Homogenization of the discrete diffusive coagulation-fragmentation equations in perforated domains

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Homogenization of the discrete diffusive

coagulation-fragmentation equations in perforated

domains.

Laurent Desvillettes1 and Silvia Lorenzani2

1 _{IMJ-PRG, Universit´}_{e Paris Diderot, France}

2 _{Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da}

Vinci 32, 20133 Milano, Italy

Abstract

The asymptotic behavior of the solution of an infinite set of Smoluchowski’s discrete coagulation-fragmentation-diffusion equations with non-homogeneous Neumann boundary conditions, defined in a periodically perforated domain, is analyzed. Our homogenization result, based on Nguetseng-Allaire two-scale convergence, is meant to pass from a microscopic model (where the physical processes are properly described) to a macroscopic one (which takes into account only the effective or averaged properties of the system). When the characteristic size of the perforations vanishes, the information given on the microscale by the non-homogeneous Neumann boundary condition is transferred into a global source term appearing in the limiting (homogenized) equations. Furthermore, on the macroscale, the geometric structure of the perforated domain induces a correction in the diffusion coefficients.

1 Introduction

This paper is devoted to the homogenization of an infinite set of Smoluchowski’s discrete coagulation-fragmentation-diffusion equations in a periodically perforated domain. The system of evolution equations considered describes the dynamics of cluster growth, that is the mechanisms allowing clusters to coalesce to form larger

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clusters or break apart into smaller ones. These clusters can diffuse in space with a diffusion constant which depends on their size. Since the size of clusters is not limited a priori, the system of reaction-diffusion equations that we consider consists of an infinite number of equations. The structure of the chosen equations, defined in a perforated medium with a non-homogeneous Neumann condition on the boundary of the perforations, is useful in investigating several phenomena arising in porous media [14], [8], [13] or in the field of biomedical research [11].

Typically, in a porous medium, the domain consists of two parts: a fluid phase where colloidal species or chemical substances, transported by diffusion, are dissolved and a solid skeleton (formed by grains or pores) on the boundary of which deposi-tion processes or chemical reacdeposi-tions take place. In recent years, the Smoluchowski equation has been also considered in biomedical research to model the aggregation and diffusion of β-amyloid peptide (Aβ) in the cerebral tissue, a process associated with the development of Alzheimer’s disease. One can define a perforated geometry, obtained by removing from a fixed domain (which represents the cerebral tissue) infinitely many small holes (the neurons). The production of Aβ in monomeric form from the neuron membranes can be modeled by coupling the Smoluchowski equation for the concentration of monomers with a non-homogeneous Neumann condition on the boundaries of the holes.

The mathematical complexity underlying the models that can be proposed to de-scribe such processes has been fully addressed in our work. Furthermore, the results of this paper constitute a generalization of some of the results contained in [14], [11], by considering an infinite system of equations where both the coagulation and fragmentation processes are taken into account. Unlike previous theoretical works, where existence and uniqueness of solutions for an infinite system of coagulation-fragmentation equations (with homogeneous Neumann boundary conditions) have been studied [19], [15], in this paper our focus lies on a distinct aspect, that is, the averaging of the system of Smoluchowski’s equations over arrays of periodically-distributed microstructures.

Our homogenization result, based on Nguetseng-Allaire two-scale convergence [17], [1], is meant to pass from a microscopic model (where the physical processes

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are properly described) to a macroscopic one (which takes into account only the effective or averaged properties of the system).

1.1 Setting of the problem

Let Ω be a bounded open set in R3 _{with a smooth boundary ∂Ω. Let Y be the}

unit periodicity cell [0, 1[3 having the paving property. We perforate Ω by removing from it a set Tǫ of periodically distributed holes defined as follows. Let us denote

by T an open subset of Y with a smooth boundary Γ, such that T ⊂ Int Y . Set Y∗ _{= Y \ T which is called in the literature the solid or material part. We define}

τ (ǫT ) to be the set of all translated images of ǫT of the form ǫ(k + T ), k ∈ Z3. Then, Tǫ := Ω ∩ τ(ǫT ).

Introduce now the periodically perforated domain Ωǫ defined by

Ωǫ = Ω \ Tǫ.

For the sake of simplicity, we make the following standard assumption on the holes [6], [9]:

there exists a ’security’ zone around ∂Ω without holes, i.e.

∃ δ > 0 such that dist (∂Ω, Tǫ) ≥ δ. (1)

Therefore, Ωǫ is a connected set. The boundary ∂Ωǫ of Ωǫ is then composed of two

parts. The first one is the union of the boundaries of the holes strictly contained in Ω. It is denoted by Γǫ and is defined by

Γǫ:= ∪

∂(ǫ(k + T )) | ǫ(k + T ) ⊂ Ω

.

The second part of ∂Ωǫ is its fixed exterior boundary denoted by ∂Ω. It is easily

seen that (see [2], Eq. (3))

lim

ǫ→0ǫ | Γǫ|N −1=| Γ|N −1

| Ω |N

| Y |N

, (2)

where | · |N is the N -dimensional Hausdorff measure.

Throughout this paper, ǫ will denote the general term of a sequence of positive real numbers which converges to zero. We will consider in the following a discrete

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coagulation-fragmentation-diffusion model for the evolution of clusters [3], [4]. De-noting by uǫ

i := uǫi(t, x) ≥ 0 the density of clusters with integer size i ≥ 1 at position

x ∈ Ωǫ and time t ≥ 0 and by di the diffusion constant for clusters of size i, the

corresponding system can be written as a family of equations in Ωǫ:

                                               ∂uǫ₁ ∂t − ∇x· (d1∇xuǫ1) + uǫ1 P∞j=1a1,juǫj = P∞ j=1B1+jβ1+j,1uǫ1+j in [0, T ] × Ωǫ, ∂uǫ₁ ∂ν ≡ ∇xuǫ1· n = 0 on [0, T ] × ∂Ω, ∂uǫ₁ ∂ν ≡ ∇xuǫ1· n = ǫ ψ(t, x,xǫ) on [0, T ] × Γǫ, uǫ₁(0, x) = U1 in Ωǫ, (3)

where ψ is a given bounded function satisfying the following conditions: (i) ψ(t, x,x_ǫ_{) ∈ C}1_{(0, T ; B) with B = C}1_{[Ω; C}1

#(Y )]

(ii) ψ(t = 0, x,x_ǫ) = 0

and U1 is a positive constant such that U1 ≤ kψkL∞_{([0,T ];B)}.

In addition, if i ≥ 2,                                                ∂uǫ_i ∂t − ∇x· (di∇xuǫi) = Qǫi+ Fiǫ in [0, T ] × Ωǫ, ∂uǫ_i ∂ν ≡ ∇xuǫi · n = 0 on [0, T ] × ∂Ω, ∂uǫ_i ∂ν ≡ ∇xuǫi · n = 0 on [0, T ] × Γǫ, uǫ_i(0, x) = 0 in Ωǫ, (4)

where the terms Qǫ

i, Fiǫdue to coagulation and fragmentation, respectively, are given

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Qǫ_i := 1 2 i−1 X j=1 ai−j,juǫi−juǫj − ∞ X j=1 ai,juǫiuǫj (5) F_iǫ:= ∞ X j=1

Bi+jβi+j,iuǫi+j− Biuǫi. (6)

The parameters Bi, βi,j and ai,j, for integers i, j ≥ 1, represent the total rate Bi

of fragmentation of clusters of size i, the average number βi,j of clusters of size j

produced due to fragmentation of a cluster of size i, and the coagulation rate ai,j of

clusters of size i with clusters of size j. These parameters represent rates, so they are always nonnegative; single particles do not fragment further, and mass should be conserved when a cluster fragments into smaller pieces, so one always imposes

ai,j = aj,i≥ 0, βi,j ≥ 0, (i, j ≥ 1) (7)

B1= 0, Bi≥ 0 (i ≥ 2) (8) i = i−1 X j=1 j βi,j (i ≥ 2) (9)

In order to prove the bounds presented in the sequel, we need to impose an additional restriction on the fragmentation coefficients: For each m ≥ 1 there exists γm > 0

such that

Bjβj,m≤ γmam,j for j ≥ m + 1. (10)

1.2 Main statement and comments

Our aim is to study the homogenization of the set of equations (3)-(4) as ǫ → 0, i.e., to study the behaviour of uǫ

i(i ≥ 1) as ǫ → 0 and obtain the equations satisfied by

the limit. There is no clear notion of convergence for the sequence uǫ_i_{(i ≥ 1)) which} is defined on a varying set Ωǫ. This difficulty is specific to the case of perforated

domains. A natural way to get rid of this difficulty is given by Nguetseng-Allaire two-scale convergence [17], [1].

Theorem 1.1. If ǫ > 0, there exists a strong solution

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to system (3) - (4), which is moreover nonnegative, that is uǫ_i_{(t, x) ≥ 0 for (t, x) ∈ (0, T ) × Ω}ǫ .

Let uǫ

i(t, x) (i ≥ 1) be a family of such strong solutions to problems (3)-(4).

The sequences euǫ

i and ]∇xuǫi (i ≥ 1) two-scale converge (up to a subsequence) to:

[χ(y) ui(t, x)] and [χ(y)(∇xui(t, x) + ∇yu1i(t, x, y))] (i ≥ 1), respectively, where tilde

denotes the extension by zero outside Ωǫand χ(y) represents the characteristic

func-tion of Y∗. The limiting functions (ui(t, x), u1i(t, x, y)) (i ≥ i) are the unique

so-lutions in L2(0, T ; H1_{(Ω)) × L}2_{([0, T ] × Ω; H}_#1(Y )/R) of the following two-scale ho-mogenized systems. If i = 1:                                        θ ∂u1 ∂t (t, x) − divx d1A ∇xu1(t, x) + θ u1(t, x)P∞j=1a1,juj(t, x) = θ P∞_j=1B1+jβ1+j,1u1+j(t, x) + d1 Z Γ ψ(t, x, y) dσ(y) _{in [0, T ] × Ω,} [A ∇xu1(t, x)] · n = 0 on [0, T ] × ∂Ω, u1(0, x) = U1 in Ω. (11) If i ≥ 2                                                θ ∂ui ∂t (t, x) − divx diA ∇xui(t, x) + θ ui(t, x)P∞j=1ai,juj(t, x)

+θ Biui(t, x) = θ2Pi−1_j=1aj,i−juj(t, x) ui−j(t, x)

+θ P∞_j=1Bi+jβi+j,iui+j(t, x) in [0, T ] × Ω,

[A ∇xui(t, x)] · n = 0 on [0, T ] × ∂Ω,

ui(0, x) = 0 in Ω.

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u1_i(t, x, y) = N X j=1 wj(y) ∂ui ∂xj(t, x) (i ≥ 1), θ = Z Y χ(y)dy = |Y ∗ |

is the volume fraction of material, and A is a matrix with constant coefficients defined by

Ajk=

Z

Y∗(∇y

wj+ ˆej) · (∇ywk+ ˆek) dy,

with ˆej being the j-th unit vector in RN, and (wj)1≤j≤N the family of solutions of

the cell problem                −divy[∇ywj+ ˆej] = 0 in Y∗ (∇ywj+ ˆej) · n = 0 on Γ y → wj(y) Y − periodic (13)

1.3 Structure of the rest of the paper

The paper is organized as follows. In Section 2 we derive all the a priori estimates needed for two-scale homogenization. In particular, in order to prove the uniform L2 -bound on the infinite sums appearing in our set of Eqs. (3)-(4), we extend to the case of non-homogeneous Neumann boundary conditions a duality method first devised by M. Pierre and D. Schmitt [18] and largely exploited afterwards [3], [4]. Then, Section 3 is devoted to the proof of our main results on the homogenization of the infinite Smoluchowski discrete coagulation-fragmentation-diffusion equations in a periodically perforated domain. Finally, Appendix A and Appendix B are introduced to summarize, respectively, some fundamental inequalities in Sobolev spaces tailored for perforated media and some basic results on the two-scale convergence method (used to perform the homogenization procedure).

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2 Estimates

We first obtain the a priori estimates for the sequences uǫ

i, ∇xuǫi, ∂tuǫi in [0, T ]×Ωǫ,

that are independent of ǫ. Lemma 2.1. Assume that

sup

i

di< +∞.

Then, for all T > 0, the weak solutions to system (3)-(4) satisfy the following bound: Z T 0 Z Ωǫ _X∞ i=1 i diuǫi(t, x) _X∞ i=1 i uǫ_i(t, x) dt dx ≤ C, (14) where C is a positive constant independent of ǫ.

Proof. Let us consider the following fundamental identity or weak formulation of the coagulation and fragmentation operators [3], [4]:

∞ X i=1 ϕiQǫi = 1 2 ∞ X i=1 ∞ X j=1 ai,juǫiuǫj(ϕi+j− ϕi− ϕj), (15) ∞ X i=1 ϕiFiǫ = − ∞ X i=2 Biuǫi ϕi− i−1 X j=1 βi,jϕj , (16)

which holds for any sequence of numbers ϕi. By choosing ϕi := i above and thanks

to (9), we have ∞ X i=1 i Qǫ_i = ∞ X i=1 i F_iǫ = 0. (17) Therefore, summing together Eq. (3) and Eq. (4) multiplied by i, taking into account the result (17), we get

∂ ∂t _X∞ i=1 i uǫ_i − ∆x _X∞ i=1 i diuǫi = 0. (18) Denoting ρǫ(t, x) = ∞ X i=1 i uǫ_i(t, x), (19) and Aǫ(t, x) = [ρǫ(t, x)]−1 ∞ X i=1 i diuǫi(t, x), (20)

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the following system can be derived from Eqs. (3), (4) and (18)                                                ∂ρǫ ∂t −∆x(Aǫρǫ) = 0 in [0, T ] × Ωǫ, ∇x(Aǫρǫ) · n = 0 on [0, T ] × ∂Ω, ∇x(Aǫρǫ) · n = d1ǫ ψ(t, x,x_ǫ) on [0, T ] × Γǫ, ρǫ_{(0, x) = U} 1 in Ωǫ. (21)

We observe that (for all t ∈ [0, T ])

kAǫ(t, ·)kL∞_(Ω

ǫ)≤ sup i

di. (22)

Multiplying the first equation in (21) by the function wǫ _{defined by the following}

dual problem:                                                − ∂wǫ ∂t + Aǫ∆xwǫ = Aǫ_ρǫ _{in [0, T ] × Ω} ǫ, ∇xwǫ· n = 0 on [0, T ] × ∂Ω, ∇xwǫ· n = 0 on [0, T ] × Γǫ, wǫ_{(T, x) = 0} _{in Ω} ǫ, (23)

and integrating by parts on [0, T ] × Ωǫ, we end up with the identity

Z T 0 Z Ωǫ Aǫ(t, x) (ρǫ(t, x))2dt dx = Z Ωǫ wǫ(0, x) ρǫ(0, x) dx + ǫ d1 Z T 0 Z Γǫ ψ(t, x,x ǫ) w ǫ_{(t, x) dt dσ} ǫ(x) := I1+ I2, (24)

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Let us now estimate the terms I1 and I2. From H¨older’s inequality we obtain I1 = Z Ωǫ wǫ(0, x) ρǫ_{(0, x) dx ≤ U}1|Ωǫ|1/2kwǫ(0, ·)kL2_(Ω ǫ). (25)

Applying once more the H¨older inequality and using the estimate (22), it holds Z Ωǫ |wǫ(0, x)|2dx = Z Ωǫ Z T 0 √ Aǫ ∂tw ǫ √ Aǫ dt 2 dx ≤ T kAǫkL∞_(Ω ǫ) Z T 0 Z Ωǫ (Aǫ)−1 _∂t∂wǫ(t, x) 2 dt dx ≤ T (sup i di) Z T 0 Z Ωǫ (Aǫ)−1 ∂ ∂tw ǫ_{(t, x)} 2 dt dx. (26)

By exploiting the dual problem (23), Eq. (26) becomes Z Ωǫ |wǫ(0, x)|2dx ≤ T (sup i di) Z T 0 Z Ωǫ (Aǫ)−1_|Aǫ∆xwǫ+ Aǫρǫ|2dt dx ≤ T (sup i di) Z T 0 Z Ωǫ (Aǫ)−1 2 (Aǫ)2(∆xwǫ)2+ 2 (Aǫ)2(ρǫ)2 dt dx. (27)

Let us now estimate the first term on the right-hand side of (27). Multiplying the first equation in (23) by (∆xwǫ), Z Ωǫ (∆xwǫ) _∂wǫ ∂t dx + Z Ωǫ Aǫ(∆xwǫ)2dx = − Z Ωǫ Aǫρǫ(∆xwǫ) dx (28)

and integrating by parts on Ωǫ, we get

−_∂t∂ Z Ωǫ |∇xwǫ|2 2 dx + Z Ωǫ Aǫ(∆xwǫ)2dx = − Z Ωǫ Aǫρǫ(∆xwǫ) dx. (29)

Then, integrating once more over the time interval [0, T ] and using Young’s inequal-ity on the right-hand side of (29), one finds that

Z Ωǫ |∇xwǫ(0, x)|2dx + Z T 0 Z Ωǫ Aǫ(∆xwǫ)2dt dx ≤ Z T 0 Z Ωǫ (ρǫ)2Aǫdt dx. (30) Since the first term on the left-hand side of (30) is nonnegative, we conclude that

Z T 0 Z Ωǫ Aǫ(∆xwǫ)2dt dx ≤ Z T 0 Z Ωǫ (ρǫ)2Aǫdt dx. (31) Inserting Eq. (31) into Eq. (27), one has

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Z Ωǫ |wǫ(0, x)|2dx ≤ 4 T (sup i di) Z T 0 Z Ωǫ Aǫ(ρǫ)2dt dx. (32) Therefore, we end up with the estimate

I1 ≤ 2 U1 |Ωǫ| T sup i di 1/2 Z T 0 Z Ωǫ Aǫ(ρǫ)2dt dx 1/2 . (33)

By using Lemma A.1 and H¨older’s inequality, the term I2 in (24) can be rewritten

as I2= ǫ d1 Z T 0 Z Γǫ ψ(t, x,x ǫ) w ǫ_{(t, x) dt dσ} ǫ(x) ≤ C d1 Z T 0 kψ(t)kB Z Ωǫ |wǫ|2dx 1/2 + ǫ Z Ωǫ |∇xwǫ|2dx 1/2 , (34)

where we have taken into account the following estimate (see Lemma B.1):

ǫ Z Γǫ |ψ(t, x,x ǫ)| 2_dσ ǫ(x) ≤ ˜C kψ(t)k2B (35)

with ˜C being a positive constant independent of ǫ and B = C1[Ω; C_#1(Y )]. Note that we do not really need the C1 _{in the estimate above, continuity would indeed}

be sufficient.

Since ψ ∈ L∞([0, T ]; B), using the Cauchy-Schwarz inequality, Eq. (34) reads

I2≤ C1d1kwǫkL2_{(0,T ;L}2_(Ω

ǫ))+ C2d1ǫ k∇xw ǫ

kL2_{(0,T ;L}2_(Ω

ǫ)) := J1+ J2, (36)

where C1 and C2 are positive constants independent of ǫ. Let us now estimate the

terms J1 and J2. Using H¨older’s inequality and estimate (22), by following the same

strategy as the one leading to (32), we have Z T 0 Z Ωǫ |wǫ(t, x)|2dt dx = Z T 0 Z Ωǫ Z T t √ Aǫ∂sw ǫ_{(s, x)} √ Aǫ ds 2 dt dx ≤ T2(sup i di) Z T 0 Z Ωǫ (Aǫ)−1 ∂wǫ(t, x) ∂t 2 dt dx ≤ 4 T2(sup i di) Z T 0 Z Ωǫ Aǫ(ρǫ)2dt dx. (37) Therefore,

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J1= C1d1 Z T 0 Z Ωǫ |wǫ(t, x)|2dt dx 1/2 ≤ 2 C1d1T (sup i di)1/2 Z T 0 Z Ωǫ Aǫ(ρǫ)2dt dx 1/2 . (38)

In order to estimate J2, we go back to Eq. (29), then integrating over [t, T ], one has

1 2 Z T t Z Ωǫ ∂ ∂s|∇xw ǫ_{(s, x)|}2_{ds dx −}Z T t Z Ωǫ Aǫ(∆xwǫ)2ds dx = Z T t Z Ωǫ Aǫρǫ(∆xwǫ) ds dx. (39)

Young’s inequality applied to the right-hand side of Eq. (39) leads to Z Ωǫ |∇xwǫ(t, x)|2dx + Z T t Z Ωǫ Aǫ(∆xwǫ)2ds dx ≤ Z T t Z Ωǫ Aǫ(ρǫ)2ds dx. (40) Taking into account that the second term on the left-hand side of (40) is nonnegative and integrating once more over time, we get

Z T 0 Z Ωǫ |∇xwǫ(t, x)|2dt dx ≤ T Z T 0 Z Ωǫ Aǫ(ρǫ)2dt dx. (41) Therefore, we conclude that

J2=C2d1ǫ Z T 0 Z Ωǫ |∇xwǫ(t, x)|2dt dx 1/2 ≤ C2d1ǫ (T )1/2 Z T 0 Z Ωǫ Aǫ(ρǫ)2dt dx 1/2 . (42)

By combining (38) and (42), we end up with the estimate

I2 ≤ d1 2 C1T r sup i di+ C2ǫ √ T Z T 0 Z Ωǫ Aǫ(ρǫ)2dt dx 1/2 . (43) Hence, inserting the estimates (33) and (43) in Eq. (24), one obtains

Z T 0 Z Ωǫ Aǫ(t, x) (ρǫ(t, x))2_{dt dx ≤ C}₃2 (44) where

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C3= max 2 U1 r |Ωǫ| T sup i di, d1[2 C1T r sup i di+ C2 √ T ] . (45) Thus, recalling the definitions of Aǫ and ρǫ, the assertion of the Lemma follows immediately.

Corollary: If we assume that infidi > 0 and

ai,j ≤ Cst (i1−ζ+ j1−ζ) (46)

for some ζ > 0, then, the estimate (14) leads to the following bound: Z T 0 Z Ωǫ ∞ X j=1 ai,juǫj(t, x) 2 dt dx ≤ C (47) with C independent of ǫ.

Lemma 2.2. Let T > 0 be arbitrary and uǫ

1 be a classical solution of (3). Then,

kuǫ1kL∞_{(0,T ;L}∞_(Ω ǫ))≤ |U1| + ku ǫ 1kL∞ (0,T ;L∞ (Γǫ))+ γ1. (48)

Proof. Let us test the first equation of (3) with the function

φ1 ≡ p (uǫ1)(p−1) p ≥ 2.

We stress that the function φ1 is strictly positive and continuously differentiable in

[0, t] × Ω, for all t > 0. Integrating, the divergence theorem yields Z t 0 ds Z Ωǫ ∂ ∂s(u ǫ 1)p(s) dx + d1p (p − 1) Z t 0 ds Z Ωǫ |∇xuǫ1|2(uǫ1)(p−2)dx = −p Z t 0 ds Z Ωǫ a1,1(uǫ1)(p+1)dx − p Z t 0 ds Z Ωǫ (uǫ₁)p ∞ X j=2 a1,juǫjdx + p Z t 0 ds Z Ωǫ (uǫ₁)(p−1) ∞ X j=2 Bjβj,1uǫjdx + ǫ d1p Z t 0 ds Z Γǫ ψ(s, x,x ǫ) (u ǫ 1)(p−1)dσǫ(x) ≤ −p Z t 0 ds Z Ωǫ ∞ X j=2 [a1,juǫ1− Bjβj,1] uǫj(uǫ1)(p−1)dx + ǫ d1p Z t 0 ds Z Γǫ ψ(s, x,x ǫ) (u ǫ 1)(p−1)dσǫ(x). (49)

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By exploiting the assumption (10), we end up with the estimate Z t 0 ds Z Ωǫ ∂ ∂s(u ǫ 1)p(s) dx + d1p (p − 1) Z t 0 ds Z Ωǫ |∇xuǫ1|2(uǫ1)(p−2)dx ≤ ǫ d1p Z t 0 ds Z Γǫ ψ(s, x,x ǫ) (u ǫ 1)(p−1)dσǫ(x) + p γ₁p Z t 0 ds Z Ωǫ ∞ X j=2 a1,juǫj dx (50)

The H¨older inequality applied to the right-hand side of (50) and the duality estimate lead to Z Ωǫ (uǫ₁(t, x))pdx + d1p (p − 1) Z t 0 ds Z Ωǫ |∇xuǫ1|2(uǫ1)(p−2)dx ≤ Z Ωǫ U₁pdx + ǫ d1p kψkL∞ (0,T ;L∞ (Γǫ)) Z t 0 ds Z Γǫ (uǫ₁)(p−1)dσǫ(x) + CT p γ1p. (51)

Since the second term on the left-hand side of (51) is nonnegative, one has Z Ωǫ (uǫ₁(t, x))p_{dx ≤} Z Ωǫ U₁pdx + ǫ d1p kψkL∞_{(0,T ;L}∞_(Γ ǫ)) Z t 0 ds Z Γǫ [1 + (uǫ₁)p] dσǫ(x) + CTp γ₁p ≤ Z Ωǫ U₁pdx + ǫ d1p kψkL∞_{(0,T ;L}∞_(Γ ǫ))T |Γǫ| + ǫ d1p kψkL∞_{(0,T ;L}∞_(Γ ǫ)) Z t 0 ds Z Γǫ (uǫ₁)pdσǫ(x) + CTp γ1p. (52)

Hence, we conclude that

sup t∈[0,T ] lim p→∞ Z Ωǫ (uǫ₁(t, x))pdx 1/p ≤ |U1| + kuǫ1kL∞ (0,T ;L∞ (Γǫ))+ γ1. (53)

The boundedness of uǫ₁(t, x) in L∞_{([0, T ]×Γ}ǫ), uniformly in ǫ, can be immediately

deduced from Lemma 2.3 below.

Lemma 2.3. Let T > 0 be arbitrary and uǫ₁ be a classical solution of (3). Then, kuǫ1kL∞_{(0,T ;L}∞_(Γ

ǫ)) ≤ C max(kψkL∞(0,T ;B), γ1) (54)

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In order to establish Lemma 2.3, we will first need the following preliminary result [11].

Theorem 2.1 ([11], Theorem 5.2). Assume that there exist positive constants T , ˆ

k = kψkL∞_{(0,T ;B)}, γ, such that for all k ≥ ˆk we have

Then

ess sup(t,x)∈[0,T ]×Γǫu ǫ

1(t, x) ≤ 2 m ˆk (56)

where the positive constant m is independent of ǫ. Proof. of Lemma 2.3

Since this proof is close to the proof of Lemma 5.2 in [11], we only sketch it. Let T > 0 and k ≥ 0 be fixed. Define: u(k)ǫ (t) := (uǫ₁(t) − k)+for t ≥ 0, with derivatives:

∂u(k)_ǫ ∂t = ∂uǫ₁ ∂t 1{uǫ1>k} (57) ∇xu(k)ǫ = ∇xuǫ11{uǫ 1>k}. (58) Moreover, u(k)_ǫ _|∂Ω= (uǫ1|∂Ω−k)+ (59) u(k)_ǫ _|Γǫ= (u ǫ 1|Γǫ −k)+ (60)

Let us assume k ≥ ˆk, where ˆk := kψkL∞

(0,T ;B). Then,

uǫ₁(0, x) = U1≤ ˆk ≤ k. (61)

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1 2 Z Ωǫ |u(k)ǫ (t)|2dx = Z t 0 d ds 1 2 Z Ωǫ |u(k)ǫ (s)|2dx ds = Z t 0 ds Z Ωǫ ∂u(k)_ǫ (s) ∂s u (k) ǫ (s) dx. (62)

Taking into account Eq. (57) and Eq. (3), we obtain that for all s ∈ [0, T1]

Z Ωǫ ∂u(k)_ǫ (s) ∂s u (k) ǫ (s) dx = Z Ωǫ ∂uǫ₁(s) ∂s u (k) ǫ (s) dx = Z Ωǫ d1∆xuǫ1− uǫ1 ∞ X j=1 a1,juǫj+ ∞ X j=1 B1+jβ1+j,1uǫ1+j u(k)_ǫ (s) dx = ǫ d1 Z Γǫ ψ s, x,x ǫ u(k)_ǫ (s) dσǫ(x) − d1 Z Ωǫ ∇xuǫ1(s) · ∇xu(k)ǫ (s) dx − Z Ωǫ (uǫ₁(s))2a1,1u(k)ǫ (s) dx − Z Ωǫ uǫ₁(s) ∞ X j=2 a1,juǫj(s) u(k)_ǫ (s) dx + Z Ωǫ _X∞ j=2 Bjβj,1uǫj(s) u(k)_ǫ (s) dx ≤ ǫ d1 Z Γǫ ψ s, x,x ǫ u(k)_ǫ (s) dσǫ(x) − d1 Z Ωǫ ∇xuǫ1(s) · ∇xu(k)ǫ (s) dx − Z Ωǫ ∞ X j=2 a1,juǫ1(s) − Bjβj,1 uǫ_j(s) u(k)_ǫ (s) dx. (63)

By using the assumption (10), Lemma A.1 and Young’s inequality, one has, taking k ≥ γ1, Z Ωǫ ∂u(k)_ǫ (s) ∂s u (k) ǫ (s) dx ≤ ǫ d1 2 Z Bǫ k(s) ψ s, x,x ǫ 2 dσǫ(x) +C1d1 2 Z Aǫ k(s) |u(k)ǫ (s)|2dx − d1 1 −C1ǫ 2 2 Z Ωǫ |∇xu(k)ǫ (s)|2dx (64)

where we denote by Aǫ_k(t) and B_kǫ(t) the set of points in Ωǫ and on Γǫ, respectively,

at which uǫ

1(t, x) > k. It holds:

|Aǫk(t)| ≤ |Ωǫ|,

|Bkǫ(t)| ≤ |Γǫ|,

where | · | is the Hausdorff measure. Inserting Eq. (64) into Eq. (62) and varying over t, we end up with the estimate:

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Introducing the following norm

kuk2Qǫ(T ):= sup 0≤t≤T Z Ωǫ |u(t)|2dx + Z T 0 dt Z Ωǫ |∇u(t)|2dx, (66) the inequality (65) can be rewritten as follows:

min 1 2, d1 1 −C1ǫ2 2 ku(k)ǫ k2Qǫ(T1)≤ C1d1 2 Z T1 0 dt Z Aǫ k(t) |u(k)ǫ (t)|2dx +ǫ d1 2 Z T1 0 dt Z Bǫ k(t) ψ t, x,x ǫ 2 dσǫ(x). (67) Let us estimate the right-hand side of (67). From H¨older’s inequality, we obtain

Z T1 0 dt Z Aǫ k(t) |u(k)ǫ (t)|2dx ≤ ku(k)ǫ kL2r1(0,T1;Lq1(Ωǫ))k1A ǫ kkLr1′(0,T1;Lq ′ 1(Ωǫ)), (68) with r′ 1 = _r₁r_{− 1}1 , q′1 = _q₁q_{− 1}1 , r1 = 2 r1, q1 = 2 q1, where, for N > 2, r1 ∈ (2, ∞)

and q₁ _{∈ (2,}_{(N −2)}2 N ) have been chosen such that 1 r1 + N 2 q₁ = N 4 . In particular, r₁′, q₁′ _{< ∞, so that (68) yields}

Z T1 0 dt Z Aǫ k(t) |u(k)ǫ (t)|2dx ≤ ku(k)ǫ kL2r1(0,T1;Lq1(Ωǫ))|Ω| 1/q′ 1_T1/r ′ 1 1 . (69)

If we choose (for ǫ small enough)

T1/r ′ 1 1 < min{1, d 1} 2C1d1c2 |Ω| −1/q′ 1 ≤ min 1 2, d1 1 −C1ǫ2 2 C1d1c2 |Ω| −1/q′ 1,

then from (113) (and c being the constant appearing in this formula) it follows that C1d1 2 Z T1 0 dt Z Aǫ k(t) |u(k)ǫ (t)|2dx ≤ 1 2min 1 2, d1 1 −C1ǫ2 2 ku(k)ǫ k2Qǫ(T1). (70)

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Analogously, from H¨older’s inequality we have, for k ≥ ˆk ǫ d1 2 Z T1 0 dt Z Bǫ k(t) ψ t, x,x ǫ 2 dσǫ(x) ≤ ǫ d1k2 2 ˆk2 k2 k1Bǫ kkL1(0,T1;L1(Γǫ)) ≤ ǫ d1k 2 2 Z T1 0 dt |B ǫ k(t)|. (71) Thus (67) yields ku(k)ǫ k2Qǫ(T1)≤ ǫ γ k 2 Z T1 0 dt |B ǫ k(t)|. (72)

Hence, by Theorem 2.1 we obtain

kuǫ1kL∞

(0,T1;L∞(Γǫ))≤ 2 m max(ˆk, γ1)

where the positive constant m is independent of ǫ. Analogous arguments are valid for the cylinder [Ts, Ts+1] × Ωǫ, s = 1, 2, . . . , p − 1 with

Ts+1− Ts 1/r′ 1 < min{1, d1} 2C1d1c2 |Ω| −1/q′ 1

and Tp≡ T . Thus, after a finite number of steps, we obtain the estimate (54).

Lemma 2.4. The sequence ∇xuǫ1 is bounded in L2([0, T ] × Ωǫ), uniformly in ǫ.

This Lemma can be easily proved by following the same arguments presented in [11] (Lemma 5.4), provided that the assumption (10) is taken into account.

Lemma 2.5. Let uǫ

i(t, x) (i ≥ 2) be a classical solution of (4). Then

kuǫikL∞_{(0,T ;L}∞_(Ω

ǫ))≤ Ki (73)

uniformly with respect to ǫ, where

Ki= 1 + _Xi−1 j=1 aj,i−jKjKi−j (Bi+ ai,i) + γi. (74)

Proof. The Lemma can be proved directly by induction following the proof reported in [19] (Lemma 2.2, p. 284). Since we have a zero initial condition for the system

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(4), we have chosen a function slightly different than what was done in [19] to test the ith equation of (4):

φi≡ p (uǫi)(p−1) p ≥ 2.

We stress that the functions φi are strictly positive and continuously differentiable

in [0, t] × Ω, for all t > 0.

Therefore, multiplying the ith equation in system (4) by φi and reorganizing the

terms appearing in the sums, we can write the estimate ||uǫi||pLp_(Ω ǫ)+ dip (p − 1) Z t 0 Z Ωǫ |∇xuǫi|2(uǫi)p−2dxds ≤ Z t 0 Z Ωǫ 1 2 i−1 X j=1

ai−j,juǫjuǫi−j− ai,i|uǫi|2− Biuǫi

p (uǫ_i)p−1dxds − Z t 0 Z Ωǫ _Xi−1 j=1 ai,juǫiuǫj + ∞ X j=i+1 (ai,juǫi− Bjβj,i) uǫj p (uǫ_i)p−1dxds.

We now work using an induction on i. Supposing that we already know that kuǫjkL∞_{(0,T ;L}∞_(Ω

ǫ)) ≤ Kj for all j < i, and using assumption (10), the previous

estimate leads to ||uǫi||pLp_(Ω ǫ)≤ Z t 0 Z Ωǫ 1 2 i−1 X j=1

ai−j,jKjKi−j − ai,i|uǫi|2− Biuǫi

p (uǫ_i)p−1dxds + Z t 0 Z Ωǫ ∞ X j=i+1 ai,j(−uǫi + γi) uǫjp (uǫi)p−1dxds =: I1+ I2.

Then, thanks for example to Young’s inequality, I1 ≤ _Xi−1 j=1 ai−j,jKjKi−j p (Bi+ ai,i)1−p |Ωǫ| T + p ai,i|Ωǫ| T, and I2 ≤ Z t 0 Z Ωǫ ∞ X j=i+1 ai,j(γi− uǫi) uǫj1{uǫ i≤γi}p (u ǫ i)p−1dxds ≤ p γip Z t 0 Z Ωǫ _X∞ j=i+1 ai,juǫj dxds ≤ C p γip(|Ωǫ| T )1/2,

where Cauchy-Schwarz inequality and the duality Lemma 2.1 (more precisely Eq. (47)) have been exploited.

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Using these estimates for bounding ||uǫi||Lp_(Ω_ǫ₎ and letting p → ∞, we end up

with the desired estimate.

Lemma 2.6. The sequence ∇xuǫi (i ≥ 2) is bounded in L2([0, T ] × Ωǫ), uniformly

in ǫ.

This Lemma can be easily proved by following the same arguments presented in [11] (Lemma 5.6), provided that the assumption (10) is taken into account.

Lemma 2.7. The sequence ∂tuǫi (i ≥ 1) is bounded in L2([0, T ] × Ωǫ), uniformly in

ǫ.

Proof. Since this proof is close to the proof of Lemma 5.9 in [11], we only sketch it. Case i = 1: Let us multiply the first equation in (3) by the function ∂tuǫ1(t, x).

Integrating, the divergence theorem yields Z Ωǫ ∂uǫ₁(t, x) ∂t 2 dx + d1 2 Z Ωǫ ∂ ∂t(|∇xu ǫ 1(t, x)|2) dx = ǫ d1 Z Γǫ ψ t, x,x ǫ ∂uǫ₁ ∂t dσǫ(x) − Z Ωǫ uǫ₁ _X∞ j=1 a1,juǫj ∂uǫ₁ ∂t dx + Z Ωǫ _X∞ j=1 B1+jβ1+j,1uǫ1+j ∂uǫ 1 ∂t dx. (75)

Using Young’s inequality and exploiting the boundedness of uǫ

1(t, x) in L∞(0, T ; L∞(Ωǫ)), one gets C1 Z Ωǫ ∂uǫ₁(t, x) ∂t 2 dx +d1 2 Z Ωǫ ∂ ∂t(|∇xu ǫ 1(t, x)|2) dx ≤ ǫ d1 Z Γǫ ψ t, x,x ǫ ∂uǫ₁ ∂t dσǫ(x) + C2 Z Ωǫ ∞ X j=1 a1,juǫj 2 dx + C3 Z Ωǫ ∞ X j=2 Bjβj,1uǫj 2 dx, (76)

where C1, C2 and C3 are positive constants independent of ǫ. Integrating over [0, t]

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C1 Z t 0 ds Z Ωǫ ∂uǫ 1 ∂s 2 dx + d1 2 Z Ωǫ |∇xuǫ1(t, x)|2dx ≤ C4 + ǫ d1 Z Γǫ ψ t, x,x ǫ uǫ₁(t, x) dσǫ(x) − ǫ d1 Z t 0 ds Z Γǫ ∂ ∂sψ s, x,x ǫ uǫ₁(s, x) dσǫ(x), (77) since ψ t = 0, x, xǫ ≡ 0.

Applying once more Young’s inequality and taking into account the estimate (35) and Lemma A.1, Eq. (77) can be rewritten as follows

C1 Z t 0 ds Z Ωǫ ∂uǫ 1 ∂s 2 dx + d1 2(1 − ǫ 2_C 5) Z Ωǫ |∇xuǫ1(t, x)|2dx ≤ C6, (78)

where the positive constants C1, C5, C6 are independent of ǫ, since ψ ∈ L∞(0, T ; B),

uǫ

1 is bounded in L∞(0, T ; L∞(Ωǫ)), ∇xuǫ1 is bounded in L2(0, T ; L2(Ωǫ)) and the

following inequality holds:

ǫ Z Γǫ ∂tψ t, x,x ǫ 2 dσǫ(x) ≤ C7k∂tψ(t)k2B ≤ C8, (79)

with C7and C8independent of ǫ. For a sequence ǫ of positive numbers going to zero:

(1 − ǫ2_C

5) ≥ 0. Then, the second term on the left-hand side of (78) is nonnegative,

and one has:

k∂tuǫ1k2L2_{(0,T ;L}2_(Ω_ǫ₎₎≤ C, (80)

where C ≥ 0 is a constant independent of ǫ.

Case i ≥ 2: Let us multiply the first equation in (4) by the function ∂tuǫi(t, x).

Integrating, the divergence theorem yields Z Ωǫ ∂uǫ_i(t, x) ∂t 2 dx + di 2 Z Ωǫ ∂ ∂t(|∇xu ǫ i(t, x)|2) dx = 1 2 Z Ωǫ _Xi−1 j=1 ai−j,juǫi−juǫj ∂uǫ i ∂t dx − Z Ωǫ uǫ_i _X∞ j=1 ai,juǫj ∂uǫ i ∂t dx + Z Ωǫ _X∞ j=1

Bi+jβi+j,iuǫi+j

∂uǫ_i ∂t dx − Z Ωǫ Biuǫi ∂uǫ_i ∂t dx. (81)

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Using Young’s inequality and exploiting the boundedness of uǫ_i(t, x) in L∞(0, T ; L∞(Ωǫ)), one gets C1 Z Ωǫ ∂uǫ_i(t, x) ∂t 2 dx + di 2 Z Ωǫ ∂ ∂t(|∇xu ǫ i(t, x)|2) dx ≤ C2+ C3 Z Ωǫ ∞ X j=1 ai,juǫj 2 dx + C4 Z Ωǫ ∞ X j=i+1 Bjβj,iuǫj 2 dx, (82)

where C1, C2, C3 and C4 are positive constants independent of ǫ.

Integrating over [0, t] with t ∈ [0, T ], thanks to (47) and (10), we end up with the estimate C1 Z t 0 ds Z Ωǫ ∂uǫ i ∂s 2 dx +di 2 Z Ωǫ |∇xuǫi(t, x)|2dx ≤ C5, (83)

with C5≥ 0 independent of ǫ. Since the second term on the left-hand side of (83) is

nonnegative, we conclude that

k∂tuǫik2L2_{(0,T ;L}2_(Ω

ǫ))≤ C, (84)

where C ≥ 0 is a constant independent of ǫ.

3 Proof of the main result

We start here the proof of our main Theorem 1.1.

3.1 Existence of solutions for a given ǫ > 0

We first explain how the estimates of the previous section can be used in the proof of existence, for a given ε, of a solution to system (3) - (4). In order to do so, we introduce a finite size truncation of this system, which writes, once the dependence

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w.r.t. ε has been eliminated for readability,                                                ∂un₁ ∂t − ∇x· (d1∇xun1) + un1 Pn j=1a1,junj = Pn−1 j=1B1+jβ1+j,1un1+j in [0, T ] × Ωǫ, ∂un₁ ∂ν ≡ ∇xun1 · n = 0 on [0, T ] × ∂Ω, ∂un₁ ∂ν ≡ ∇xun1 · n = ǫ ψ(t, x,xǫ) on [0, T ] × Γǫ, un₁(0, x) = U1 in Ωǫ, (85) and, if i = 2, .., n,                                                ∂un_i ∂t − ∇x· (di∇xuni) = Qni + Fin in [0, T ] × Ωǫ, ∂un_i ∂ν ≡ ∇xuni · n = 0 on [0, T ] × ∂Ω, ∂un_i ∂ν ≡ ∇xuni · n = 0 on [0, T ] × Γǫ, un_i(0, x) = 0 in Ωǫ, (86)

where the truncated coagulation and breakup kernels Qn_i, F_inwrite

Qn_i := 1 2 i−1 X j=1 ai−j,juni−junj − n X j=1 ai,juni unj (87) F_in:= n−i X j=1

Bi+jβi+j,iuni+j− Biuni. (88)

We observe, then, that the duality lemma (that is, Lemma 2.1) is still valid in this setting (with a proof that exactly follows the proof written above), so that we end up with the a priori estimate

Z T 0 Z Ωε n X i=1 i un_i(t, x) 2 dtdx ≤ C, (89)

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where C is a constant which does not depend on n.

Using now a proof analogous to that of Lemma 2.5, we can obtain the a priori estimate

||uni||L∞

([0,T ]×Ωε)≤ C,

where C is a constant which also does not depend on n (in fact we will not use the uniformity w.r.t. n of this bound in the sequel).

At this point, we use standard theorems for systems of reaction-diffusion equa-tions in order to get the existence and uniqueness of a smooth solution to system (85) - (86) (for a given n ∈ IN − {0}). We refer to [10], Prop. 3.2 p. 97 and Thm. 3.3 p. 105 for a complete description of a case with a slightly different boundary condition (homogeneous Neumann instead of Neumann) and a different right-hand side (but having the same crucial property, that is leading to an L∞ a priori bound on the components of the unknown).

We now briefly explain how to pass to the limit when n → ∞ in such a way that the limit of un

i satisfies the system (3) - (4). First, we notice that thanks to the

duality estimate (89), each component sequence (un_i)n≥i is bounded in L2([0, T ] ×

Ωε). As a consequence, we can extract a subsequence from (uni)n≥i still denoted by

(un_i)n≥i (the extraction is done diagonally in such a way that it gives a subsequence

which is common for all i) which converges in L2_{([0, T ] × Ω}

ε) weakly towards some

function ui ∈ L2([0, T ] × Ωε). Using then the a priori estimate (consequence of the

duality lemma, the assumptions on the coagulation and fragmentation coefficients, and natural bound on un_i in L1 coming from a direct integration of the equations)

||∂u n i ∂t − ∇x· (di∇xu n i)||L1_{([0,T ]×Ω}_ε₎ ≤ C_i,

where Ci may depend on i but not on n, we see that the convergence in fact holds

in L2_{([0, T ] × Ω}ε) strong. This is sufficient to pass to the limit in system (85) - (86)

and get system (3) - (4).

3.2 Homogenization

Now that existence for a given ǫ is obtained, we provide the proof of the homog-enization part of Theorem 1.1.

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In view of Lemmas 2.2-2.6 the sequences euǫ

i and ]∇xuǫi (i ≥ 1) are bounded

in L2_{([0, T ] × Ω), and by application of Theorem B.1 and Theorem B.3, they} two-scale converge, up to a subsequence, to: [χ(y) ui(t, x)] and [χ(y)(∇xui(t, x) +

∇yu1i(t, x, y))] (i ≥ 1) (cf. [1]). Similarly, in view of Lemma 2.7, it is possible to prove

that the sequence g∂uǫi

∂t

(i ≥ 1) two-scale converges to:

χ(y) ∂ui

∂t (t, x)

(i ≥ 1). We can now find the homogenized equations satisfied by ui(t, x) and u1i(t, x, y).

In the case i = 1, let us multiply the first equation of (3) by the test function

φǫ≡ φ(t, x) + ǫ φ1 t, x,x ǫ ,

where φ ∈ C1([0, T ]×Ω) and φ1 ∈ C1([0, T ]×Ω; C#∞(Y )). Integrating, the divergence

theorem yields Z T 0 Z Ωǫ ∂uǫ₁ ∂t φǫ(t, x, x ǫ) dt dx + d1 Z T 0 Z Ωǫ ∇xuǫ1· ∇φǫdt dx + Z T 0 Z Ωǫ uǫ₁ ∞ X j=1 a1,juǫjφǫdt dx = ǫ d1 Z T 0 Z Γǫ ψ t, x,x ǫ φǫdt dσǫ(x) + Z T 0 Z Ωǫ ∞ X j=1 B1+jβ1+j,1uǫ1+jφǫdt dx. (90)

Passing to the two-scale limit, thanks to Theorem B.2 and Theorem B.5, we get Z T 0 Z Ω Z Y∗ ∂u1 ∂t (t, x) φ(t, x) dt dx dy + d1 Z T 0 Z Ω Z Y∗[∇ xu1(t, x) + ∇yu11(t, x, y)] · [∇xφ(t, x) + ∇yφ1(t, x, y)] dt dx dy + Z T 0 Z Ω Z Y∗ u1(t, x) ∞ X j=1 a1,juj(t, x) φ(t, x) dt dx dy = d1 Z T 0 Z Ω Z Γ ψ(t, x, y) φ(t, x) dt dx dσ(y) + Z T 0 Z Ω Z Y∗ ∞ X j=1 B1+jβ1+j,1u1+j(t, x) φ(t, x) dt dx dy. (91) The passage to the limit in the infinite sums can be performed since thanks to the assumptions on ai,j, Bj and βi,j, and to the duality lemma (using Cauchy-Schwarz

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inequality), Z T 0 Z Ωǫ ∞ X j=K B1+jβ1+j,1uǫ1+jφǫdt dx ≤ Z T 0 Z Ωǫ ∞ X j=K γ1a1,1+j uǫ1+j dt dx ||φǫ||∞ ≤ CTK−ζ.

An integration by parts shows that (91) is a variational formulation associated to the following homogenized system:

−divy[d1(∇xu1(t, x) + ∇yu11(t, x, y))] = 0 in [0, T ] × Ω × Y∗, (92) [∇xu1(t, x) + ∇yu11(t, x, y)] · n = 0 on [0, T ] × Ω × Γ, (93) θ∂u1 ∂t (t, x) − divx d1 Z Y∗(∇ xu1(t, x) + ∇yu11(t, x, y))dy + θ u1(t, x) ∞ X j=1 a1,juj(t, x) = d1 Z Γ ψ(t, x, y) dσ(y) + θ ∞ X j=1 B1+jβ1+j,1u1+j(t, x) in [0, T ] × Ω, (94) Z Y∗(∇ xu1(t, x) + ∇yu11(t, x, y)) dy · n = 0 on [0, T ] × ∂Ω, (95) where θ = Z Y χ(y)dy = |Y ∗ |

is the volume fraction of material. Furthermore, by continuity, we have that

u1(0, x) = U1 in Ω.

Taking advantage of the constancy of the diffusion coefficient d1, Eqs. (92) and (93)

can be reexpressed as follows:

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∇yu11(t, x, y) · n = −∇xu1(t, x) · n on [0, T ] × Ω × Γ. (97)

Then, u1

1(t, x, y) satisfying (96)-(97) can be written as

u1₁(t, x, y) = N X j=1 wj(y) ∂u1 ∂xj (t, x), (98)

where (wj)1≤j≤N is the family of solutions of the cell problem:

               −∇y· [∇ywj + ˆej] = 0 in Y∗, (∇ywj+ ˆej) · n = 0 on Γ, y → wj(y) Y − periodic. (99)

By using the relation (98) in Eqs. (94) and (95), the system (11) can be immediately derived (cf. [1]).

In the case i ≥ 2, let us multiply the first equation of (4) by the test function

φǫ≡ φ(t, x) + ǫ φ1 t, x,x ǫ , where φ ∈ C1_{([0, T ]×Ω) and φ}

1 ∈ C1([0, T ]×Ω; C_#∞(Y )). Integrating, the divergence

theorem yields Z T 0 Z Ωǫ ∂uǫ_i ∂t φǫ(t, x, x ǫ) dt dx + di Z T 0 Z Ωǫ ∇xuǫi · ∇φǫdt dx = − Z T 0 Z Ωǫ uǫ_i ∞ X j=1 ai,juǫjφǫdt dx + 1 2 Z T 0 Z Ωǫ i−1 X j=1 aj,i−juǫjuǫi−jφǫdt dx + Z T 0 Z Ωǫ ∞ X j=1

Bi+jβi+j,iuǫi+jφǫdt dx −

Z T 0 Z Ωǫ Biuǫiφǫdt dx. (100)

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Z T 0 Z Ω Z Y∗ ∂ui ∂t (t, x) φ(t, x) dt dx dy + di Z T 0 Z Ω Z Y∗[∇ xui(t, x) + ∇yu1i(t, x, y)] · [∇xφ(t, x) + ∇yφ1(t, x, y)] dt dx dy = − Z T 0 Z Ω Z Y∗ ui(t, x) ∞ X j=1 ai,juj(t, x) φ(t, x) dt dx dy +1 2 Z T 0 Z Ω Z Y∗ i−1 X j=1 aj,i−juj(t, x) ui−j(t, x) φ(t, x) dt dx dy + Z T 0 Z Ω Z Y∗ ∞ X j=1

Bi+jβi+j,iui+j(t, x) φ(t, x) dt dx dy

− Z T 0 Z Ω Z Y∗ Biui(t, x) φ(t, x) dt dx dy. (101) The passage to the limit in the infinite sums can be performed since thanks to the assumptions on ai,j, Bj and βi,j, and to the duality lemma,

Z T 0 Z Ωǫ ∞ X j=K

Bi+jβi+j,iuǫi+jφǫdt dx

≤ Z T 0 Z Ωǫ ∞ X j=K γiai,i+j uǫi+j dt dx ||φǫ||∞ ≤ CT,iK−ζ.

An integration by parts shows that (101) is a variational formulation associated to the following homogenized system:

−divy[di(∇xui(t, x) + ∇yu1i(t, x, y))] = 0 in [0, T ] × Ω × Y∗, (102) [∇xui(t, x) + ∇yu1i(t, x, y)] · n = 0 on [0, T ] × Ω × Γ, (103) θ∂ui ∂t (t, x) − divx di Z Y∗(∇ xui(t, x) + ∇yu1i(t, x, y))dy = −θ ui(t, x) ∞ X j=1 ai,juj(t, x) + θ 2 i−1 X j=1 aj,i−juj(t, x) ui−j(t, x) + θ ∞ X j=1

Bi+jβi+j,iui+j(t, x) − θ Biui(t, x) in [0, T ] × Ω,

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Z Y∗(∇ xui(t, x) + ∇yu1i(t, x, y)) dy · n = 0 on [0, T ] × ∂Ω, (105) where θ = Z Y χ(y)dy = |Y ∗ | is the volume fraction of material. Moreover, by continuity

ui(0, x) = 0 in Ω.

Taking advantage of the constancy of the diffusion coefficient di, Eqs. (102) and

(103) can be reexpressed as follows

△yu1i(t, x, y) = 0 in [0, T ] × Ω × Y∗, (106)

∇yu1i(t, x, y) · n = −∇xui(t, x) · n on [0, T ] × Ω × Γ. (107)

Then, u1_i(t, x, y) satisfying (106)-(107) can be written as

u1_i(t, x, y) = N X j=1 wj(y) ∂ui ∂xj (t, x) (108)

where (wj)1≤j≤N is the family of solutions of the cell problem (99). By using the

relation (108) in Eqs. (104) and (105), the system (12) can be immediately derived (cf. [1]).

A

Appendix A

Lemma A.1. The following estimate holds: If v ∈ Lip (Ωǫ), then

kvk2L2_(Γ ǫ) ≤ C1 ǫ−1 Z Ωǫ |v|2dx + ǫ Z Ωǫ |∇xv|2dx , (109)

where C1 is a constant which does not depend on ǫ.

The inequality (109) can be easily obtained from the standard trace theorem by means of a scaling argument [2].

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Lemma A.2. Suppose that the domain Ωǫ is such that assumption (1) is satisfied.

Then there exists a family of linear continuous extension operators

Pǫ : W1,p(Ωǫ) → W1,p(Ω)

and a constant C > 0 independent of ǫ such that

Pǫv = v in Ωǫ, and Z Ω|Pǫv| p dx ≤ C Z Ωǫ |v|pdx , (110) Z Ω|∇(P ǫv)|pdx ≤ C Z Ωǫ |∇v|pdx (111) for each v ∈ W1,p(Ωǫ) and for any p ∈ (1, +∞).

For the proof of this Lemma see for instance [5].

As a consequence of the existence of extension operators one can derive the Sobolev inequalities in W1,p(Ωǫ) with a constant independent of ǫ.

Lemma A.3 (Anisotropic Sobolev inequalities in perforated domains). (i) For arbitrary v ∈ H1_{(0, T ; L}2_(Ω

ǫ)) ∩ L2(0, T ; H1(Ωǫ)) and q1 and r1 satisfying

the conditions        1 r1 + N2q1 = N4 , r1∈ [2, ∞], q1∈ [2,_{N −2}2N ] for N > 2, (112)

the following estimate holds:

kvkLr1_{(0,T ;L}q1_(Ω_ǫ₎₎≤ c kvk_Q_ǫ_{(T )}, (113)

where c is a positive constant independent of ǫ, and

kvk2Qǫ(T ):= sup 0≤t≤T Z Ωǫ |v(t)|2dx + Z T 0 dt Z Ωǫ |∇v(t)|2dx; (114)

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(ii) For arbitrary v ∈ H1(0, T ; L2(Ωǫ)) ∩ L2(0, T ; H1(Ωǫ)) and q2 and r2 satisfying the conditions        1 r2 + (N − 1) 2q2 = N4 , r2∈ [2, ∞], q2∈ [2,2(N −1)_{(N −2)}] for N ≥ 3, (115) the following estimate holds:

kvkLr2_{(0,T ;L}q2_(Γ_ǫ₎₎≤ c ǫ− N 2− (1−N) q2 _kvk Qǫ(T ), (116)

where c is a positive constant independent of ǫ and the norm kvkQǫ(T ) is defined as

in (114).

For the proof of this Lemma, see [11].

B

Appendix B

Let us summarize some definitions and results on two-scale convergence [1], [2], [17], [7], [12], [16].

Definition B.1. A sequence of functions vǫ in L2_{([0, T ] × Ω) two-scale converges} to v0∈ L2([0, T ] × Ω × Y ) if lim ǫ→0 Z T 0 Z Ω vǫ(t, x) φ t, x,x ǫ dt dx = Z T 0 Z Ω Z Y v0(t, x, y) φ(t, x, y) dt dx dy, (117) for all φ ∈ C1([0, T ] × Ω; C#∞(Y )).

Theorem B.1. If vǫ is a bounded sequence in L2_{([0, T ] × Ω), then there exists a} function v0(t, x, y) in L2([0, T ] × Ω × Y ) such that, up to a subsequence, vǫ two-scale

converges to v0.

The following theorem is useful in obtaining the limit of the product of two two-scale convergent sequences.

Theorem B.2. Let vǫ _{be a sequence of functions in L}2_{([0, T ] × Ω) which two-scale}

converges to a limit v0 ∈ L2([0, T ] × Ω × Y ). Suppose furthermore that

lim ǫ→0 Z T 0 Z Ω|v ǫ_{(t, x)|}2_{dt dx =}Z T 0 Z Ω Z Y |v0(t, x, y)| 2_{dt dx dy.} ₍₁₁₈₎

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Then, for any sequence wǫ in L2_{([0, T ] × Ω) that two-scale converges to a limit} w0 ∈ L2([0, T ] × Ω × Y ), we have lim ǫ→0 Z T 0 Z Ω vǫ(t, x) wǫ(t, x) φ t, x,x ǫ dt dx = Z T 0 Z Ω Z Y v0(t, x, y) w0(t, x, y) φ(t, x, y) dt dx dy (119) for all φ ∈ C1([0, T ] × Ω; C#∞(Y )).

Remark: Note that, in the setting of this paper, identity (118) can be obtained by standard computations, used in problems with perforated domains. One uses the properties of the extension operators Pǫ stated in Lemma A.2. For instance,

using vǫ_{(t, x) := u}ǫ

1, we see that Pǫvǫ converges strongly in L2 towards v0 := u1.

As a consequence, |Pǫvǫ|2 converges towards v20 strongly in L1, and therefore it also

2-scales converges towards the same quantity. Finally, the properties of Pǫ enable

us to obtain identity (118).

The next theorems yield a characterization of the two-scale limit of the gradients of bounded sequences vǫ. This result is crucial for applications to homogenization problems.

We identify H1(Ω) = W1,2(Ω), where the Sobolev space W1,p(Ω) is defined by W1,p(Ω) = v|v ∈ Lp(Ω), ∂v ∂xi ∈ L p_{(Ω), i = 1, . . . , N} , and we denote by H_#1(Y ) the closure of C_#∞(Y ) for the H1-norm.

Theorem B.3. Let vǫ be a bounded sequence in L2(0, T ; H1(Ω)) that converges weakly to a limit v(t, x) in L2_{(0, T ; H}1_{(Ω)). Then, v}ǫ _{two-scale converges to v(t, x),}

and there exists a function v1(t, x, y) in L2([0, T ] × Ω; H#1(Y )/R) such that, up to a

subsequence, ∇vǫ _{two-scale converges to ∇}

xv(t, x) + ∇yv1(t, x, y).

Theorem B.4. Let vǫ _{and ǫ∇v}ǫ _{be two bounded sequences in L}2_{([0, T ] × Ω). Then,}

there exists a function v1(t, x, y) in L2([0, T ]×Ω; H_#1(Y )/R) such that, up to a

subse-quence, vǫ _{and ǫ∇v}ǫ _{two-scale converge to v}

1(t, x, y) and ∇yv1(t, x, y), respectively.

The main result of two-scale convergence can be generalized to the case of se-quences defined in L2_{([0, T ] × Γ}ǫ).

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Theorem B.5. Let vǫ be a sequence in L2_{([0, T ] × Γ}ǫ) such that ǫ Z T 0 Z Γǫ |vǫ(t, x)|2dt dσǫ(x) ≤ C, (120)

where C is a positive constant, independent of ǫ. There exist a subsequence (still denoted by ǫ) and a two-scale limit v0(t, x, y) ∈ L2([0, T ]×Ω; L2(Γ)) such that vǫ(t, x)

two-scale converges to v0(t, x, y) in the sense that

lim ǫ→0ǫ Z T 0 Z Γǫ vǫ(t, x) φ t, x,x ǫ dt dσǫ(x) = Z T 0 Z Ω Z Γ v0(t, x, y) φ(t, x, y) dt dx dσ(y) (121) for any function φ ∈ C1_{([0, T ] × Ω; C}∞

#(Y )).

The proof of Theorem B.5 is very similar to the usual two-scale convergence theorem [1]. It relies on the following lemma [2]:

Lemma B.1. Let B = C[Ω; C#(Y )] be the space of continuous functions φ(x, y) on

Ω × Y which are Y -periodic in y. Then, B is a separable Banach space which is dense in L2(Ω; L2_{(Γ)), and such that any function φ(x, y) ∈ B satisfies}

ǫ Z Γǫ φ(x, x ǫ) 2 dσǫ(x) ≤ C kφk2B, (122) and lim ǫ→0 ǫ Z Γǫ φ x,x ǫ 2 dσǫ(x) = Z Ω Z Γ|φ(x, y)| 2_{dx dσ(y).} ₍₁₂₃₎

Acknowledgements

S.L. is supported by GNFM of INdAM, Italy.

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