u ε α . In other words we make the following change of unknowns: .
The above change of unknowns is valid, thanks to the positivity result in Proposition 5. Now we state a result that gives the factorized system satisfied by the new unknown (v ε α ) 1≤α≤N .
are given by The proof of Lemma 7 is just a matter of simple algebra, using (4.4), and we refer to [10], [16] for more details, keeping in mind the following chain rule formulae: Remark 8. The divergence of the convective fieldsb α satisfy It follows that N α=1 div ybα = 0.
Remark 9. The factorized system (4.7) still has large lower order terms. But, as noticed in [3,10], the terms are transformed as "difference terms". This factorization is the key for getting a priori estimate on the differences (v ε α − v ε β ). The following lemma gives the a priori estimates on the new unknown.
be a weak solution of (4.7)-(4.8). There exists a constant C, independent of ε, such that Proof. To derive the a priori estimates, we multiply (4.7) by v ε α followed by integrating over R d and sum the obtained expressions over 1 ≤ α ≤ N : To simplify the above expressions, we now use the scaled spectral problems (4.4)- In the above expression, we recognize the scaled adjoint cell problem (4.5). We also recognize the scaled expression of (4.9) for the convective fieldb α . Taking all these into consideration, we have the following: Summing over α, we have: (4.14) Now, let us employ (4.14) in the estimate (4.13) which leads to: The above expression is nothing but the following energy estimate: Each one of the integrands in the above estimate is positive because of the positivity assumption (2.3), coercivity assumption (2.5) and the cooperative assumption (2.6). Integrating the energy estimate (4.15) over (0, T ) yields the a priori estimates (4.12).

Two-scale Compactness
The homogenization procedure is to consider the weak formulation of (4.7)-(4.8) with appropriately chosen test functions and passing to the limit as ε → 0. The usual approach is to obtain two-scale limits using a priori estimates of Lemma 10 by employing some compactness theorems. As it has been noticed in [20,14,7], the classical notion of two-scale convergence from [1,23] needs to be modified in order to address the homogenization of parabolic problems in strong convection regime. We recall this modified notion of two-scale convergence with drift, as first defined in [20].
We denote this convergence by u ε Now we state a compactness theorem, again borrowed from [20], which guarantees the existence of two-scale limits with drift for certain sequences.
Proposition 12. [20,2] Let b * be a constant vector in R d and let the sequence u ε be uniformly bounded in L 2 ((0, T ) × R d ). Then, there exist a subsequence, still denoted by ε, and a function u 0 (t, x, y) Remark 13. Note that the case b * = 0 coincides with the classical notion of twoscale convergence from [1,23]. It should also be noted that the two-scale limits obtained according to Proposition 12 depend on the chosen drift velocity b * ∈ R d . These issues are addressed in [16]. Unfortunately, the notion of convergence in Definition 11 does not carry over to the case when the drift velocity b * varies in space.
If the sequence {u ε } has additional bounds, then the result of Proposition 12 can be improved. The following result addresses this issue when the sequence has uniform H 1 bounds in space.
Proposition 14. [20,2] Let b * be a constant vector in R d and let the sequence u ε be uniformly bounded in L 2 ((0, T ); H 1 (R d )). Then, there exist a subsequence, still denoted by ε, and functions u 0 (t, x) ∈ L 2 ((0, T ); Having given the notion of convergence, we shall state a result that gives the two-scale limits corresponding to solution sequences for (4.7)-(4.8).
) N of the system (4.7)-(4.8), twoscale converge with drift b * , as ε → 0, in the following sense: Proof. Consider the a priori bounds (4.12) on v ε α obtained in Lemma 10. It follows from Proposition 14 that there exist a subsequence (still indexed by ε) and twoscale limits, say v α ∈ L 2 ((0, T ); for every 1 ≤ α ≤ N . Also from the a priori estimates (4.12) we have: The estimate (5.4) implies that the two-scale limits obtained in the first line of (5.3) do match i.e., v α = v for every 1 ≤ α ≤ N . However, the limit of the coupled term isn't straightforward. Since we have the existence of a subsequence and a function q(t, x, y) ∈ L 2 ((0, T ) × R d ; L 2 # (Y )) from Proposition 12 such that .
Let us pass to the limit in (5.6) as ε → 0. The first term on the right hand side vanishes as the limits of v ε α and v ε β match. To pass to the limit in the second term of the right hand side, we shall use (5.5). Considering the two-scale limit in the second line of (5.3), upon passing to the limit as ε → 0 in (5.6) we have: ¿From (5.7) we deduce that (v 1,α −v 1,β ) and q(t, x, y) differ by a function of (t, x), say l(t, x). As v 1,α and v 1,β are also defined up to the addition of a function solely dependent on (t, x), we can get rid of l(t, x) and we recover indeed the following limit q(t, x, y) = v 1,α − v 1,β .

Homogenization Result
This section deals with the homogenization of the coupled system (4.7)-(4.8). To begin with, we state a Fredholm alternative for solving the cell problem, which is a key ingredient in the homogenization result.
Consider the following cooperative system: for every 1 ≤ α ≤ N , where the coefficients (b α ,D α ) are as in (4.9)-(4.10) and the hypotheses (2.4)-(2.7) hold. Then there exists a unique solution (ζ α ) 1≤α≤N ∈ (H 1 # (Y )) N /(R × 1) to (6.1), where 1 = (1, · · · , 1) ∈ R N , if and only if the following compatibility condition holds true: Proof. To prove that condition (6.2) is necessary, let us integrate the left hand side of (6.1) over the unit cell. Exploiting the periodic boundary conditions, we will be left with: Substituting for the divergence term in the above expression from (4.11) and summing over α indeed guarantees that the condition (6.2) on the source term is necessary.
To prove sufficiency, let us assume that (6.2) is satisfied. Consider the following norm on the quotient space H (Y ) := (H 1 # (Y )) N /(R × 1): (It is easy to show that (6.3) is a norm on H (Y ) since the zero set of (6.3) is the subspace spanned by 1.) The variational formulation of (6.1) in H (Y ) is: find The compatibility condition (6.2) implies that (f α ) 1≤α≤N is orthogonal to 1 in L 2 and consequently that the linear form L(η) in (6.4) is continuous. By performing similar computations as in the proof of Lemma 10, we can show that the bilinear form in (6.4) is coercive in H (Y ) i.e., To show that the bilinear form in (6.4) is continuous on H (Y )×H (Y ) we remark that, first,´Y Q(η)·1 dy = 0 for any η ∈ H (Y ) (this is precisely the computation which yields the compatibility condition (6.2)) and, second, Q(η − c1) = 0 for any η ∈ H (Y ) and any c ∈ R. Therefore, for any ζ, η ∈ H (Y ), we have the following: . We can thus apply the Lax-Milgram lemma in H (Y ) to obtain the existence and uniqueness of a solution to (6.1). In the previous section, using the a priori estimates, we have obtained twoscale limits with drift for the solution sequence. Now, by choosing an appropriate drift constant b * , we shall characterize the two scale limits. Contrary to the compactness result of Theorem 15 which gives the convergence up to a subsequence, the next result guarantees that the entire sequence v ε α converges to v for every 1 ≤ α ≤ N . The main result of this article is the following.
Theorem 18. Let (v ε α ) 1≤α≤N be the sequence of solutions to the system (4.7)-(4.8). The entire sequence v ε α converges, in the sense of Theorem 15, to the limits v ∈ L 2 ((0, T ); H 1 (R d )) and v 1,α ∈ L 2 ((0, T ) × R d ; H 1 # (Y )) for every 1 ≤ α ≤ N (see (5.2) for details). The two-scale limits v 1,α are explicitly given by satisfy the cell problem: for every 1 ≤ i ≤ d, where the drift velocity b * is given by Further, the two-scale limit v(t, x) is the unique solution of the scalar diffusion equation: with the elements of the dispersion matrix D given by Remark 19. The irreducibility assumption (2.7) on the coupling matrix Π ensures microscopic equilibrium among all v ε α resulting in a single homogenized limit v(t, x) i.e., if the coupling matrix Π ≡ 0 (say), we get N different homogenized limits.
Remark 20. Our main homogenization result (Theorem 18) holds only for weakly coupled cooperative parabolic systems. Our approach does not answer the homogenization of general weakly coupled parabolic systems, not to mention fully coupled systems. We heavily rely upon the cooperative assumption on the coupling matrix as the positivity and spectral theorems are known only in the cooperative case. Remark 21. The homogenized limit v(t, x) is proven to satisfy a scalar diffusion equation (6.8), which is a bit deceptive by its simplicity. However, if we make the following change of functions: we remark thatṽ(t, x) indeed satisfies the following scalar convection-diffusionreaction equation: Therefore, b * /ε is precisely the effective drift while λ/ε 2 is the effective reaction rate. Remark that because of the large drift ε −1 b * , we cannot work in bounded domains.

Remark 22.
The assumption of pure periodicity on the coefficients of (2.1) is crucial for the results obtained in this article. The natural thought for generalizing the results of this article is to explore the possibility of considering "locally periodic" coefficients i.e., coefficients of the type b(x, x/ε), where the function is Y -periodic in the second variable. If the convective fields b ε α were locally periodic, then it is clear that the drift vector b * (x) should depend on x. However, in such a case, we have no idea on how to extend the method of two-scale asymptotic expansion, not to mention the even greater difficulties in generalizing the notion of two-scale convergence with non-constant drift (as already mentioned in Remark 13). Such a generalization still remains as an outstanding open problem in the theory of Taylor dispersion.

Remark 23. This article only addresses the homogenization of linear systems.
We have also considered only diagonal diffusion models. Cross diffusion phenomena occurs naturally in the physics of multicomponent gaseous mixtures, population dynamics and porous media (cf. [8] and references therein). The natural nonlinear transport model to consider is the Maxwell-Stefan's equations. A complete mathematical study of the Maxwell-Stefan laws is still missing. There have been some recent studies in this direction (cf. [8,12,19,13] for example). One approach would be to consider the "parabolically" scaled Maxwell-Stefan's equations and arrive at an homogenization result. The obvious questions to ask is the following: Is there a scalar diffusion limit even in case of nonlinear Maxwell-Stefan's equations? This problem might involve mathematical techniques quite different from the ones used here as the spectral problems (which is the crux of the Factorization method) in the nonlinear counterpart have not been well understood. We hope to return to this question in subsequent publications.
Before we present the proof of Theorem 18, we state a lemma that gives some qualitative information on the dispersion matrix.
Lemma 24. The dispersion matrix D given by (6.9) is symmetric positive definite.
Proof. The symmetric part is obvious. By the hypothesis on the coupling matrix Π and the positivity of the first eigenvector functions, the factor Π αβ ϕ * α ϕ β is always non-positive for α = β. By the hypothesis (2.5), we know that the diffusion matrices D α are coercive with coercivity constants c α > 0. For ξ ∈ R d , we define Then, Now, we need to show that Dξ · ξ > 0 for all ξ = 0. Suppose that Dξ · ξ = 0 which in turn implies that ω αξ + ξ · y ≡ C α for some constant C α . As the cell solutions (ω i,α ) 1≤α≤N are Y -periodic, they cannot be affine. Thus the above equalities are possible only when ξ = 0 which implies the positive definiteness of D.
Proof of Theorem 18. In the sequel we use the notations The idea is to test the factorized equation (4.7) with φ ε α = φ ε + ε φ ε 1,α , where φ(t, x) and φ 1,α (t, x, y) are smooth functions with compact support in x, which vanish at the final time T and are Y -periodic with respect to y. We get Substituting for φ ε α in the above variational formulation and integrating by parts leads to (6.10) In a first step we choose φ ε ≡ 0 in (6.10) and pass to the limit as ε → 0 which yields: The above expression is the variational formulation for the following PDE: for every 1 ≤ α ≤ N . By the Fredholm result of Lemma 16, we have the existence and uniqueness of (v 1,α ) 1≤α≤N ∈ L 2 ((0, T ) × R d ; H (Y )) if and only if the compatibility condition (6.2) is satisfied. Writing down the compatibility condition for (6.12) yields the expression (6.7) for the drift velocity b * . Also by linearity of (6.12), we deduce that we can separate the slow and fast variables in v 1,α as in (6.5) with (ω i,α ) 1≤α≤N satisfying the coupled cell problem (6.6).
In a second step we choose φ ε 1,α ≡ 0 in (6.10) and substitute (4.8) for the initial data v ε α (0, x), which yields (6.13) Using the expression (4.11) for the divergence ofb α allows us to obtain Thanks to (6.14) all terms of order O(ε −2 ) in (6.13) cancel each other. There are, however, terms of O(ε −1 ) in (6.13) which still prevent us to pass to the limit as ε → 0. In order to remedy the situation, we introduce the following auxiliary problem posed in the unit cell: The above auxiliary problem is well-posed, thanks to our choice (6.7) of the drift velocity and the chosen normalization (4.3). We scale (6.15) to the entire domain via the change of variables y → ε −1 x. The vector-valued function Ξ ε (x) = Ξ(x/ε) satisfies (6.16) Getting back to the variational formulation (6.13), let us regroup the problematic terms of order O(ε −1 ): where we have used the scaled auxiliary problem (6.16). We can now pass to the limit in (6.18) since the sequences (v ε β − v ε α )/ε are bounded. Taking into consideration (6.14) and (6.18), the variational formulation (6.13) rewrites as (6.19) Using the compactness results from Theorem 15, we pass to the limit as ε → 0 in the above variational formulation leading to: Substituting (6.5) for v 1,α in (6.20), we obtain Using the information from the auxiliary cell problem (6.15) in (6.21) and making a rearrangement similar to that of (6.17), we deduce that (6.21) is nothing but the variational formulation for a scalar diffusion equation (6.8) for v(t, x) with the entries of the diffusion matrix given by By integration by parts, it is clear that the diffusion matrix D is contracted with the Hessian matrix ∇∇v, which is symmetric. Thus the non-symmetric part of D does not contribute to the homogenized equation (6.8). So, the above expression for the diffusion matrix is symmetrized: To obtain the desired expression (6.9) for the diffusion matrix, we consider the variational formulation for the cell problem (6.6) with test functions (ψ α ) 1≤α≤N (6.23) In (6.23) we first choose the test function (ψ α ) = (ω j,α ). Similarly, in (6.23) for j instead of i, we choose the test function (ψ α ) = (ω i,α ). This leads to (6.24) Using formula (4.11) for the divergence ofb α in (6.24), its right hand side simplifies as (6.25) Plugging (6.25) in the symmetrized formula (6.22) leads to the desired equation (6.9). Eventually the scalar homogenized equation (6.8) has a unique solution since, by virtue of Lemma 24, the dispersion matrix is positive definite. This guarantees that the entire sequence v ε α converges to v, for 1 ≤ α ≤ N , and not merely a subsequence as in Theorem 15.

Adsorption in Porous Media
In this section, we give a generalization of our previous result in a more applied context. Our goal is to upscale a model of multicomponent transport in an highly heterogeneous porous medium in presence of adsorption reaction at the fluid-pore interface. In [7], the authors study the homogenization of one single scalar convection-diffusion-reaction equation posed in an ε-periodic infinite porous medium: Typically, an ε-periodic infinite porous medium is built out of R d (d = 2 or 3, being the space dimension) by removing a periodic distribution of solid obstacles which, after rescaling, are all similar to the unit obstacle Σ 0 . More precisely, let Y =]0, 1[ d be the unit periodicity cell. Let us consider a smooth partition Y = Σ 0 ∪ Y 0 where Σ 0 is the solid part and Y 0 is the fluid part. The fluid part (extended by periodicity) is assumed to be a smooth connected open subset whereas no particular assumptions are made on the solid part. For each multiindex j ∈ Z d , we define Y j ε := ε(Y 0 + j), Σ j ε := ε(Σ 0 + j), S j ε := ε(∂Σ 0 + j), the periodic porous medium Ω ε := ∪ j∈Z d Y j ε and the (d − 1)−dimensional surface ∂Ω ε := ∪ j∈Z d S j ε . In this section, we generalize the results of [7] to the multicomponent case. We consider the following weakly coupled cooperative parabolic system with Neumann boundary condition at the fluid-pore interface.
Remark 25. Note the different scaling in front of the surface reaction terms. It is of order ε −1 because it balances a flux rather than a diffusive term, as in the previous model of Section 2. As usual, by the change of variable (τ, y) → (ε −2 t, ε −1 x) all singular powers of ε disappears in (7.2) written in the (τ, y) variables.
The hypotheses on the coefficients in (7.2) are exactly the same as in Section 2. As before it is impossible to obtain uniform (in ε) estimates on the solutions u ε α of (7.2). As was done in Section 4, we employ the method of factorization by introducing a new unknown: where (λ, ϕ α ) and (λ, ϕ * α ) are the principal eigenpairs associated with the (new) following spectral problems respectively: Proposition 5, which guarantees the existence of principal eigenpairs for the spectral problems (4.1)-(4.2), carries over to the above spectral problems (7.3)-(7.4) as well. This is apparent from the proofs in [24,22]. The normalization (ensuring uniqueness of the eigenfunctions) that we choose is: As in Section 4 it is a matter of simple algebra to obtain the factorized system for (7.2) with the new unknown which is, for each 1 ≤ α ≤ N , where the convective fieldsb α and diffusion matricesD α are given by the same formulae (4.9) and (4.10). A proof, completely similar to that of Lemma 10, yields the following a priori estimates (7.5): Remark 26. Since the (d−1) dimensional measure of the periodic surface ∂Ω ε is of order O(ε −1 ), a bound of the type √ ε z ε L 2 (∂Ωε) ≤ C means that the sequence z ε is bounded on the surface ∂Ω ε .
In the a priori estimates (7.6), we have bounds in function spaces defined on the periodic surface ∂Ω ε . In order to speak of the convergence of sequences in such function spaces, we need to generalize the Definition 11 of two-scale convergence with drift for periodic surfaces. This generalization was introduced in [16]. We state this definition together with the corresponding compactness result (the proof of which is similar to that of Theorem 9.1 in [2]).
Taking w = 1 ε v ε α − v ε β , we deduce from (7.14) and a priori estimates (7.6) that The above estimate (7.15) implies that the limits obtained in (7.13) do match i.e., v α = v for every 1 ≤ α ≤ N . The H 1 a priori estimate in space as in (7.6) does imply that v ∈ L 2 ((0, T ); H 1 (R d )) and that there exist limits v 1,α ∈ L 2 ((0, T ) × R d ; H 1 # (Y 0 )) such that The rest of the proof is completely similar to the proof of Theorem 18. We safely leave it to the reader.
We finish by stating a technical lemma which was useful in the proof of Theorem 29.
The existence and uniqueness of ξ ∈ H 1 # (Y 0 )/R is guaranteed for the above problem as (7.17) is indeed the compatibility condition from the Fredholm alternative. Choosing θ = ∇ y ξ gives one possible solution for (7.18).