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Derivation and analysis of a fluid-dynamical model in thin and long elastic vessels

  • Received: 01 July 2006 Revised: 01 December 2006
  • Primary: 76Z05, 35L50; Secondary: 74F10, 35Q30.

  • Starting from the three-dimensional Newtonian and incompressible Navier-Stokes equations in a compliant straight vessel, we derive a reduced one-dimensional model by an averaging procedure which takes into consideration the elastic properties of the wall structure. In particular, we neglect terms of the first order with respect to the ratio between the vessel radius and length. Furthermore, we consider that the viscous effects are negligible with respect to the propagative phenomena. The result is a one-dimensional nonlinear hyperbolic system of two equations in one space dimension, which describes the mean longitudinal velocity of the flow and the radial wall displacement. The modelling technique here applied to straight cylindrical vessels may be generalized to account for curvature and torsion. An analysis of well posedness is presented which demonstrates, under reasonable hypotheses, the global in time existence of regular solutions.

    Citation: Debora Amadori, Stefania Ferrari, Luca Formaggia. Derivation and analysis of a fluid-dynamical model in thin and long elastic vessels[J]. Networks and Heterogeneous Media, 2007, 2(1): 99-125. doi: 10.3934/nhm.2007.2.99

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  • Starting from the three-dimensional Newtonian and incompressible Navier-Stokes equations in a compliant straight vessel, we derive a reduced one-dimensional model by an averaging procedure which takes into consideration the elastic properties of the wall structure. In particular, we neglect terms of the first order with respect to the ratio between the vessel radius and length. Furthermore, we consider that the viscous effects are negligible with respect to the propagative phenomena. The result is a one-dimensional nonlinear hyperbolic system of two equations in one space dimension, which describes the mean longitudinal velocity of the flow and the radial wall displacement. The modelling technique here applied to straight cylindrical vessels may be generalized to account for curvature and torsion. An analysis of well posedness is presented which demonstrates, under reasonable hypotheses, the global in time existence of regular solutions.


    We consider a system of reaction-diffusion equations in a domain Ω consisting of two bulk regions Ω+ϵ and Ωϵ, which are separated by a thin layer ΩMϵ with periodic heterogeneous structure. The thickness of the thin layer as well as the period and the size of its heterogeneities are of order ϵ>0. Here, the parameter ϵ is much smaller than the length scale of the domain Ω. The equations in the layer depend on ϵ, the diffusion coefficients having the size ϵγ with γ[1,1]. On the interface S±ϵ between the bulk region Ω±ϵ and the thin layer ΩMϵ, we consider nonlinear transmission conditions formulated in terms of normal fluxes which are given by nonlinear functions of the solutions on both sides of S±ϵ. Our aim is to derive effective models in the limit ϵ0.

    In the limit, the thin layer reduces to an interface Σ between the bulk domains Ω±. The processes in the bulk are again described by a system of nonlinear reaction-diffusion equations, and the main challenges of the paper are related to the derivation of effective interface conditions across Σ. To a certain extent, the effective interface conditions preserve the form of the microscopic ones in the sense that they involve the normal fluxes of the macroscopic solutions at Σ. These fluxes are given by nonlinear functions of the solutions in the respective bulk domain and the homogenized limit of the microscopic solutions in the thin layer. It turns out that the equations satisfied by the latter depend on the values of the parameter γ. More precisely, for γ=1 the homogenized problem in the layer is formulated on the standard periodicity cell, for γ(1,1), we obtain a nonlinear system of ordinary differential equations on Σ, whereas for γ=1 a system of nonlinear reaction-diffusion equations on Σ arises. In all three cases, the homogenized problem in the layer is coupled to the macroscopic solutions in the bulk domains Ω±.

    The rigorous derivation of interface conditions for multi-scale and multi-physics problems is a field that is just at the beginning. More and better analytical multi-scale tools are required to treat the arising nonlinear systems in many applications (e.g. biology, material sciences or geosciences, to mention just few of them). Our paper is one of the first steps in this process. The effective model derived here is a non-standard micro-macro strongly coupled system of nonlinear equations, involving the bulk-regions, the separating interface and the so called cell problem which takes care of the microscopic processes in the thin layer.

    The derivation of effective interface conditions is based on two-scale convergence for thin heterogeneous layers, and a Kolmogorov-type compactness result for Banach valued functions, applied to the unfolded sequence in the layer. Compared to previous contributions (see [14,15] for γ=1 and [10] for γ[1,1)), where continuous transmission conditions for the microscopic solutions and their normal fluxes were considered at the interfaces S±ϵ, we deal here with additional difficulties induced by nonlinear transmission conditions at S±ϵ. More precisely, we have nonlinear terms on the interfaces S±ϵ, and less regularity in time for the solutions. To cope with these problems, additionally to the techniques developed in [14,15,10], we make use of the auxiliary function given by the average with respect to the n-th spatial variable in the layer, and the averaging operator for thin domains, which is the adjoint of the unfolding operator in the layer. The averaged function is used in case γ[1,1) as an approximation for the solution in the layer. Its properties allow the application of classical compactness results (Aubin-Lions-lemma for γ=1, and classical Kolmogorov-criterion for γ(1,1)). However, to apply the latter, we have to introduce equivalent norms on the Sobolev space H1(ΩMϵ), which are well adapted to the thin layer and to different choices of γ. The averaging operator for thin domains is used in case γ=1 to prove that the time derivative of the unfolded sequence in the layer exists in a weak sense, and that it can be controlled by the time derivative of the solution itself. This allows to show that the assumptions of the Kolmogorov-type compactness result for Banach valued functions, see [8], are fulfilled, which gives strong convergence of the unfolded sequence. We emphasize that also in this case, the scaled Sobolev spaces mentioned above are needed.

    The investigation of processes in domains separated by thin layers with periodic microstructure can also be found in elasticity problems, see e.g., [11,13] where the heterogeneous structure is described by periodically varying constitutive properties, or [12], where two domains separated by a thin layer made of periodic vertical beams are considered. Further applications can be encountered in fluid flow through thin filters, built up by an array of obstacles, see e.g., [3]. In [17], a reactive transport model with an additional convective contribution in the thin layer and a nonlinear transmission condition of Dirichlet type at the bulk-layer interface was considered. In the latter, however, a thin homogeneous layer was considered. In [4], long and horizontally arranged inclusions, only connected in one direction, were considered for a linear reaction-diffusion problem, and a concept of two-scale convergence adapted to this special structure was introduced.

    This paper is organized as follows: In Section 2, we introduce the microscopic model, and establish existence and uniqueness of a weak solution. In Section 4, we derive estimates for the microscopic solutions necessary for the derivation of strong compactness results. The averaged function is defined in Section 5 and some basic properties are established. In Section 6 the averaging operator for thin domains is defined and the commuting property of the generalized time derivative and the unfolding operator is proved. Section 7 contains our main results. First, we prove compactness results for the microscopic solutions, especially the strong convergences in the thin layer. These are then used for the derivation of macroscopic models, including the effective interface conditions. In the A, we briefly recapitulate the concepts of two-scale convergence and unfolding operator for thin domains, together with related basic results.

    We consider the domain Ω:=Σ×(H,H)Rn with fixed H>0, n2, and Σ is a bounded and connected Lipschitz domain in Rn1. Further, let ϵ>0 be a sequence with ϵ1N. The set Ω consists of three subdomains given by

    Ω+ϵ:=Σ×(ϵ,H),ΩMϵ:=Σ×(ϵ,ϵ),Ωϵ:=Σ×(H,ϵ),

    see Figure 1. The domains Ω±ϵ and ΩMϵ are separated by an interface S±ϵi.e.,

    S+ϵ:=Σ×{ϵ} and Sϵ:=Σ×{ϵ},
    Figure 1.  The microscopic domain containing the thin layer ΩMϵ with periodic structure for n=2. The heterogeneous structure of the membrane is modeled by the diffusion coefficient DM.

    hence, we have Ω=Ω+ϵΩϵΩMϵS+ϵSϵ. As mentioned above, for ϵ0 the membrane ΩMϵ reduces to an interface Σ×{0}, which we also denote by Σ suppressing the n-th component, and we define

    Ω+:=Σ×(0,H) and Ω:=Σ×(H,0).

    To describe the microscopic structure of ΩMϵ, we introduce the standard cell Z given by

    Z:=Y×(1,1) with Y:=(0,1)n1,

    and denote the upper and lower boundary of Z by

    S+:=Y×{1}=(0,1)n1×{1} and S:=Y×{1}=(0,1)n1×{1}.

    The vector valued functions u±ϵ:(0,T)×Ω±ϵRm and uMϵ:(0,T)×ΩMϵRm are the solutions of

    tu±i,ϵD±iΔu±i,ϵ=f+i(u±ϵ) in (0,T)×Ω±ϵ, (1a)
    D±iu±i,ϵν=h±i(u±ϵ,uMϵ) on (0,T)×S±ϵ, (1b)
    D±iu±i,ϵν=0 on (0,T)×Ω±ϵS±ϵ, (1c)
    u±ϵ(0)=U±0 in Ω±ϵ, (1d)

    for i=1,,m, and

    1ϵtuMi,ϵϵγ(DMi(xϵ)uMi,ϵ)=1ϵgi(xϵ,uMϵ) in (0,T)×ΩMϵ, (1e)
    ϵγDMi(ϵ)uMi,ϵν=hM,±i(ˉxϵ,uMϵ,u±ϵ) on (0,T)×S±ϵ, (1f)
    ϵγDMi(ϵ)uMi,ϵν=0 on (0,T)×ΩMϵS±ϵ, (1g)
    uMϵ(0)=UM0(ˉ,xnϵ) in ΩMϵ, (1h)

    for γ[1,1] and i=1,,m.

    Thin heterogeneous layers occur in many applications, e.g., biology and engineering, and for every specific situation appropriate transmission conditions between the layer domain and the bulk regions are required. In previous contributions (see [14,15] for γ=1 and [10] for γ[1,1)), the authors considered continuous transmission conditions for the microscopic solutions and their normal fluxes at the interfaces S±ϵ. However, in many applications the concentrations are discontinuous at the interfaces between different regions and the normal fluxes across these interfaces are controlled by traces of the concentrations in the neighboring regions. In the present paper, this situation is modelled, with the help of the nonlinear transmission conditions at S±ϵ. Furthermore, as can be seen from the methods used in our paper, we now have the opportunity to mix up the nonlinear Neumann transmission conditions and the continuous transmission conditions on S±ϵ for different species, and thus to extend the range of applications which can be investigated using multiscale techniques, see also Remark 5 (ⅱ). Finally, let us also mention that the scaling of the equations in the layer allows for a wide range of diffusion coefficients, and is motivated by the different roles played by the layers in different applications.

    Assumptions on the data:

    A1) For i=1,,m it holds that D±i>0 and DMiLper(Y,L(1,1)). Further, we set DMi,ϵ(x):=DMi(xϵ) and DMi is strictly positive almost everywhere.

    A2) The function f±=(f±1,,f±m):RmRm is Lipschitz continuous. This ensures the existence of a constant C>0 such that

    |f±i(z)|C(1+|z|) for all zRm.

    A3) The function g=(g1,,gm):¯Y×[1,1]×RmRm is continuous, uniformly Lipschitz continuous with respect to the last variable, and Y-periodic with respect to the first variable. Especially,

    |gi(ˉy,yn,z)|C(1+|z|)for all (ˉy,yn,z)¯Y×[1,1]×Rm.

    We shortly write y=(ˉy,yn) and define gϵ(t,x,z):=g(t,xϵ,z).

    A4) The function h±=(h±1,,h±m):Rm×RmRm is Lipschitz continuous, especially, there exists a constant C>0 such that

    |h±(z±,zM)|C(1+|z±|+|zM|) (z±,zM)Rm×Rm.

    A5) The function hM,±=(hM,±1,,hM,±m):¯Y×Rm×RmRm is continuous, Y-periodic with respect to the first variable, and uniformly Lipschitz continuous with respect to the second and the third variablei.e., for a constant C>0 it holds that

    |hM,±(ˉy,zM,z±)|C(1+|z±|+|zM|) (ˉy,zM,z±)¯Y×Rm×Rm.

    A6) For the initial functions we assume U±0L2(Ω±)m and UM0L2(Σ×(1,1))m.

    In the following, we denote for an arbitrary open set URn the duality pairing ,H1(U),H1(U) by ,U, and the inner product (,)L2(U) by (,)U.

    Definition 2.1. We call uϵ:=(u+ϵ,uMϵ,uϵ) with u±ϵ:(0,T)×Ω±ϵRm and uMϵ:(0,T)×ΩMϵRm a weak solution of the Problem (1), if

    u±ϵL2((0,T),H1(Ω±ϵ))mH1((0,T),H1(Ω±ϵ))m,uMϵL2((0,T),H1(ΩMϵ))mH1((0,T),H1(ΩMϵ))m,

    and for all test-functions ϕ±H1(Ω±ϵ) and ϕMH1(ΩMϵ), and almost every t(0,T) it holds that

    tu±i,ϵ,ϕ±Ω±ϵ+D±i(u±i,ϵ,ϕ±)Ω±ϵ=(f±i(u±ϵ),ϕ±)Ω±ϵ+(h±i(u±ϵ,uMϵ),ϕ±)S±ϵ,1ϵtuMi,ϵ,ϕMΩMϵ+ϵγ(DMi(ϵ)uMi,ϵ,ϕM)ΩMϵ=1ϵ(gi(ϵ,uMϵ),ϕM)ΩMϵ+(hM,+i(ˉxϵ,uMϵ,u+ϵ),ϕM)S+ϵ+(hM,i(ˉxϵ,uMϵ,uϵ),ϕM)Sϵ, (2)

    together with the initial conditions (1d) and (1h).

    First of all, we establish the existence of a unique solution using a fix-point argument. The idea is standard, therefore we only give a short sketch of the proof.

    Proposition 1. For every γ[1,1] there exists a unique weak solution uϵ of the Problem (1).

    Proof. Uniqueness follows by standard energy estimates. For the existence, we use Schäfer's fixed point theorem on the space

    X:=X+×XM×X

    with X:=L2((0,T),Hβ(Ωϵ))m for {+,,M} and β(12,1) fixed, where on X we define the operator F:XX by F(ˉu)=u and u is the unique weak solution of the linearized problem of (1), where we replace the functions in the arguments of the nonlinearities on the right-hand side by the function ˉu. The existence of u is ensured by the Galerkin-method. The continuity and compactness of the operator F is based on the compactness of the embedding

    L2((0,T),H1(Ωϵ))H1((0,T),H1(Ωϵ))L2((0,T),Hβ(Ωϵ))

    and similar estimates as in Lemma 4.2 below.

    Our aim is to derive macroscopic approximations for the microscopic solutions uϵ by passing to the limit ϵ0 in the variational equation (2). Hereby, we use multi-scale techniques adapted to the thin layers with microscopic structure, like the two scale-convergence and the unfolding operator for thin domains. The definitions and a brief overview on results related to this concepts are given in Appendix A.

    In this section, we point out main steps in this process, and indicate the challenging aspects together with our original contributions. Eventually, we present the macroscopic models (which differ for different values of the parameter γ) obtained in the limit ϵ0.

    To pass to the asymptotic limit for ϵ0, we use compactness results based on estimates for uϵ. Firstly, we prove energy estimates, see Lemma 4.2, by using rather standard techniques. More challenging are, however, the estimates for the difference between the solutions and their shifts, which are used to prove strong convergence results for the solution uMϵ in the thin layer and its traces on S±ϵ, by means of Kolmogorov-type theorems. The estimates for shifts with respect to the spatial variable x are given in Lemma 4.3. Similar estimates can be found in previous works of the authors, e.g., [15,10]. To estimate the shifts with respect to the time variable, see Theorem 7.3, we introduce the equivalent norm

    vϵ2Hϵ,γ:=1ϵvϵ2L2(ΩMϵ)+ϵγvϵ2L2(ΩMϵ). (3)

    on H1(ΩMϵ), and estimate the solution in the layer and its time deriative with respect to this norm, see Lemma 4.4.

    Our next step is to prove compactness results for the microscopic solutions, especially in the thin layer. We treat differently the cases γ[1,1) and γ=1. For γ[1,1), we approximate the solution in the layer by its average with respect to the variable xn, i.e., we define for uϵL2((0,T)×ΩMϵ) the function ˉuϵL2((0,T)×Σ) via

    ˉuϵ(t,ˉx):=12ϵϵϵuϵ(t,ˉx,xn)dxn.

    In Section 5, we provide estimates for the averaged function in thin domains and estimate the difference between the averaged function and the microscopic solution uMϵ on the domain ΩMϵ and on the boundaries S±ϵ. Based on these results, we establish the relation between the weak limit of ˉuϵ and the two-scale limit of uMϵ in the layer, see (13) and show a regularity result with respect to time for the two-scale limit u0, see Proposition 3. Eventually, we prove a strong compactness result for γ=1, see Proposition 4. In conclusion, we can say that the averaged function provides an elegant and powerful tool for showing compactness results with respect to strong two-scale convergence in the thin layer. We emphasize however, that a main ingredient in the proof of the compactness result was the fact that for γ[1,1), the two-scale limit of uMϵ is independent of the microscopic variable y.

    In the critical case γ=1, the two-scale limit of uMϵ in general depends on y, and the averaged function in the thin layer is not any longer a good approximation for uMϵ. To show strong compactness results in this case, we use the unfolded sequence in the layer together with a Kolmogorov-typ compactness result for Banach valued functions, see [8]. This requires some regularity of the time derivative of the unfolded sequence TMϵuMϵ. To get rid with this challenging task, in Section 6 we introduce the so called averaging operator for thin domains UMϵ, see also [6] for general domains. It turns out that 1ϵUMϵ is the formal adjoint of the unfolding operator TMϵ. Based on the properties of the averaging operator, we are able to show a regularity result with respect to time of the unfolded sequence in the layer, see Proposition 7, and then to estimate tTMϵuMϵ by the time derivative of the function uMϵ. Here again the equivalent norm (3) is required, see Proposition 8.

    Using the results from the previous sections, in Section 7 we give the proofs for our main results: The convergences of the microscopic solutions, see Theorem 7.1, 7.3, and 7.5, as well as the macroscopic models. The latter consist of equations in the bulk regions Ω± and effective interface laws at Σ, and can be found in Theorem 3.1 (for the case γ=1), Theorem 3.2 (for the case γ(1,1)) and Theorem 3.3 (for the case γ=1), which we formulate in the following:

    Theorem 3.1. Let γ=1 and let uϵ be the solution of Problem (1). Let u±0 and uM0 be the limit functions from Proposition 9 and Theorem 7.1. Then

    u±0L2((0,T),H1(Ω±))mH1((0,T),H1(Ω±))m,uM0L2((0,T),H1(Σ))mH1((0,T),H1(Σ))m,

    and (u±0,uM0) is the unique weak solution of

    tu±i,0D±iΔu±i,0=f±i(u±0)in(0,T)×Ω±,D±iu±i,0ν=0on(0,T)×Ω±Σ,D±iu±i,0ν=h±i(u±0,uM0)on(0,T)×Σ,u±0(t)=U±0inΩ±,

    for i=1,,m, and

    |Z|tuMi,0ˉx(DM,iˉxuMi,0)=Zgi(y,uM0(t,ˉx))dy+α{±}YhM,αi(ˉy,uM0(t,ˉx),uα0(t,ˉx,0))dˉyin(0,T)×Σ,DMiyuMi,0ν=0on(0,T)×Σ,uM0(0,ˉx)=ZUM0(ˉx,yn)dyinΣ,

    where the homogenized diffusion matrix DM,iR(n1)×(n1) is given by

    (DM,i)kl=ZDMi(y)(wi,k+ek)(wi,l+el)dy,

    and the wi,j are the solutions of the cell-problems

    (DMi(wi,j+ej))=0inZ,DMi(wi,j+ej)ν=0onS+S,wi,jisYperiodicwithZwi,jdy=0. (4)

    Theorem 3.2. For γ(1,1), let uϵ be the solution of Problem (1), u±0 and uM0 are the limit functions from Proposition 9 and Theorem 7.3. Then

    u±0L2((0,T),H1(Ω±))mH1((0,T),H1(Ω±))m,uM0H1((0,T),L2(Σ))m,

    and (u±0,uM0) is the unique weak solution of

    tu±i,0D±iΔu±i,0=f±i(u±0)in(0,T)×Ω±,D±iu±i,0ν=0on(0,T)×Ω±Σ,D±iu±i,0ν=h±i(u±0,uM0)on(0,T)×Σ,u±0(0)=U±0inΩ±,

    for i=1,,m, and

    |Z|tuMi,0=Zgi(y,uM0(t,ˉx))dy+α{±}YhM,αi(ˉy,uM0(t,ˉx),uα0(t,ˉx,0))dˉyin(0,T)×Σ,uM0(0,ˉx)=11UM0(ˉx,yn)dyninΣ.

    Theorem 3.3. Let uϵ be the solution of Problem (1) for γ=1, u±0 and uM0 are the limit functions from Proposition 9 and Theorem 7.5. Then

    u±0L2((0,T),H1(Ω±))mH1((0,T),H1(Ω±))m,uM0L2((0,T),L2(Σ,Hper))mH1((0,T),L2(Σ,Hper))m,

    and (u±0,uM0) is the unique weak solution of

    tu±i,0D±iΔu±i,0=f±i(u±0)in(0,T)×Ω±,D±iu±i,0ν=0on(0,T)×Ω±Σ,D±iu±i,0ν=Yh±i(u±0,uM0(t,ˉx,ˉy,±1)dˉyon(0,T)×Σ,u±0(t)=U±0inΩ±,

    for i=1,,m, and (±Z:=Z(S+S))

    tuMi,0y(DMiyuMi,0)=gi(y,uM0)in(0,T)×Σ×Z,DMiyuMi,0ν=hM,±i(ˉy,uM0(t,ˉx,ˉy,±1),u±0(t,ˉx,0))on(0,T)×Σ×S±,DMiyuMi,0ν=0on(0,T)×Σ×±Z,uM0(0,ˉx,y)=UM0(ˉx,yn)inΣ×Z,uM0isYperiodicwithrespecttothelastvariable.

    Our aim is to derive macroscopic approximations for the microscopic solutions uϵ by passing to the limit ϵ0 in the variational equation (2). For this purpose, we use compactness results based on estimates for uϵ. A main issue in the proof of these estimates is to exhibit the precise dependence on the parameters ϵ and δ.

    Throughout this paper, we will frequently use the following trace estimate for thin domains.

    Lemma 4.1. For uϵH1(ΩMϵ) and every θ>0 it holds that

    uϵL2(S±ϵ)C(θ)ϵuϵL2(ΩMϵ)+θϵuϵL2(ΩMϵ),

    with a constant C(θ)>0.

    Proof. There exists an extension ˜uϵH1(Rn1×(ϵ,ϵ)) of uϵ, such that

    ˜uϵL2(Rn1×(ϵ,ϵ))CuϵL2(ΩMϵ),˜uϵH1(Rn1×(ϵ,ϵ))CuϵH1(ΩMϵ),

    with a constant C>0 independent of ϵ. This follows from the fact that we extend the function uϵ only with respect to the first (n1) components and fix the last one. Now, define Rϵ:=R×(ϵ,ϵ), such that RRn1 is a rectangle with integer corner points and ΣR. By a decomposition argument for Rϵ, we obtain for arbitrary θ>0

    uϵL2(S±ϵ)˜uϵL2(R×{±ϵ})C(θ)ϵ˜uϵL2(Rϵ)+θϵ˜uϵL2(Rϵ)C(θ)ϵuϵL2(ΩMϵ)+θCϵuϵL2(ΩMϵ).

    In a first step, we obtain the following estimates:

    Lemma 4.2. The solution uϵ of Problem (1) fulfills the following a priori estimates for a constant C>0 independent of ϵ

    u±i,ϵL((0,T),L2(Ω±ϵ))+u±i,ϵL2((0,T),L2(Ω±ϵ))C, (5a)
    1ϵuMi,ϵL((0,T),L2(ΩMϵ))+ϵγ2uMi,ϵL2((0,T),L2(ΩMϵ))C, (5b)
    1ϵtuMi,ϵL2((0,T),H1(ΩMϵ))+tu±i,ϵL2((0,T),H1(Ω±ϵ))C, (5c)

    for i=1,,m.

    Proof. Test the variational equation (2) for u±i,ϵ with u±i,ϵ and use the growth conditions for f±i and h±i from our assumptions to obtain almost everywhere in (0,T)

    tu±i,ϵ,u±i,ϵΩ±ϵ+D±iu±i,ϵ2L2(Ω±ϵ)=(f±i(u±ϵ),u±i,ϵ)Ω±ϵ+(h±i(u±ϵ,uMϵ),u±i,ϵ)S±ϵC(1+u±i,ϵ2L2(Ω±ϵ)+u±i,ϵ2L2(S±ϵ)+uMϵ2L2(S±ϵ))C(1+u±ϵ2L2(Ω±ϵ)+1ϵuMϵ2L2(ΩMϵ))+δϵuMi,ϵ2L2(ΩMϵ)+δu±ϵ2L2(Ω±ϵ)),

    for an arbitrary δ>0, where we have used the scaled trace estimate from Lemma 4.1. Now, testing the equation (2) for uMi,ϵ with uMi,ϵ and using similar arguments as above we get

    1ϵtuMi,ϵ,uMi,ϵΩMϵ+ϵγ(DMi(ϵ)uMi,ϵ,uMi,ϵ)ΩMϵ=1ϵ(gi(ϵ,uMϵ),uMi,ϵ)ΩMϵ+(hM,+i(ˉxϵ,uMϵ,u+ϵ),uMi,ϵ)S+ϵ+(hM,i(ˉxϵ,uMϵ,uϵ),uMi,ϵ)SϵC(1+1ϵuMϵ2L2(ΩMϵ)+uMϵ2L2(S+ϵSϵ)+u+ϵ2L2(S+ϵ)+uϵ2L2(Sϵ))C(1+1ϵuMϵ2L2(ΩMϵ)+u+ϵ2L2(Ω+ϵ)+uϵ2L2(Ωϵ))+δ(ϵuMϵ2L2(ΩMϵ)+u+ϵ2L2(Ω+ϵ)+uϵ2L2(Ωϵ))

    for an arbitrary δ>0. Using the identity tu±i,ϵ,u±i,ϵΩ±ϵ=12ddtu±i,ϵ2L2(Ω±ϵ) and a similar result for the function uMi,ϵ, we obtain from the both inequalities above by summing over i=1,,m, choosing δ small enough (so the terms including δ can be absorbed by the left-hand side), the positivity of DMi, and ϵϵγ for small ϵ, the inequality

    ddt(1ϵuMϵ(t)2L2(ΩMϵ)+u+ϵ(t)2L2(Ω+ϵ)+uϵ(t)2L2(Ωϵ))+ϵγuMϵ(t)2L2(ΩMϵ)+u+ϵ(t)2L2(Ω+ϵ)+uϵ(t)2L2(Ωϵ)C(1+1ϵuMϵ(t)2L2(ΩMϵ)+u+ϵ(t)2L2(Ω+ϵ)+uϵ(t)2L2(Ωϵ)).

    Integrating with respect to time from 0 to t(0,T) gives us

    1ϵuMϵ(t)2L2(ΩMϵ)+u+ϵ(t)2L2(Ω+ϵ)+uϵ(t)2L2(Ωϵ)+ϵγuMϵ2L2((0,t)×ΩMϵ)+u+ϵ2L2((0,t)×Ω+ϵ)+uϵ2L2((0,t)×Ωϵ)C(1+1ϵuMϵ2L2((0,t)×ΩMϵ)+u+ϵ2L2((0,t)×Ω+ϵ)+uϵ2L2((0,t)×Ωϵ))+1ϵUM0(ˉx,xnϵ)2L2(ΩMϵ)+U+02L2(Ω+ϵ)+U02L2(Ωϵ).

    The assumptions for the initial conditions and Gronwall's inequality imply

    1ϵuMϵ2L((0,T),L2(ΩMϵ))+u+ϵ2L((0,T),L2(Ω+ϵ))+uϵ2L((0,T),L2(Ωϵ))C,

    and together with the inequality above, we obtain the estimates for the gradients in (5a) and (5b). It remains to prove the estimates for the time derivative. Therefore we test the variational equation (2) for u±i,ϵ with v±H1(Ω±ϵ), such that v±H1(Ω±ϵ)1 and obtain with similar arguments as above

    tu±i,ϵ,v±Ω±ϵ=D±i(u±i,ϵ,v±)Ω±ϵ+(f±i(u±ϵ),v±)Ω±ϵ+(h±i(u±ϵ,uMϵ),v±)S±ϵC(u±i,ϵL2(Ω±ϵ)v±L2(Ω±ϵ)+v±L2(Ω±ϵ)+u±ϵL2(Ω±ϵ)v±L2(Ω±ϵ)+v±L2(S±ϵ)+u±ϵL2(S±ϵ)v±L2(S±ϵ)+uMϵL2(S±ϵ)v±L2(S±ϵ)).

    Using again the trace estimate from Lemma 4.1, the boundedness of v±, and the definition of the operator norm in H1(Ω±ϵ), we get for almost every t(0,T)

    tu±i,ϵ(t)H1(Ω±ϵ)C(1+u±ϵH1(Ω±ϵ)+1ϵuMϵL2(ΩMϵ)+ϵuMϵL2(ΩMϵ)).

    Integration with respect to time and the inequalities (5a) and (5b) imply the second inequality in (5c). For the first inequality, we choose vϵH1(ΩMϵ) with vϵH1(ΩMϵ)1 as a test function for the variational equation in the membrane and obtain

    1ϵtuMi,ϵ,vϵΩMϵ=ϵγ(DMi(ϵ)uMi,ϵ,vϵ)ΩMϵ+1ϵ(gi(ϵ,uMϵ),vϵ)ΩMϵ+(hM,+i(ˉxϵ,uMϵ,u+ϵ),vϵ)S+ϵ+(hM,i(ˉxϵ,uMϵ,uϵ),vϵ)SϵC(ϵγuMi,ϵL2(ΩMϵ)vϵL2(ΩMϵ)+1ϵvϵL2(ΩMϵ)+1ϵuMϵL2(ΩMϵ)vϵL2(ΩMϵ)+vϵL2(S+ϵ)+uMϵL2(S+ϵ)vϵL2(S+ϵ)+u+ϵL2(S+ϵ)vϵL2(S+ϵ)+vϵL2(Sϵ)+uMϵL2(Sϵ)vϵL2(Sϵ)+uϵL2(Sϵ)vϵL2(Sϵ)). (6)

    Then, using the trace inequality and the boundedness of vϵ, we get

    tuMi,ϵ2H1(ΩMϵ)C(ϵ+ϵ2+2γuMi,ϵ2L2(ΩMϵ)+uMϵ2L2(ΩMϵ)+ϵu+ϵ2L2(Ω+ϵ)+ϵuϵ2L2(Ωϵ)).

    Integration with respect to time and the a priori estimates (5a) and (5b) give

    tuMi,ϵ2L2((0,T),H1(ΩMϵ))C(ϵ+ϵ2+2γuMi,ϵ2L2((0,T)×ΩMϵ))C(ϵ+ϵ2+γ)Cϵ.

    This gives us the last inequality and the proof is complete.

    The above estimates are not sufficient for the derivation of appropriate strong convergence results for the solution uMϵ in the thin layer and its traces on S±ϵ. Such results are, however, needed to pass to the limit ϵ0 in the nonlinear terms in the variational equation (2). We will show strong convergence by means of Kolmogorov-type theorems. These rely on the estimates for the difference between the solutions and their shifts given in the next lemma.

    Since we are operating in bounded domains, we have to make sure that the shifts are well-defined. For an arbitrary domain URd, we define for h>0 the set

    Uh:={xU:dist(U,x)>h}. (7)

    Further, we write

    ΩMϵh:=Σh×(ϵ,ϵ),Ω+ϵh:=Σh×(ϵ,H),Ωϵh:=Σh×(H,ϵ). (8)

    In the same way as in (21) (see Appendix A), we define the sets Kϵ,h, ˆΣϵ,h, Λϵ,h, ˆΩMϵ,h, and ΛMϵ,h, by replacing Σ with Σh. For an arbitrary function vL2(Ω) we define for lZn1 small and almost every xΣh×(H,H)

    δlv(x):=v(x+ϵ(l,0))v(x). (9)

    In the following, we suppress the index l on the left-hand side for an easier notation, but we should keep in mind the dependence on this parameter. The following estimate for the shifts δuMϵ hold:

    Lemma 4.3. Let uϵ be the solution of the Problem (1) with γ[1,1]. Then for every 0<h1, lZn1 with ϵ|l|<h there exists a constant C=C(h), such that for almost every t(0,T) the following estimate is valid:

    1ϵδuMϵ(t)2L2(ΩMϵth)+ϵγδuMi,ϵ2L2((0,T)×ΩMϵth)C(δu+ϵ2L2(0,T)×Ω+ϵh)+δuϵ2L2((0,T)×Ωϵh)+ϵγ+1+1ϵδuMϵ(0)2L2(ΩMϵh)+δu+ϵ(0)2L2(Ω+ϵh)+δuϵ(0)2L2(Ωϵh)).

    Proof. Let ηC0(Σh) be a cut-off function with 0η1 and η=1 in Σ2h. For all ϕH1(ΩMϵ) and almost everywhere in (0,T), we have the following variational equality for δuMi,ϵ:

    1ϵtδuMi,ϵ,η2ϕΩMϵ+ϵγΩMϵDMi(xϵ)δuMi,ϵ(η2ϕ)dx=1ϵΩMϵδgi(xϵ,uMϵ)η2ϕdx+α±SαϵδhM,αi(ˉxϵ,uMϵ,uαϵ)η2ϕdσ,

    with

    δgi(xϵ,uMϵ):=gi(xϵ,uMϵl)gi(xϵ,uMϵ),δhM,±i(ˉxϵ,uMϵ,u±ϵ):=hM,±i(ˉxϵ,uMϵl,u±ϵl)hM,±i(ˉxϵ,uMϵ,u±ϵ),

    where uϵ,l(t,x):=uϵ(t,x+ϵ(l,0)) for {+,,M}. Here, we extended uMϵ by zero outside ΩMϵ, but this has no influence due to the cut-off function η. Now, choose ϕ=δuMi,ϵ as a test function, to obtain for a constant c0>0

    12ϵddtηδuMi,ϵ2L2(ΩMϵ)+c0ϵγηδuMi,ϵ2L2(ΩMϵ)1ϵΩMϵδgi(xϵ,uMϵ)η2δuMi,ϵdx2ϵγΩMϵηδuMi,ϵDMi(xϵ)δuMi,ϵηdx+α±SαϵδhM,αi(ˉxϵ,uMϵ,uαϵ)η2δuMi,ϵdσ=:I1ϵ+I2ϵ+I+ϵ+Iϵ.

    Now, we integrate with respect to time and estimate the terms on the right-hand side. From the Lipschitz continuity of g we obtain for the first term

    t0I1ϵdtCϵηδuMϵ2L2((0,t)×ΩMϵ).

    For the second term, our a priori estimates imply

    t0I2ϵdtCϵγηδuMi,ϵL2((0,t)×ΩMϵ)δuMi,ϵL2((0,t)×ΩMϵh)C(1ϵηδuMi,ϵ2L2((0,t)×ΩMϵ)+ϵγ+1).

    Using the Lipschitz continuity of hM,±i, the scaled trace estimates for thin domains from Lemma 4.1, and the a priori estimates, we obtain for arbitrary θ>0

    t0I±ϵdtCt0S±ϵ|ηδu+ϵm|2+|ηδuMϵ|2dσdtC(ηδu+ϵm2L2((0,t)×Ω±ϵ)+1ϵηδuMϵ2L2((0,t)×ΩMϵ)+δu+ϵm2L2((0,t)×Ω±ϵ,h)+ϵ2)+θ(ηδu+ϵm2L2((0,t)×Ω±ϵ)+ϵηδuMϵ2L2((0,t)×ΩMϵ)).

    Altogether, we obtain for arbitrary θ>0 and almost every t(0,T) the estimate

    1ϵηδuMi,ϵ(t)2L2(ΩMϵ)1ϵηδuMi,ϵ(0)2L2(ΩMϵ)+ϵγηδuMi,ϵ2L2((0,t)×ΩMϵ)C(α±ηδu+ϵm2L2((0,t)×Ω±ϵ)+1ϵηδuMϵ2L2((0,t)×ΩMϵ)+δu+ϵm2L2((0,t)×Ω±ϵ,h)+ϵγ+1)+θ(α±ηδu+ϵm2L2((0,t)×Ω±ϵ)+ϵηδuMϵ2L2((0,t)×ΩMϵ))=:Δ.

    In a similar way, we get

    ηδu±i,ϵ(t)2L2(Ω±ϵ)ηδu±i,ϵ(0)2L2(Ω±ϵ)+ηδu±i,ϵ2L2((0,t)×Ω±ϵ)Δ.

    Adding up all these inequalities, choosing θ small enough, such that the terms on the right-hand side containing θ can be absorbed by the left-hand side, and using Gronwall's inequality, we get the desired result.

    Finally, we give further estimates for the solution in the thin layer and its time derivative with respect to equivalent norms, which are introduced on H1(ΩMϵ). These norms are well adapted to the thin layer structure and the special choice of γ, and the estimates in these norms are used especially to to control the time derivative of the unfolded sequence, see Proposition 8, and to estimate the difference of solutions and their shifts with respect to time, see Theorem 7.3. Thus, let us define

    Hϵ,γ:={vϵL2(ΩMϵ):vϵL2(ΩMϵ)n},

    together with the inner product

    (vϵ,wϵ)Hϵ,γ:=1ϵ(vϵ,wϵ)ΩMϵ+ϵγ(vϵ,wϵ)ΩMϵ,

    i.e., the norm vϵ2Hϵ,γ=1ϵvϵ2L2(ΩMϵ)+ϵγvϵ2L2(ΩMϵ). Of course, the norm Hϵ,γ is equivalent to the usual norm H1(ΩMϵ) with equivalent-constants depending on ϵ.

    Remark 1. We consider the Gelfand-triple Hϵ,γL2(ΩMϵ)Hϵ,γ under the natural embedding of Hϵ,γ in L2(ΩMϵ). Then for the microscopic solution in the layer, we have uMϵH1((0,T),Hϵ,γ) with tuMϵ,ϕϵHϵ,γ,Hϵ,γ=tuMϵ,ϕϵΩMϵ for all ϕϵHϵ,γ.

    For the solution uMϵ as a function in L2((0,T),Hϵ,γ)H1((0,T),Hϵ,γ), we obtain the following a priori estimates which are independent of γ.

    Lemma 4.4. Let uϵ be the solution of Problem (1) for γ[1,1]. Then, it holds that

    uMϵL2((0,T),Hϵ,γ)C, (10a)
    tuMi,ϵL2((0,T),Hϵ,γ)Cϵ. (10b)

    Proof. The first inequality follows directly from Lemma 4.2. For the second inequality, we choose vϵHϵ,γ with vϵHϵ,γ=1i.e., we have

    vϵL2(ΩMϵ)ϵ,vϵL2(ΩMϵ)ϵγ2,vϵL2(S±ϵ)C.

    Inequality (6) from the proof of Lemma 4.2 is still valid for this vϵ. Using the inequalities above, we obtain

    |tuMi,ϵ,vϵHϵ,γ,Hϵ,γ|C(ϵ1+γ2uMϵL2(ΩMϵ)+ϵuMϵL2(ΩMϵ)+ϵu±ϵH1(Ω±ϵ)+ϵ).

    Squaring, integration with respect to time, and Lemma 4.2 give us the desired result.

    The aim of this section is to provide general results for the averaged function in thin domains, obtained by taking the average over the n-th component. This function will be used as an auxiliary function to obtain appropriate strong convergence results for the solution uMϵ in the case γ[1,1). We define for uϵL2((0,T)×ΩMϵ) the function ˉuϵL2((0,T)×Σ) via

    ˉuϵ(t,ˉx):=12ϵϵϵuϵ(t,ˉx,xn)dxn.

    Here, we consider a sequence uϵL2((0,T),H1(ΩMϵ))H1((0,T),H1(ΩMϵ)), such that

    1ϵuϵL2((0,T)×ΩMϵ)+ϵγ2uϵL2((0,T)×ΩMϵ)+1ϵtuϵL2((0,T),H1(ΩMϵ))C. (11)

    The following estimates hold for the averaged sequence ˉuϵ and its derivatives.

    Lemma 5.1. It holds that ˉuϵL2((0,T),H1(Σ))H1((0,T),H1(Σ)), such that

    ˉuϵL2((0,T)×Σ)12ϵuϵL2((0,T)×ΩMϵ)C, (12a)
    ˉxˉuϵL2((0,T)×Σ)12ϵˉxuϵL2((0,T)×ΩMϵ)Cϵγ12, (12b)
    tˉuϵL2((0,T),H1(Σ))12ϵtuϵL2((0,T),H1(ΩMϵ))C. (12c)

    Further, we have

    1ϵuϵˉuϵL2(ΩMϵ)2ϵnuϵL2((0,T)×ΩMϵ)Cϵ1γ2. (12d)

    Proof. It is clear that ˉuϵL2((0,T),H1(Σ)), so we have to check inequalities (12a) and (12b). We have

    ˉuϵ2L2((0,T)×Σ)=T0Σ|12ϵϵϵuϵ(t,ˉx,xn)dxn|2dˉxdt=1(2ϵ)2T0Σ|ϵϵuϵ(t,ˉx,xn)dxn|2dˉxdt12ϵuϵ2L2((0,T)×ΩMϵ)(11)C.

    In a similar way, we obtain

    ˉxˉuϵ2L2(Σ)=1(2ϵ)2T0Σ|ϵϵˉxuϵ(t,ˉx,xn)dxn|2dˉxdt12ϵˉxuϵ2L2((0,T)×ΩMϵ)(11)Cϵγ1.

    Now, we consider the time derivative of ˉuϵ. First of all, we have for all ψD(0,T), and ϕH1(Σ) with constant extension in xn-direction:

    T0Σˉuϵ(t,ˉx)ϕ(ˉx)ψ(t)dˉxdt=12ϵT0ΩMϵuϵ(t,x)ϕ(ˉx)ψ(t)dxdt=12ϵT0uϵ(t),ϕΩMϵψ(t)dt,

    i.e., we have tˉuϵL2((0,T),H1(Σ)) with

    tˉuϵ,ϕΣ=12ϵtuϵ,ϕΩMϵϕH1(Σ).

    Additionally, since 2ϵϕH1(Σ)=ϕH1(ΩMϵ), we immediately obtain the following estimates almost everywhere in (0,T)

    |tˉuϵ,ϕΣ|12ϵtuϵH1(ΩMϵ)ϕH1(ΩMϵ)12ϵtuϵH1(ΩMϵ)ϕH1(Σ).

    Squaring, integration with respect to time, and (11) gives inequality (12c).

    It remains to prove estimate (12d). We obtain with the fundamental theorem of calculus

    uϵˉuϵ2L2((0,T)×ΩMϵ)=1(2ϵ)2T0ΩMϵ|ϵϵuϵ(t,ˉx,xn)uϵ(t,ˉx,˜xn)d˜xn|2dxdtT0ΩMϵ(ϵϵ|nuϵ(t,ˉx,s)|ds)2dxdt(2ϵ)2nuϵ2L2((0,T)×ΩMϵ)(11)Cϵ2γ.

    In the following proposition, we compare the L2-norm on Σ of the traces of uϵ on S±ϵ and the averaged function ˉuϵ.

    Lemma 5.2. It holds that

    uϵ|S±ϵˉuϵL2(Σ)Cϵ1γ2.

    Proof. We only consider the trace on S+ϵ. Again from the fundamental theorem of calculus, we obtain

    uϵ|S+ϵˉuϵ2L2((0,T)×Σ)=1(2ϵ)2T0Σ|ϵϵuϵ(ˉx,ϵ)uϵ(ˉx,xn)dxn|2dˉxdtCϵnuϵ2L2((0,T)×Σ)(11)Cϵ1γ.

    Remark 2. From Lemma 5.1, we immediately obtain the existence of a function ˉuoL2((0,T)×Σ)H1((0,T),H1(Σ)), such that up to a subsequence it holds that

    ˉuϵˉuo weakly in L2((0,T)×Σ),tˉuϵtˉuo weakly in L2((0,T),H1(Σ)).

    In Section 7, we will use the averaged function to prove convergence results for the solution uMϵ in the layer. For this purpose, the following two propositions will be helpful.

    Let u0L2((0,T)×Σ×Z) be the two-scale limit of uϵ, which exists (up to a subsequence) due to (11) and Proposition 10.

    Proposition 2. The following relation holds between the weak limit ˉuo of ˉuϵ and the two-scale limit u0 of uϵ: For almost every (t,ˉx)(0,T)×Σ, we have

    ˉuo(t,ˉx)=1|Z|Zu0(t,ˉx,y)dy. (13)

    Proof. For all ϕL2(Σ) and ψD(0,T), we have

    T0ΣZu0(t,ˉx,y)ϕ(ˉx)ψ(t)dydˉxdt=limϵ01ϵT0ΩMϵuϵ(t,x)ϕ(ˉx)ψ(t)dxdt=limϵ0|Z|T0Σˉuϵ(t,ˉx)ϕ(ˉx)ψ(t)dˉxdt=|Z|T0Σˉuo(t,ˉx)ϕ(ˉx)ψ(t)dˉxdt.

    From ˉuoH1((0,T),H1(Σ)), we get the differentiability respect to time of the mean of u0 over Z:

    Proposition 3. We have 1|Z|Zu0(t,ˉx,y)dyH1((0,T),H1(Σ)) with

    t(1|Z|Zu0(t,ˉx,y)dy),ϕΣ=tˉuo(t),ϕΣϕH1(Σ),a.e.t(0,T).

    Proof. This follows easily from partial integration with respect to time. In fact, for all ϕH1(Σ), ψD(0,T), we get

    T0tˉuo,ϕΣψ(t)dt=limϵ0T0tˉuϵ(t),ϕΣψ(t)dt=limϵ012ϵT0ΩMϵuϵ(t,x)ϕ(ˉx)ψ(t)dxdt=1|Z|T0ΣZu0(t,ˉx,y)ϕ(ˉx)ψ(t)dydˉxdt.

    A straightforward consequence of Proposition 2 and Proposition 3 is given in the following

    Corollary 1. If the two-scale limit u0 does not depend on y, we have

    ˉuo=u0,tˉuo=tu0.

    For γ=1, we obtain the following strong compactness result:

    Proposition 4. Let uϵL2((0,T),H1(ΩMϵ))H1((0,T),H1(ΩMϵ)), such that

    uϵL2((0,T)×ΩMϵ)+uϵL2((0,T)×ΩMϵ)+tuϵL2((0,T),H1(ΩMϵ))Cϵ.

    Then, there exist u0L2((0,T),H1(Σ))H1((0,T),H1(Σ)) and u1L2((0,T)×Σ,Hpero), such that up to a subsequence it holds that

    uϵu0stronglyinthetwoscalesenseuϵˉxu0+yu1inthetwoscalesense,uϵ|S±ϵu0inL2((0,T)×Σ),tˉuϵtu0weaklyinL2((0,T),H1(Σ)).

    Proof. Proposition 10 implies that, up to a subsequence, uϵ converges in two-scale sense to a limit u0, which is independent of the microscopic variable y. On the other hand, due to Lemma 5.1, the sequence ˉuϵ is bounded in L2((0,T),H1(Σ))H1((0,T),H1(Σ)). The Aubin-Lions Lemma implies the strong convergence of ˉuϵ and the limit coincides with the two-scale limit u0, see Corollary 1. Using now estimate (12d), it follows that uϵ converges even strongly in the two-scale sense to the limit u0. The convergence of uϵ follows from Proposition 10(ⅲ). The strong convergence of the traces is a direct consequence of Lemma 5.2, and the weak convergence of tˉuϵ is obtained from the boundedness of ˉuϵ in H1((0,T),H1(Σ)).

    Lemma A.2 shows that the unfolded sequence TMϵuϵ inherits the regularity with respect to the microscopic variable yZ from the function uϵ. Now, we want to analyze if this is also true for the regularity with respect to the time variable. If we have uϵH1((0,T),L2(ΩMϵ)), then an easy integration by substitution shows TMϵuϵH1((0,T),L2(Σ×Z)) and we have almost everywhere in (0,T)×Σ×Z the identity tTMϵuϵ(t,ˉx,y)=TMϵ(tuϵ)(t,ˉx,y). In our case, however, the microscopic solution uMϵH1((0,T),H1(ΩMϵ)) and it is not clear, whether the time derivative of the unfolded sequence TMϵuMϵ exists in a weak sense, since a pointwise definition with respect to the spatial variable is not possible. In this section, we show that also in this case a time derivative exists in some weaker sense and it can be controlled by the time derivative of the function uMϵ itself. Therefore, we introduce the so called averaging operator for thin domains UMϵ, see also [6] for general domains, and show that under a suitable restriction of the domain of definition, the averaging operator UMϵ preserves the spatial regularity.

    For ˆΩMϵ and ΛMϵ given in (21), see Appendix, and {x}:=x[x], we define

    UMϵ:L2((0,T)×Σ×Z)L2((0,T)×ΩMϵ),UMϵ(ϕ)(t,x)={Yϕ(t,ϵ(ˉz+[ˉxϵ]),({ˉxϵ},xnϵ))dˉz for xˆΩMϵ,0 for xΛMϵ.

    The following Lemma shows that 1ϵUMϵ is the formal adjoint of the unfolding operator TMϵ:

    Proposition 5. Let uϵL2((0,T)×ΩMϵ) and ϕL2((0,T)×Σ×Z), then we have

    T0ΣZTMϵuϵ(t,ˉx,y)ϕ(t,ˉx,y)dydˉxdt=1ϵT0ΩMϵuϵ(t,x)UMϵ(ϕ)(t,x)dxdt.

    Further, we have the identity UMϵ(TMϵuϵ)=uϵ for all uϵL2((0,T)×ΩMϵ).

    Proof. The proof uses the same arguments as in [6,Proposition 4.3] and is skipped here.

    As a corollary we immediately obtain the following inequality:

    Corollary 2. For ϕL2((0,T)×Σ×Z) we have

    UMϵ(ϕ)L2((0,T)×ΩMϵ)ϵϕL2((0,T)×Σ×Z).

    From the definition of the operator UMϵ, we see that for a function ϕL2((0,T)×Σ,H1(Z)), the function UMϵ(ϕ) is not an element of the space L2((0,T),H1(ΩMϵ)). Already for n=2 it is easy to find a counterexample. If we want more regularity of UMϵ(ϕ) with respect to the spatial variable, we have to restrict the domain of definition. We define the space

    H0:=¯C0(Y×[1,1])H1(Z), (14)

    together with the usual H1(Z)-normi.e., H0 is the space of functions from H1(Z) with trace equal to 0 on the lateral boundary Z(S+S).

    Proposition 6. It holds UMϵ:L2((0,T)×Σ,H0)L2((0,T),H1(ΩMϵ)), and for ϕL2((0,T)×Σ,H0), we have almost everywhere in (0,T)×ΩMϵ

    ϵUMϵ(ϕ)(t,x)=UMϵ(yϕ)(t,x).

    Proof. Let ϕL2((0,T)×Σ,H0) and uϵD((0,T)×ΩMϵ). Then, with Proposition 5 and integration by parts, we obtain for i=1,,n

    T0ΩMϵUMϵ(ϕ)(t,x)xiuϵ(t,x)dxdt=T0ΣZϕ(t,ˉx,y)yiTMϵuϵ(t,ˉx,y)dydˉxdt=T0ΣZyiϕ(t,ˉx,y)TMϵuϵ(t,ˉx,y)dydˉxdt+T0ΣZϕ(t,ˉx,y)TMϵuϵ(t,ˉx,y)νidσydˉxdt.

    The last term is equal to 0, since ϕ vanishes on the lateral boundary of Z and since uϵ has compact support in ΩMϵi.e., TMϵuϵ=0 on S±. Hence, we obtain using again Proposition 5

    T0ΩMϵUMϵ(ϕ)(t,x)xiuϵ(t,x)dxdt=1ϵT0ΩMϵUMϵ(yiϕ)(t,x)uϵ(t,x)dxdt,

    which gives us the desired result.

    We are now able to state our regularity result with respect to time for the unfolding operator.

    Proposition 7. Let uϵL2((0,T)×ΩMϵ)H1((0,T),H1(ΩMϵ)). Then we have

    TMϵuϵL2((0,T),L2(Σ×Z))H1((0,T),L2(Σ,H0)),

    and it holds that

    tTMϵuϵ(t),ϕL2(Σ,H0),L2(Σ,H0)=1ϵtuϵ(t),UMϵ(ϕ)ΩMϵ

    for almost every t(0,T) and all ϕL2(Σ,H0).

    Proof. Let ϕL2(Σ,H0) and ψD(0,T). Due to Proposition 6, we have UMϵ(ϕ)H1(ΩMϵ). We obtain

    T0ΣZTMϵuϵ(t,ˉx,y)ϕ(ˉx,y)ψ(t)dydˉxdt=1ϵT0ΩMϵuϵ(t,x)UMϵ(ϕ)(x)ψ(t)dxdt=1ϵT0tuϵ(t),UMϵ(ϕ)ΩMϵψ(t)dt,

    what gives us the claim.

    Our aim now is to estimate the norm of tTMϵuϵ by a suitable norm of tuϵ. Due to the properties of UMϵ from Corollary 2 and Proposition 6, we can see that it is appropriate to work with the scaled norm on Hϵ,1. For this space, the averaging operator UMϵ is a linear operator between the space L2((0,T)×Σ,H0) and L2((0,T),Hϵ,1), such that its operator norm is independent of ϵ:

    Lemma 6.1. Let ϕL2((0,T)×Σ,H0), then it holds

    UMϵ(ϕ)L2((0,T),Hϵ,1)ϕL2((0,T)×Σ,H0).

    Proof. The claim follows immediately from Corollary 2 and Proposition 6.

    As an easy consequence, we obtain the following estimate for the time derivative tTMϵuϵ.

    Proposition 8. Let uϵL2((0,T)×ΩMϵ)H1((0,T),H1(ΩMϵ)), then we have

    tTMϵuϵL2((0,T),L2(Σ,H0))1ϵtuϵL2((0,T),Hϵ,1).

    Proof. Due to Lemma 6.1, we have UMϵ(ϕ)Hϵ,11 for all ϕL2(Σ,H0) with ϕL2(Σ,H0)=1. Hence, for almost every t(0,T) we obtain for all ϕL2(Σ,H0) with ϕL2(Σ,H0)=1

    tTMϵuϵ(t),ϕL2(Σ,H0),L2(Σ,H0)=1ϵtuϵ(t),UMϵ(ϕ)Hϵ,1,Hϵ,11ϵtuϵ(t)Hϵ,1UMϵ(ϕ)Hϵ,11ϵtuϵ(t)Hϵ,1,

    i.e., tTMϵuϵ(t)L2(Σ,H0)1ϵtuϵ(t)Hϵ,1. Squaring and intgration with respect to time gives us the desired result.

    In this section, we derive convergence results for the sequences u±ϵ and uMϵ, which we then use for the derivation of the macroscopic problems. Compared with the results in [10,15], new challenges appear concerning the convergence of the solutions in the thin layer. Firstly, the time derivative of the solutions is no more bounded in the L2-norm, but is only a functional pointwise in time. This causes difficulties especially for the derivation of strong compactness results, which are needed for passing to the limit in the nonlinear terms. Concerning the nonlinear terms, other than in [10,15], here, we have nonlinear terms at the interfaces S±ϵ, which require the strong two-scale convergence of the traces uMϵ|S±ϵ.

    For the derivation of the convergence results for uMϵ in the layer, we will use different approaches: For γ=1 and γ(1,1), when the limit function uM0 is independent of the microscopic variable y, we approximate the solution in the layer by its average ˉuMϵ, see (15) below. In the first case, the latter is bounded in L2((0,T),H1(Σ))H1((0,T),H1(Σ)), and thus converges strongly, due to the Aubin-Lions Lemma. In the second case, the gradient ˉxˉuMϵ is no longer bounded uniformly with respect to ϵ. Here, we use the Kolmogorov compactness result, based on estimates for the shifts with respect to t and ˉx. In the critical case γ=1, when the two-scale limit in general depends on y, we use the unfolded sequence together with a Kolmogorov-type compactness result for Banach valued functions. To pass to the limit in the microscopic problem (2), we use the two-scale convergence for thin domains with heterogeneous structure, see Definition A.1 in A.

    Let us start with the convergence results in the bulk domains. These are similar to those in [10,15], except for the time derivative, which is in this case only a functional pointwise in time. For the convergence of the time derivative, we transform the fixed domain Ω± to the ϵ-dependent set Ω±ϵ via the transformation

    Φ±ϵ:Ω±Ω±ϵ,Φ±ϵ(x)=(ˉx,HϵHxn±ϵ),

    and define ˜u±ϵ(t,x):=u±ϵ(t,Φ±ϵ(x)) for almost every (t,x)(0,T)×Ω±. Integration by substitution gives us ˜u±ϵL2((0,T),H1(Ω±))mH1((0,T),H1(Ω±))m with

    t˜u±i,ϵ(t),ϕΩ±=HHϵtu±i,ϵ(t),ϕ(Φ±ϵ)1Ω±ϵ,

    and especially we obtain with our a priori estimates from Lemma 4.2

    t˜u±i,ϵL2((0,T),H1(Ω±))Ctu±i,ϵL2((0,T),H1(Ω±ϵ))C.

    Proposition 9. Let uϵ be the solution of Problem (1) for γ[1,1]. Then there exists u±0L2((0,T),H1(Ω±))mH1((0,T),H1(Ω±))m, such that up to a subsequence

    χΩ±ϵu±i,ϵu±i,0stronglyinL2((0,T)×Ω±),χΩ±ϵu±i,ϵu±i,0weaklyinL2((0,T)×Ω±),˜u±i,ϵ|Σu±i,0stronglyinL2((0,T)×Σ),t˜u±i,ϵtu±i,0weaklyinL2((0,T),H1(Ω±)).

    Further, for all ϕL2((0,T),H1(Ω±)) it holds that

    limϵ0T0tu±i,ϵ(t),ϕ|Ω±ϵ(t)Ω±ϵdt=T0tu±i,0(t),ϕ(t)Ω±dt.

    Proof. The convergences of χΩ±ϵu±i,ϵ, χΩ±ϵu±i,ϵ, the trace ˜u±i,ϵ, and the time derivative t˜u±i,ϵ follow the same lines as in [15]. The last convergence follows easily from the representation of t˜u±i,ϵ above.

    After having established the convergence results for the sequences u±ϵ, we concentrate on the sequence uMϵ in the layer. Since different approaches are used for different values of the parameter γ, we consider each of these cases in a separate subsection, where after establishing the convergence results, we also derive the corresponding macroscopic problem.

    Remark 3. In the following, we consider the traces uMϵ|S±ϵ and u±ϵ|S±ϵ as functions in L2((0,T)×Σ) instead of the ϵ-dependent space L2((0,T)×S±ϵ), especially, we have u±ϵ|S±ϵ=˜u±i,ϵ|Σ.

    To prove the convergence results for the sequence uMϵ and to pass to the limit in the microscopic equations, we will use as an auxiliary function the averaged function

    ˉuMϵ(t,ˉx):=12ϵϵϵuMϵ(t,ˉx,xn)dxn, (15)

    introduced in Section 5.

    Theorem 7. For γ=1, there exist functions uM0L2((0,T),H1(Σ))mH1((0,T),H1(Σ))m and uM1L2((0,T)×Σ,Hpero)m, such that up to a subsequence it holds that

    uMi,ϵuMi,0stronglyinthetwoscalesenseuMi,ϵˉxuMi,0+yuMi,1inthetwoscalesense,uMi,ϵ|S±ϵuMi,0inL2((0,T)×Σ),tˉuMi,ϵtuMi,0weaklyinL2((0,T),H1(Σ)).

    Proof. This follows directly from Lemma 4.2 and Proposition 4.

    From the strong convergences above, we immediately obtain the two-scale convergences of the nonlinear terms:

    Corollary 3. For γ=1, it holds up to a subsequence

    f±(χΩ±ϵu±ϵ)f±(u±0)inL2((0,T)×Ω±),g(ϵ,uMϵ)g(,uM0)inthetwoscalesense,h±(u±ϵ|S±ϵ,uMϵ|S±ϵ)h±(u±0|Σ,uM0)inthetwoscalesenseonΣ,hM,±(ϵ,uMϵ|S±ϵ,u±ϵ|S±ϵ)hM,±(,uM0,u±0|Σ)inthetwoscalesenseonΣ.

    Proof. The first convergence follows from Proposition 9. To show the second convergence, we start from

    1ϵT0ΩMϵg(xϵ,uMϵ)ϕ(t,ˉx,xϵ)dxdt=1ϵT0ΩMϵ[g(xϵ,uMϵ)g(xϵ,ˉuMϵ)]ϕ(t,ˉx,xϵ)dxdt+1ϵT0ΩMϵ[g(xϵ,ˉuMϵ)g(xϵ,uM0)]ϕ(t,ˉx,xϵ)dxdt+1ϵT0ΩMϵg(xϵ,uM0)ϕ(t,ˉx,xϵ)dxdt=:Iϵ1+Iϵ2+Iϵ3,

    with ϕC0([0,T]ׯΣ,C0per([0,1]n1,C0([1,1]))). Using the Lipschitz-continuity of the function g and inequality (12d) corresponding to uMϵ, we obtain limϵ0Iϵ1=0. To estimate Iϵ2, we proceed as follows

    Iϵ2=1ϵT0ΩMϵ[g(xϵ,ˉuMϵ)g(xϵ,uM0)]ϕ(t,ˉx,xϵ)dxdtCϵT0ΩMϵ|g(xϵ,ˉuMϵ)g(xϵ,uM0)|dxdtCϵΩMϵ|ˉuMϵuM0|dxdtC||ˉuMϵuM0||L1((0,T)×Σ). (16)

    Thus, the strong convergence of ˉuMϵ to uM0 in L2((0,T)×Σ) implies limϵ0Iϵ2=0. Finally, the two-scale convergence of g(ϵ,uM0) to g(,uM0) yields the desired result. In a similar way the third and the last convergence follow, by using additionally the strong convergence of u±ϵ|S±ϵ in L2((0,T)×Σ) from Proposition 9.

    Now, based on the above convergence results, we derive the macroscopic model from Theorem 3.1.

    Proof of Theorem 3.1. To obtain the equations in the bulk domains, we test the variational equation (2) in the bulk with ϕ(x)ψ(t), where ϕD(¯Ω±) and ψD(0,T). Integration with respect to time, using Proposition 9, Theorem 7.1, Corollary 3, and a density argument, we can go to the limit ϵ0, and obtain the weak formulation for u±0.

    Testing the variational equation (2) of uMi,ϵ with ϕϵ(t,ˉx):=ϵψ(t)ϕ(ˉx)θ(xϵ) and ψD(0,T), ϕD(¯Σ), and θHper, we get after integrating with respect to time

    1ϵT0tuMi,ϵ(t),ϕϵ(t)ΩMϵdt+T0ΩMϵDMi(xϵ)uMi,ϵ(t,x)[θ(xϵ)ˉxϕ(ˉx)+1ϵϕ(ˉx)yθ(xϵ)]ψ(t)dxdt=1ϵT0ΩMϵgi(xϵ,uMϵ)ϕϵ(t,x)dxdt+α±T0SαϵhM,αi(ˉxϵ,uMϵ,uαϵ)ϕϵ(t,ˉx,α1)dˉxdt.

    Due to our a priori estimates, all the terms except the diffusion therm involving yθ are of order ϵ. Hence, from the two-scale convergence of uMi,ϵ in Theorem 7.1, we obtain for ϵ0

    T0ΣZDMi(y)[ˉxuMi,0(t,ˉx)+yuMi,1(t,ˉx,y)]ψ(t)ϕ(ˉx)yθ(y)dydˉxdt=0.

    This implies almost everywhere in (0,T)×Σ×Z

    uMi,1(t,ˉx,y)=n1j=1juMi,0(t,ˉx)wi,j(y), (17)

    where wi,j is the unique weak solution of the cell problem (4).

    Now, we choose as a test function ψ(t)ϕ(ˉx), with ψ and ϕ as above. After integration with respect to time and using the properties of ˉuϵ from Section 5, we obtain

    T0tˉuMi,ϵ(t),ϕΣψ(t)dt+1ϵT0ΩMϵDMi(xϵ)uMi,ϵ(t,x)ˉxϕ(ˉx)ψ(t)dxdt=1ϵT0ΩMϵgi(xϵ,uMϵ)ϕ(ˉx)ψ(t)dxdt+α±T0SαϵhM,αi(ˉxϵ,uMϵ,uαϵ)ϕ(ˉx)ψ(t)dˉxdt. (18)

    From Theorem 7.1, Corollary 3, and the identity (17), we get for ϵ0

    |Z|T0tuMi,0(t),ϕΣψ(t)dt+T0ΣDM,iˉxuMi,0(t,ˉx)ˉxϕ(ˉx)ψ(t)dˉxdt.=T0Σ(Zgi(y,uM0)dy+α{±}YhM,αi(ˉy,uM0,uα0)dˉy)ϕ(ˉx)ψ(t)dˉxdt.

    Since D(¯Σ) is dense in H1(Σ), this gives us the weak formulation for the problem of uM0. The initial condition is obtained by similar arguments, where we have to choose test function ψD([0,T)) and use integration by parts in the term involving the time derivative. Uniqueness is standard.

    Again, we use the averaged function ˉuMϵ. However, now, the gradient ˉxˉuMϵ is no longer bounded uniformly with respect to ϵi.e., we are not able to apply the Aubin-Lions Lemma to obtain the strong convergence of ˉuMϵ. Instead, we apply the Kolmogorov compactness theorem [5,Theorem 4.26]. To cope with the shifts with respect to time, we make use of the following embedding of Nikolskii-type, applied to the Hilbert space Hϵ,γ with its corresponding norm.

    Lemma 7.2. Let V and H be Hilbert spaces and we assume that (V,H,V) is a Gelfand triple. Let vL2((0,T),V)H1((0,T),V). Then, for every ϕV and almost every t(0,T), h(T,T), such that t+h(0,T), we have

    |(v(t+h)v(t),ϕ)H||h|ϕVtvL2((t,t+h),V).

    Especially, it holds that

    v(t+h)v(t)2H|h|v(t+h)v(t)VtvL2((t,t+h),V).

    Proof. The proof follows the same lines as the proof for the special case V=H1(Ω) and H=L2(Ω), see [9,Lemma 9].

    Theorem 7.3. For γ(1,1), let uϵ be the solution of Problem (1). Then, there exists uM0L2((0,T)×Σ)mH1((0,T),H1(Σ))m, such that up to a subsequence it holds for all p[1,2), that

    uMi,ϵuMi,0inthetwoscalesenseˉuMi,ϵuMi,0inLp((0,T)×Σ),uMi,ϵ|S±ϵuMi,0inLp((0,T)×Σ),tˉuMi,ϵtuMi,0weaklyinL2((0,T),H1(Σ)).

    Proof. As in Proposition 7.1, Lemma 4.2 and Proposition 10 imply that, up to a subsequence, uMϵ converges in two-scale sense to a limit uM0, which is independent of the microscopic variable y. Concerning the averaged sequence ˉuMϵ, we have that due to Lemma 5.1, it is bounded in L2((0,T)×Σ). Thus, up to a subsequence, ˉuMϵ converges weakly and the limit coincides with the two-scale limit uM0. The next step in the proof is to show the strong convergence of ˉuMϵ in Lp((0,T)×Σ) by using the Kolmogorov compactness criterion.

    We extend all functions outside their domain of definition by zero. Since ˉuMϵ is bounded in L2((0,T)×Σ)Lp((0,T)×Σ), to apply the Kolmogorov compactness result, see [5,Theorem 4.26], it is enough to show

    ˉuMϵ(+s,+ˉξ)ˉuMϵLp((0,T)×Σ)0for (s,ˉξ)0 (19)

    uniformly with respect to ϵ. Therefore, we show the following two statements:

    (ⅰ) For all h>0 fixed, we have

    supϵˉuMϵ(+s,+ˉξ)ˉuMϵLp((0,T)h×Σh)0for (s,ˉξ)0.

    (ⅱ) It holds that

    supϵˉuMϵLp((0,T)(0,T)h×ΣΣh)0for h0.

    For the definition of the domains (0,T)h,Σh and ΩMϵh, see (7) and (8). Condition (ⅱ) is an easy consequence of the Hölder-inequality and the boundedness of ˉuMϵ in L2((0,T)×Σ). In fact, we have

    ˉuMϵLp((0,T)(0,T)h×ΣΣh)C|(0,T)(0,T)h×ΣΣh|2p2ph00.

    For condition (ⅰ), due to the triangle inequality, it is enough to consider shifts separately with respect to the variable t and ˉx. We fix h>0, and obtain for |ˉξ|<h

    ˉuMϵ(,+ˉξ)ˉuMϵL2((0,T)×Σh)12ϵuMϵ(,+(ˉξ,0))uMϵL2((0,T)×ΩMϵh)12ϵuMϵ(,+(ˉξ,0))uMϵ(,+ϵ([ˉξϵ],0))L2((0,T)×ΩMϵh)+12ϵuMϵ(,+ϵ([ˉξϵ],0))uMϵL2((0,T)×ΩMϵh).

    Due to the mean-value theorem and the a priori estimates for uMϵ, the first term is of order ϵ1γ2 and therefore tends to zero for ϵ0. The second term goes to zero, due to Lemma 4.3 for ϵ,ˉξ0. The uniform convergence with respect to ϵ is then obtained from the convergence of the difference of the shifts to 0 for ˉξ0 for every fixed ϵ.

    Now, we have to consider shifts with respect to time. From Lemma 4.4, we have

    uMϵL2((0,T),Hϵ,γ)C,tuMi,ϵL2((0,T),Hϵ,γ)Cϵ.

    Hence, Lemma 7.2 with V=Hϵ,γ and H=L2(ΩMϵ) implies for |s|<h

    ˉuMϵ(+s,)ˉuMϵ2L2((0,T)h×Σ)12ϵuMϵ(+s,)uMϵ2L2((0,T)h×ΩMϵ)h2ϵuMϵ(+s,)uMϵL2((0,T)h,Hϵ,γ)tuMϵL2((0,T),Hϵ,γ)Ch.

    This gives us the strong convergence of ˉuMϵ in Lp((0,T)×Σ). Then, the strong converges of the traces uMϵ|S±ϵ follow by Lemma 5.2 and the triangle inequality. The convergence of the time derivative follows since the sequence is bounded in L2((0,T),H1(Σ)).

    Again, we obtain the convergences for the nonlinearities.

    Corollary 4. For γ(1,1), it holds up to a subsequence

    f±(χΩ±ϵu±ϵ)f±(u±0)inL2((0,T)×Ω±),g(ϵ,uMϵ)g(,uM0)inthetwoscalesense,h±(u±ϵ|S±ϵ,uMϵ|S±ϵ)h±(u±0|Σ,uM0)inthetwoscalesenseonΣ,hM,±(ϵ,uMϵ|S±ϵ,u±ϵ|S±ϵ)hM,±(,uM0,u±0|Σ)inthetwoscalesenseonΣ.

    Proof. We use the same arguments as in Corollary 3. The only difference is that terms like (16) are estimated by the Lp-norm, with p<2, of ˉuMϵuM0 instead of the L2-norm, and then the strong convergence of ˉuMϵ to uM0 in Lp((0,T)×Σ) is used.

    Passing to the limit ϵ0, we obtain Theorem 3.2:

    Proof of Theorem 3.2. The arguments are quite similar to those in Theorem 3.1 for the case γ=1, so we only point out the main differences. The variational equation (18) is still valid for all ϕD(¯Σ) and ψD(0,T) if we replace 1ϵ by ϵγ in front of the diffusion term. In the limit ϵ0 this terms vanishes, since

    |ϵγT0ΩMϵDMi(xϵ)uMi,ϵ(t,x)ˉxϕ(ˉx)ψ(t)dxdt|Cϵ1+γ2.

    Finally, the L2-regularity of the time derivative tuM0 follows from the regularity of the right-hand side of the differential equation of uM0.

    Here, we treat the critical case γ=1. Since in this case the two-scale limit uM0 depends on the macroscopic and the microscopic variable, it is no longer sufficient to work with the averaged function ˉuϵ. To prove the convergences of the nonlinear terms, we use a Kolmogorov type compactness result for Banach valued functions, see [8], applied to the unfolded sequence TMϵuMϵLp(Σ,L2((0,T),Hβ(Z))). A main ingredient in this proof is the following lemma.

    Lemma 7.4. For all ϕMϵL2((0,T)×ΩMϵ), h>0, and ˉξRn1 with |ˉξ|<h, it holds for ϵ small enough that

    TMϵϕMϵ(,+ˉξ,)TMϵϕMϵ2L2((0,T)×Σ2h×Z)1ϵj{0,1}n1δlϕMϵ2L2((0,T)×ΩMϵh)

    with δlϕMϵ defined in (9) with l=l(ϵ,ˉξ,j)=j+[ˉξϵ].

    Proof. The proof is based on a special decomposition of Z and can be found in [15,page 709], where here, we additionally use the fact that ΩMϵ,2hˆΩMϵ,hΩMϵh for ϵ small enough.

    Theorem 7.5. For γ=1, let uϵ be the solution of Problem (1). Then, there exists uM0L2((0,T)×Σ,Hper)mH1((0,T),L2(Σ,H0))m, such that for p[1,2) and β(12,1) up to a subsequence it holds that

    uMi,ϵuMi,0inthetwoscalesense,ϵuMi,ϵyuMi,0inthetwoscalesense,TMϵuMi,ϵuMi,0inLp(Σ,L2((0,T),Hβ(Z))),TΣϵ(uMi,ϵ|S±ϵ)uMi,0|S±inLp(Σ,L2((0,T)×S±))

    Remark 4. Due to the boundedness of the sequence TMϵuMϵ in the space L2((0,T)×Σ,H1(Z))H1((0,T),L2(Σ,H0)), see Lemma 4.2, 4.4, and Proposition 8, we additionally obtain the weak convergence of TMϵuMϵ in L2((0,T)×Σ,H1(Z)) and tTMϵuMϵ in L2((0,T),L2(Σ,H0)). However, the convergence of the time derivative will not be used in the derivation of the macroscopic problem, since the structure of the limit problem gives us even more regularity with respect to time.

    Proof of Theorem 7.5. The two-scale convergences of uMϵ and ϵuMϵ follow from the a priori estimates for uMϵ and Proposition 10.

    We prove the strong convergence of TMϵuMϵ. We make use of the Kolmogorov-type compactness result from [8,Theorem 2.2]. We have to show (all functions are extended by zero)

    (ⅰ) For every AΣ measurable, the sequence

    vϵA(t,y):=ATMϵuMϵ(t,ˉx,y)dˉx

    is relatively compact in L2((0,T),Hβ(Z)).

    (ⅱ) It holds that

    supϵTMϵuMϵ(,+ˉξ,)TMϵuMϵLp(Σ,L2((0,T),Hβ(Z)))0for ˉξ0.

    First of all, we have vϵAL2((0,T),H1(Z))H1((0,T),H0) with time derivative

    tvϵA(t),ϕH0,H0=tTMϵuMi,ϵ(t),χA(ˉx)ϕ(y)L2(Σ,H0),L2(Σ,H0),

    since for every ψD(0,T) and ϕH0, we have

    T0ZvϵA(t,y)ϕ(y)ψ(t)dydt=T0ΣZTMϵuMi,ϵ(t,ˉx,y)ϕ(y)χA(ˉx)ψ(t)dydˉxdt=T0tTMϵuMi,ϵ(t),χA(ˉx)ϕ(ˉy)L2(Σ,H0),L2(Σ,H0)ψ(t)dt.

    Hence, Lemma 4.2, 4.4, and Proposition 8 imply the boundedness of vϵA in the space L2((0,T),H1(Z))H1((0,T),H0). For β(12,1) the embedding H1(Z)Hβ(Z) is compact and the embedding Hβ(Z)H0 is continuous. Therefore, we can apply the Aubin-Lions Lemma and obtain that the sequence vϵA is relatively compact in L2((0,T),Hβ(Z)), what proves (ⅰ). Since for all h>0 and |ˉξ|h, we have

    TMϵuMϵ(,+ˉξ,)TMϵuMϵLp(ΣΣ2h,L2((0,T),Hβ(Z)))C|h|2p2pTMϵuMϵL2((0,T)×Σ,H1(Z))Ch2p2p,

    by the same arguments as in the proof of Theorem 7.3, it is enough to show that for all h>0, it holds that

    supϵTMϵuMϵ(,+ˉξ,)TMϵuMϵLp(Σ2h,L2((0,T),Hβ(Z)))0for ˉξ0.

    So, we fix h>0 and obtain with Lemma 7.4

    TMϵuMϵ(,+ˉξ,)TMϵuMϵL2(Σ2h,L2((0,T),Hβ(Z)))CTMϵuMϵ(,+ˉξ,)TMϵuMϵL2(Σ2h,L2((0,T),H1(Z)))Cj{0,1}n1(1ϵδuMϵ2L2((0,T)×ΩMϵh)+ϵδuMϵ2L2((0,T)×ΩMϵh)).

    Due to Lemma 4.3, the right-hand side converges to zero for ϵ,ˉξ0. The uniform convergence is again obtained by the convergences of the shifts to 0 for every fixed ϵ.

    The last statement of the theorem follows from the strong convergence of TMϵuMϵ, together with the continuity of the trace operator from Hβ(Z) into Hβ12(Z) for β(12,1) and the fact that TΣϵ(uMϵ|S±ϵ)=(TMϵuMϵ)|S±.

    Corollary 5. For γ=1, it holds up to a subsequence

    f±(χΩ±ϵu±ϵ)f±(u±0)inL2((0,T)×Ω±)m,g(ϵ,uMϵ)g(,uM0)inthetwoscalesense,h±(u±ϵ|S±ϵ,uMϵ|S±ϵ)h±(u±0|Σ,uM0|S±)inthetwoscalesenseonΣ,hM,±(ϵ,uMϵ|S±ϵ,u±ϵ|S±ϵ)hM,±(,uM0|S±,u±0|Σ)inthetwoscalesenseonΣ.

    Proof. We only show the last convergence. From the strong convergence of u±ϵ|S±ϵ to u±0|Σ in L2((0,T)×Σ), see Proposition 9, it follows that TΣϵ(u±ϵ|S±ϵ) converges strongly to u±0|Σ in L2((0,T)×Σ×Y). We have

    TΣϵ(hM,±(ϵ,uMϵ|S±ϵ,u±ϵ|S±ϵ))=hM,±(ˉy,TΣϵ(uMϵ|S±ϵ),TΣϵ(u±ϵ|S±ϵ)).

    The strong convergences of TΣϵ(u±ϵ|S±ϵ) and TΣϵ(uMϵ|S±ϵ) imply, up to a subsequence, the pointwise convergence of the right-hand side almost everywhere in (0,T)×Σ×Y to hM,±(,uM0|S±,u±0|Σ). Further, this sequence is bounded in L2((0,T)×Σ×Y)i.e., it converges weakly in L2((0,T)×Σ×Y) to the same limit. Now, Lemma A.3 gives us the result.

    Finally, the compactness results from Theorem 7.5 and Corollary 5 allow us to pass to the limit ϵ0 in the microscopic problem and to obtain the macroscopic model in Theorem 3.3.

    Proof of Theorem 3.3. The equations for the bulk-domains Ω± can be obtained by similar arguments as for the cases γ=1 and γ(1,1).

    To derive the limit equation for the membrane, we test the variational equation of uMϵ in (2) with ϕ(ˉx,xϵ)ψ(t), where ϕD(¯Σ,Cper(¯Y,C1([1,1]))), ψD((0,T)), integrate over time and use integration by parts in the term involving the time derivative to obtain

    1ϵT0ΩMϵuMi,ϵ(t,x)ϕ(ˉx,xϵ)ψ(t)dxdt+ϵT0ΩMϵDMi(xϵ)uMϵ(t,x)[ˉxϕ(ˉx,xϵ)+1ϵyϕ(ˉx,xϵ)]ψ(t)dxdt=1ϵT0ΩMϵgi(xϵ,uMϵ)ϕ(ˉx,xϵ)ψ(t)dxdt+α±T0SαϵhM,αi(ˉxϵ,uMϵ,uαϵ)ϕ(ˉx,xϵ)ψ(t)dˉxdt.

    For ϵ0, we obtain from Theorem 7.5 and Corollary 5

    T0ΣZuM0(t,ˉx,y)ϕ(ˉx,y)ψ(t)dydˉxdt=T0ΣZDMi(y)yuMi,0(t,ˉx,y)yϕ(ˉx,y)ψ(t)dydˉxdt+T0ΣZgi(y,uM0)ϕ(ˉx,y)ψ(t)dydˉxdt+α±T0ΣYhM,αi(ˉy,uM0(t,ˉx,ˉy,α),uα0(t,ˉx,0))ϕ(ˉx,ˉy,α)ψ(t)dˉydˉxdt.

    By a density argument the equation holds for all ϕL2(Σ,Hper). The regularity of the terms on the right-hand side imply tuM0L2((0,T),L2(Σ,Hper)). The initial condition can be established in the usual way, where we have to use the two-scale convergence of UM0(ˉx,xnϵ) to U0(ˉx,yn). Uniqueness is again standard.

    Remark 5.

    (ⅰ) Due to the uniqueness of the solutions of the effective models in Theorems 3.1, 3.2, and 3.3, it follows that the whole sequence converges.

    (ⅱ) The results also hold, if the parameter γ is different for every speciesi.e., we have to replace γ by γi in the microscopic problem. Further, the results from this paper and [10] remain valid if we mix up the nonlinear Neumann-transmission conditions and the continuous transmission conditions on S±ϵ for different species.

    In this paper, we derived effective models for reaction-diffusion processes through thin layers with nonlinear transmission conditions at the bulk-layer interface, and diffusivities of order ϵγ,γ[1,1] in the layer region. It turns out, that for all γ[1,1], the effective bulk solution u±0 is described by the same system of reaction-diffusion equations in Ω±, with parameters inherited from the microscopic system. At the interface Σ, we obtain interface laws, which strongly depend on the choice of the parameter γ.

    In the critical case γ=1, the effective solution u0 and its normal flux may exhibit jumps across Σ. The normal flux of u±0 on Σ is given by a nonlinear Neumann boundary condition which involves the homogenized solution uM0 in the layer. The evolution of uM0 is described by a reaction-diffusion system on the standard cell Z, where xΣ plays the role of a parameter. This system is strongly coupled to the effective equations for u±0 in Ω± via a nonlinear Neumann boundary condition on S±.

    For γ[1,1), the solution u0 is continuous across Σ, and the common trace is equal to uM0, which in this case is independent of the microscopic variable y. The normal flux of u±0 on Σ is again given by a nonlinear Neumann boundary condition which depends on uM0. The interface laws satisfied by uM0 are, however, different for γ(1,1) and γ=1. For γ(1,1), the function uM0 is the solution of an ordinary differential equation which also depends on the parameter xΣ. In the case γ=1, we obtain an additional surface diffusion on Σi.e., uM0 is the solution of a reaction-diffusion equation on Σ. In both cases, the jump in the normal flux of u0 across Σ enters the equation for uM0 as a sink/source term.

    The methods developed in this paper can also be used for more complex multi-physics processes. Furthermore, the effective models obtained here rise new challenges also to numerical approaches, which have to take into account the special micro-macro features of the model.

    We repeat the definition of the two-scale convergence and the unfolding operator for thin domains, and briefly summarize some results related to these approaches, which are needed frequently throughout the paper. The two-scale convergences for domains was introduced in [1,16], and extended to thin layers with heterogeneous structure in [15].

    Definition A.1. We say that a sequence uϵL2((0,T)×ΩMϵ) converges (weakly) in the two-scale sense to the limit function u0L2((0,T)×Σ×Z), if for all ϕC0([0,T]ׯΣ,C0per([0,1]n1,C0([1,1]))) it holds that

    limϵ01ϵT0ΩMϵuϵ(t,x)ϕ(t,ˉx,xϵ)dxdt=T0ΣZu0(t,ˉx,y)ϕ(t,ˉx,y)dydˉxdt.

    Further, we say that a (weakly) two-scale convergent sequence uϵ converges strongly in the two-scale sense to the limit u0, if we additionally have

    limϵ01ϵuϵL2((0,T)×ΩMϵ)=u0L2((0,T)×Σ×Z).

    We remark that in [4] a definition of the two-scale convergence was given for a thin domain with a more particular geometry.

    In the following, we consider functions uϵL2((0,T),H1(ΩMϵ)), such that

    1ϵuϵL2((0,T)×ΩMϵ)+ϵγ2uϵL2((0,T)×ΩMϵ)C,

    with γ[1,1]. The behavior in the limit ϵ0 depends on the choice of γ, and we have to distinguish the three cases γ=1, γ(1,1), and γ=1. To state the compactness results for uϵ and uϵ for different γ, we make use of the following spaces:

    Hper:={uH1(Z):u is Y-periodic},Hpero:={uHper:Zudy=0}, (20)

    equipped with the H1(Z)-norm.

    Proposition 10. Let uϵ be a sequence in L2((0,T),H1(ΩMϵ)) such that

    1ϵuϵL2((0,T)×ΩMϵ)+ϵγ2uϵL2((0,T)×ΩMϵ)C,

    with a constant C>0 independent of ϵ. Then, we have the following convergence results:

    (i) For γ=1, there exists a subsequence and a limit function u0L2((0,T)×Σ,Hper), such that

    uϵu0inthetwoscalesense,ϵuϵyu0inthetwoscalesense.

    (ii) For γ(1,1), there exist u0L2((0,T)×Σ) and u1L2((0,T)×Σ,Hpero), such that up to a subsequence

    uϵu0inthetwoscalesense,ϵγ+12uϵyu1inthetwoscalesense.

    (iii) For γ=1, there exist u0L2((0,T),H1(Σ)) and u1L2((0,T)×Σ,Hpero), such that up to a subsequence

    uϵu0inthetwoscalesense,uϵˉxu0+yu1inthetwoscalesense,

    where ˉxu0:=(1u0,,n1u0,0).

    Proof. The proof of (ⅰ) can be found in [15], and the proof of (ⅱ) and (ⅲ) in [10].

    Next, we define the unfolding operator TMϵ for thin domains introduced in [15], which is an extension of the unfolding operator in fixed or perforated domains, see e.g., [2,6,18]. Here, we consider thin domains with a general lateral boundary Σ×(ϵ,ϵ), see also [7]. For the definition, we need the following notations:

    Kϵ:={ˉkZn1:ϵ(ˉk+Y)Σ},ˆΣϵ:=intˉkKϵϵ(ˉk+¯Y),Λϵ:=int(ΣˆΣϵ),ˆΩMϵ:=ˆΣϵ×(ϵ,ϵ),ΛMϵ:=Λϵ×(ϵ,ϵ). (21)

    Then, we define the unfolding operator as follows:

    TMϵ:L2((0,T)×ΩMϵ)L2((0,T)×Σ×Z),TMϵuϵ(t,ˉx,y)={uϵ(t,ϵ([ˉxϵ],0)+ϵy) for ˉxˆΣϵ,0 for ˉxΛϵ.

    The unfolding operator TMϵ has the following basic properties:

    Lemma A.2. [15] For uϵ,vϵL2((0,T)×ΩMϵ), we have

    (TMϵuϵ,TMϵvϵ)L2((0,T)×Σ×Z)=1ϵ(uϵ,vϵ)L2((0,T)׈ΩMϵ),TMϵuϵL2((0,T)×Σ×Z)1ϵuϵL2((0,T)×ΩMϵ).

    If additionally it holds that uϵL2((0,T),H1(ΩMϵ)), then TMϵuϵL2((0,T)×Σ,H1(Z)) and almost everywhere in (0,T)×Σ×Z it holds that

    yTMϵuϵ=ϵTMϵ(uϵ).

    Finally, we have the following relation between the unfolding operator TMϵ and the two-scale convergence in thin domains.

    Lemma A.3. [15] Let uϵL2((0,T)×ΩMϵ) be a sequence with

    uϵL2((0,T)×ΩMϵ)Cϵ,

    with a constant C>0 independent of ϵ. Then the following statements are equivalent:

    (i) uϵu0 weakly/strongly in the two-scale sense,

    (ii) TMϵuϵu0 weakly/strongly in L2((0,T)×Σ×Z).

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