Upscaling in nonlinear thermal diffusion problems

Claudia Timofte∗

Faculty of Physics, University of Bucharest, P.O.Box MG-11, Bucharest-Magurele, Romania.

The aim of this paper is to study the homogenization of a nonlinear problem arising in the modelling of thermal diffusionin a two-component composite. We consider, at the microscale, a periodic structure formed by two materials with differentthermal properties. We assume that we have nonlinear sources and that at the interface between our two materials the fluxis continuous and depends in a nonlinear way on the jump of the temperature field. We shall be interested in describing theasymptotic behavior, as the small parameter which characterizes the sizes of our two regions tends to zero, of the temperaturefield in the periodic composite.

1 Introduction and setting of the problem

The general question which will make the object of this paper is the homogenization of some nonlinear problems arising in themodelling of thermal diffusion in a periodic structure formed by two materials with different thermal features. We assume thatwe have nonlinear sources acting in one component and that at the interface between our two materials the flux is continuousand depends in a nonlinear way on the jump of the temperature field.

Let Ω be a bounded domain in �n (n ≥ 3), having the boundary of class C2. We consider that Ω is a periodic structureformed by two components, Ωε and Πε, representing two materials with different thermal features, separated by an interfaceSε. We assume that both Ωε and Πε = Ω \ Ωε are connected, but only Ωε reaches the external fixed boundary of the domainΩ. Here, ε represents a small parameter related to the characteristic size of the our two regions.

More precisely, let E1 and E2 = �n \E1 be two open, connected and periodic subsets of �n, with smooth boundaries and

let Γ be their common boundary, Γ = ∂E1 ∩ ∂E2. Let Y = (0, 1)n be the unit reference cell of periodicity and Y1 = Y ∩ E1

an open connected subset, with boundary of class C2. Let Y2 = Y \Y1. We assume that both Y1 and Y2 have positive measure.So, the 1-dimensional measure of the interface between them is also positive. We define Ωε = Ω∩ εE1 and Πε = Ω \Ωε andwe set θ =

∣∣Y \ Y2

∣∣.Let us consider a family of inhomogeneous media occupying the region Ω, parameterized by ε and represented by n × n

matrices Aε(x) of real-valued coefficients defined on Ω. We shall deal with periodic structures, defined by Aε(x) = A(xε).

Here A = A(y) is a smooth matrix-valued function on �n which is Y -periodic. We use the symbol # to denote periodicityproperties. We shall assume that

⎧⎨⎩

A ∈ L∞

# (Ω)n×n,

A is a symmetric matrix,For some 0 < γ < λ, γ |ξ|

2≤ A(y)ξ · ξ ≤ λ |ξ|

2∀ξ, y ∈ �n

and we shall denote the matrix A by A1 in Y1 and by A2, respectively, in Y2.If we denote by (0, T ) the time interval of interest, we shall analyze the asymptotic behavior of the solutions of the following

nonlinear system:

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

−div (Aε1∇uε) + β(uε) = f in Ωε × (0, T ),

−div (Aε2∇vε) = f, in Πε × (0, T ),

Aε1∇uε · ν = Aε

2∇vε · ν on Sε × (0, T ),

Aε1∇uε · ν + αε

∂uε

∂t= aεg(vε − uε) on Sε × (0, T ),

uε = 0 on ∂Ω × (0, T ),uε(0, x) − vε(0, x) = c0(x), on Sε.

(1)

Here, ν is the exterior unit normal to Ωε, a > 0, f ∈ L2(0, T ; L2(Ω)), c0 ∈ H10 (Ω), α > 0 and β and g are continuous

functions, monotonously non-decreasing and such that β(0) = 0 and g(0) = 0. We shall suppose that there exist a positiveconstant C and an exponent q, with 0 ≤ q < n/(n− 2), such that

|β(v)| ≤ C(1 + |v|q), |g(v)| ≤ C(1 + |v|

q). (2)

∗ Corresponding author: e-mail: claudiatimofte@yahoo.com, Phone: +4072 149 8466, Fax: +4021 457 4521

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

PAMM · Proc. Appl. Math. Mech. 7, 4080031–4080032 (2007) / DOI 10.1002/pamm.200700817

These two general situations are well illustrated, for instance, by the following important practical examples: g(v) =δv

1 + γv, δ, γ > 0 (Langmuir kinetics) and β(v) = |v|p−1v, 0 < p < 1 (Freundlich kinetics).

Notice that, in fact, due to the compactness injection theorems in Sobolev spaces, it would be enough to assume that βsatisfies the corresponding growth condition (2) for some 0 ≤ q < (n + 2)/(n − 2).

By classical existence and uniqueness results (see [3], [7] and [10]), we know that (1) is a well-posed problem.Our goal is to obtain the asymptotic behavior, when ε → 0, of the solution of problem (1). Using Tartar’s method of

oscillating test functions (see [8]), coupled with monotonicity methods and results from the theory of semilinear problems (see[3] and [5]), we can prove that the asymptotic behavior of the solution of our problem is governed by a new nonlinear system,similar to the famous Barenblatt’s model (see [2], [6] and [9]), with extra terms capturing the effect of the interfacial barrier,of the dynamical boundary condition and of the nonlinear sources. Our results constitute a generalization of those obtainedin [2] and [6], by considering nonlinear sources, nonlinear dynamical transmission conditions and different techniques in theproofs (for detailed proofs we refer to [9]). Similar problems have been considered, using different techniques, in [1], [7] and[10], for studying electrical conduction in biological tissues.

2 The main result

Using well-known extension results (see [4]) and suitable test functions, we can pass to the limit in the variational formulationof problem (1) and we obtain the effective behavior of the solution of our microscopic model. Thus, the main result of thispaper can be formulated as follows:

Theorem 2.1 One can construct two extensions P εuε and P εvε of the solutions uε and uε of problem (1) such thatP εuε ⇀ u, P εvε ⇀ v, weakly in L2(0, T ; H1

0(Ω)), where⎧⎪⎪⎪⎨⎪⎪⎪⎩

| Γ |∂(u − v)

∂t− div (A

1∇u) + θβ(u) − a | Γ | g(v − u) = θf in Ω × (0, T ),

| Γ |∂(v − u)

∂t− div (A

2∇v) + a | Γ | g(v − u) = (1 − θ)f in Ω × (0, T ),

u(0, x) − v(0, x) = c0(x) on Ω.

Here, A1

and A2

are the homogenized matrices, defined by:

A1

ij =

∫Y1

(aij + aik

∂χ1j

∂yk

)dy,

A2

ij =

∫Y2

(aij + aik

∂χ2j

∂yk

)dy,

in terms of the functions χ1k

∈ H1per(Y1)/�, χ

2k∈ H1

per(Y2)/�, k = 1, ..., n, weak solutions of the cell problems

⎧⎨⎩

−∇y · ((A1(y)∇yχ1k

) = ∇yA1(y)ek, y ∈ Y1,

(A1(y)∇yχ1k

) · ν = −A1(y)ek · ν, y ∈ Γ,

⎧⎨⎩

−∇y · ((A2(y)∇yχ2k

) = ∇yA2(y)ek, y ∈ Y2,

(A2(y)∇yχ2k

) · ν = −A2(y)ek · ν, y ∈ Γ.

Thus, in the limit, we obtain a system similar to the so-called Barenblatt model. Alternatively, such a system is similar tothe so-called bidomain model, appearing in the context of electrical activity of the heart.

Acknowledgments This work was supported by the CNCSIS Grant Ideas 992, under contract 31/2007.

References

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[10] M. Veneroni, Math. Meth. Appl. Sci. 29(14), 1631 (2006).

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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