Cent. Eur. J. Phys. 1-10 [610085]

Cent. Eur. J. Phys. •1-10
Author version
Central European Journal of Physics
Multiscale Analysis in Nonlinear Thermal Diffusion
Problems in Composite Structures
Research Article
Claudia Timofte1∗
1 Faculty of Physics, University of Bucharest,
P.O. Box MG-11, Bucharest-Magurele, Romania
Abstract: The aim of this paper is to analyze the asymptotic behavior of the solution of a nonlinear problem arising
in the modelling of thermal diffusion in a two-component composite. We consider, at the microscale,
a periodic structure formed by two materials with different thermal features. We assume that we have
nonlinear sources and that at the interface between our two materials the flux is continuous and depends in
a dynamical nonlinear way on the jump of the temperature field. We shall be interested in describing the
asymptotic behavior, as the small parameter which characterizes the sizes of our two regions tends to zero,
of the temperature field in the periodic composite. We prove that the effective behavior of the solution
of this system is governed by a new system, similar to the famous Barenblatt’s model, with extra terms
capturing the effect of the interfacial barrier, of the dynamical boundary condition and of the nonlinear
sources.
PACS (2008): 02.60.Lj, 44.35.+c
Keywords: homogenization, composite materials, interfacial barrier, dynamical boundary conditions
©Versita Warsaw and Springer-Verlag Berlin Heidelberg.
1. Introduction and setting of the problem
The purpose of this paper is to analyze the effective behavior of the solution of a nonlinear problem arising in the
modelling of thermal diffusion in a periodic structure formed by two materials with different thermal properties,
separated by an active interface. We assume that we have nonlinear sources acting in one component and that
at the interface between our two materials the flux is continuous and depends in a nonlinear way on the jump of
the temperature field.
Recent theoretical and computational advanced studies investigating the effective behavior of composite materials
are based on a model which conceives the composite material, despite of its discrete structure, as the coupling
of two continuous superimposed domains. Homogenization theory will allow us to justify such a model and to
∗E-mail: [anonimizat]
1

Multiscale Analysis in Nonlinear Thermal Diffusion Problems in Composite Structures
give a meaning to the effective properties of highly heterogeneous materials, modeled by equations with rapidly
oscillating periodic coefficients.
Let Ω be a bounded domain in Rn(n≥3), with the boundary ∂Ω being a Lipschitz manifold, made by a finite
number of connected components. We consider that Ω is a periodic structure formed by two components, Ωεand
Πε, representing two materials with different thermal features, separated by an interface Sε. We assume that
both Ωεand Πε= Ω\Ωεare connected, but only Ωεreaches the external fixed boundary of the domain Ω. Here,
εrepresents a small parameter related to the characteristic size of the our two regions.
More precisely, following H. I. Ene and D. Polisevski [ 12], letY1be a Lipschitz open connected subset of the unit
cubeY= (0,1)n. LetY2=Y\Y1. We suppose that Y2has a locally Lipschitz boundary. Moreover, we assume
that the intersections of the boundary of Y2with the boundary of the unit cell Yare identically reproduced on
opposite faces of the cube, which are denoted, for any 1 ≤i≤n, by
Σi={y∈∂Y|yi= 1}
and
Σ−i={y∈∂Y|yi= 0}.
We suppose that repeating Yby periodicity, the union of all the sets Y1is connected and has a locally C2
boundary. Also, we assume that the origin of the coordinate system is set in a ball contained in this union.
Let
Zε={k∈Zn|εk+εY⊆Ω}
and
Iε={k∈Zε|εk±εei+εY⊆Ω,∀i=1,n},
whereeiare the elements of the canonical basis of Rn.
We define
Πε= int([
k∈Iε(εk+εY2))
and
Ωε= Ω\Πε
and we set θ=Y\Y2.
Let us consider a family of inhomogeneous media occupying the region Ω, parameterized by εand represented
byn×nmatricesAε(x) of real-valued coefficients defined on Ω. We shall deal with periodic structures, defined
byAε(x) =A(x
ε). HereA=A(y) is a smooth matrix-valued function on Rnwhich isY-periodic. We use the
2

Claudia Timofte
symbol # to denote periodicity properties. We shall assume that
8
>>>><
>>>>:A∈L∞
#(Ω)n×n,
Ais a symmetric matrix,
For some 0 <γ <λ, γ |ξ|2≤A(y)ξ·ξ≤λ|ξ|2∀ξ, y∈Rn
and we shall denote the matrix AbyA1inY1and byA2, 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:
8
>>>>>>>>>>>>>><
>>>>>>>>>>>>>>:−div (Aε
1∇uε) +β(uε) =fin Ωε×(0,T),
−div (Aε
2∇vε) =f, in Πε×(0,T),
Aε
1∇uε·ν=Aε
2∇vε·νonSε×(0,T),
Aε
1∇uε·ν+αε∂
∂t(uε−vε) =εg(vε−uε) onSε×(0,T),
uε= 0 on∂Ω×(0,T),
uε(0,x)−vε(0,x) =c0(x),onSε.(1)
Here,νis the exterior unit normal to Ωε,f∈L2(0,T;L2(Ω)),c0∈H1
0(Ω) andα > 0. The function βis
continuous, monotonously non-decreasing and such that β(0) = 0 and the function gis continuously differentiable,
monotonously non-decreasing and with g(0) = 0. We shall suppose that there exist a positive constant Cand an
exponentq, with 0 ≤q<n/ (n−2), such that
|β(v)| ≤C(1 +|v|q),dg
dv≤C(1 +|v|q). (2)
As particular examples of such functions we can consider, for instance, the following important practical ones:
β(v) =δv
1 +γv, δ,γ > 0 (Langmuir kinetics), β(v) =|v|p−1v,0<p< 1 (Freundlich kinetics), g(v) =avor
g(v) =av3, witha>0.
Well posedness of problem (1) in suitable function spaces and proper energy estimates have been obtained in [ 1],
[4] and [ 13].
Since the period of the structure is small compared to the dimension of Ω, or in other words, since the nonhomo-
geneities are small compared to the global dimension of the structure, an asymptotic analysis becomes necessary.
Two scales are important for a suitable description of the given structure: one which is comparable with the
dimension of the period, called the microscopic scale and denoted by y=x/ε, and another one which is of the
same order of magnitude as the global dimension of our system, called the macroscopic scale and denoted by x.
The main goal of the homogenization method is to pass from the microscopic scale to the macroscopic one; more
precisely, using the homogenization method, we try to describe the macroscopic properties of our nonhomoge-
neous system in terms of the properties of its microscopic structure. Intuitively, the nonhomogeneous real system,
3

Multiscale Analysis in Nonlinear Thermal Diffusion Problems in Composite Structures
having a very complicated microstructure, is replaced by a fictitious homogeneous one, whose global characteris-
tics represent a good approximation of the initial system. Hence, the homogenization method provides a general
framework for obtaining these macroscale properties, eliminating the difficulties related to the explicit determi-
nation of a solution of the problem at the microscale and offering a less detailed description, but one which is
applicable to much more complex systems.
Also, from the point of view of numerical computation, the homogenized equations, defined on a fixed domain Ω
and describing the effective behavior of our system, will have constant coefficients, called effective or homogenized
coefficients (see Section 2), and, hence, it will be easier to be solved numerically than the original equations,
having rapidly oscillating coefficients, being defined on a complicated domain and satisfying nonlinear conditions
on the boundaries. The dependence on the real microstructure is given through the homogenized coefficients.
Hence, 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 [ 14]), coupled with monotonicity methods and results from the theory of
semilinear problems (see [ 4] and [ 9]), 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 [ 3] and [ 12]), 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 obtained in [ 3] and [ 12], by considering nonlinear sources,
nonlinear dynamical transmission conditions and different techniques in the proofs. Similar problems have been
considered, using different techniques, in [ 2] and [ 13], for studying electrical conduction in biological tissues.
In conclusion, at a macroscopic level, despite of the discrete cellular structure, the composite material can be
represented by a continuous model, describing the averaged properties of the complex structured composite
material. The resulting macroscopic model describes the composite material as the superimposition of two interpe-
netrating continuous media, coexisting at every point of the domain. Also, let us notice that the above model is
a degenerate parabolic system, as the time derivatives involve the unknown v−u, instead of the unknowns uand
voccurring in the second-order conduction term.
Problems closed to this one have been considered by D. Cioranescu and P. Donato [ 5], D. Cioranescu, P. Donato
and H.I. Ene [ 6], C. Conca and P. Donato [ 10], C. Conca, J.I. D´ ıaz and C. Timofte [ 9], C. Conca, F. Murat and C.
Timofte [ 11], C. Timofte [ 16], [17] and [ 18], H. Ene and D. Polisevski [ 12], C. Timofte [ 15], for the deterministic
case, W. Wang and J. Duan [ 20] for the stochastic one.
The plan of the paper is as follows: in the second section we give the main convergence result of this paper. The
last section is devoted to the proof of the convergence result.
2. The main result
Using well-known extension results (see, for instance, [ 8], [5] and [ 13]) and suitable test functions, we can pass to
the limit in the variational formulation of problem (1) and we obtain the effective behavior of the solution of our
4

Claudia Timofte
microscopic model. Therefore, the main result of this paper can be formulated as follows:
Theorem 2.1.
One can construct two extensions PεuεandPεvεof the solutions uεandvεof problem (1) such that Pεuε/arrowrighttophalf
u, Pεvε/arrowrighttophalfv, weakly in L2(0,T;H1
0(Ω)) , where
8
>>>>>><
>>>>>>:α|Γ|∂
∂t(u−v)−div(A1∇u) +θβ(u)− |Γ|g(v−u) =
=θfinΩ×(0,T),
α|Γ|∂
∂t(v−u)−div(A2∇v)+|Γ|g(v−u) =
= (1−θ)finΩ×(0,T),
u(0,x)−v(0,x) =c0(x)onΩ.(3)
Here,A1andA2are the homogenized matrices, defined by:
A1
ij=Z
Y1
aij+aik∂χ1j
∂yk
dy,
A2
ij=Z
Y2
aij+aik∂χ2j
∂yk
dy,
in terms of the functions χ1k∈H1
per(Y1)/R, χ2k∈H1
per(Y2)/R,k= 1,…,n, weak solutions of the cell problems
8
<
:−∇ y·((A1(y)∇yχ1k) =∇yA1(y)ek, y∈Y1,
(A1(y)∇yχ1k)·ν=−A1(y)ek·ν, y ∈Γ,
8
<
:−∇ 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 to the so-called bidomain model , appearing in the context of electrical activity of the heart.
3. Proof of the main result
From [ 1] and [ 13], we know that there exists a unique weak solution ( uε,vε) of (1), with uε∈L2(0,T;H1(Ωε)),
uε= 0 on∂Ω andvε∈L2(0,T;H1(Πε)).
Using our assumptions on the data and Cauchy-Schwartz, Poincar´ e’s, Young’s and Gronwall’s inequalities, we
can obtain suitable energy estimates, independent of ε, for our solution (see [ 2], [13], [9], [15] and [ 20]). More
precisely, if we multiply the first equation in (1) by uε, the second one by vεand we integrate formally by parts,
we obtain, for 0 <t<T ,
Zt
0Z
ΩεAε
1∇uε· ∇uεdxdτ +Zt
0Z
ΠεAε
2∇vε· ∇vεdxdτ +Zt
0Z
Ωεβ(uε)uεdxdτ +
εα
2Z
Sε(vε−uε)2dσ+εZt
0Z
Sεg(vε−uε)(vε−uε)dσdτ =
5

Multiscale Analysis in Nonlinear Thermal Diffusion Problems in Composite Structures
εα
2Z
Sε(c0)2dσ+Zt
0Z
Ωεfuεdxdτ +Zt
0Z
Πεfvεdxdτ. (4)
Therefore, using the conditions imposed for βandgand our hypotheses on the data, we have:
Zt
0Z
ΩεAε
1∇uε· ∇uεdxdτ +Zt
0Z
ΠεAε
2∇vε· ∇vεdxdτ +εZ
Sε(vε−uε)2dσ≤C, (5)
whereCis independent of ε.
The above computations can be made perfectly rigorous by passing to the limit in the analogous stability estimates
for a suitable regularized or discretized system (see [ 13]).
As a matter of fact, one can obtain the needed a priori estimates by writing the variational formulation of problem
(1) in a compact form. To this end, let us introduce the following space:
Vε={wε= (uε,vε)∈(H1(Ωε)∩H1
0(Ω))×H1(Πε)|vε−uε∈L2(Sε)}
and let us take w= (u,v)∈ Vεand the test function w= (u,v)∈ Vε. If we introduce the bilinear forms
aε(w,w) =Z
ΩεAε
1∇u· ∇udx+Z
ΠεAε
2∇v· ∇vdx,
bε(w,w) =εZ
Sε(v−u)(v−u)dσ,
and
Fε(w,w) =εZ
Sεg(v−u)(v−u)dσ+Z
Ωεβ(u)udx,
and we adopt the point of view to consider uεandvεas time dependent functions with values in a space dependent
functional space (see [ 13]), we can write the variational formulation of problem (1) in the following compact form:
d
dtbε(wε,w) +aε(wε,w) +Fε(wε,w) =Z
Ωεfudx+Z
Πεfvdx,
bε(wε(0),w) =bε(wε
0,w),
wherewε= (uε,vε),w∈ Vε, with Λε(w)<+∞andwε
0= (uε
0,vε
0)∈ Vε,(uε
0−vε
0)|Sε=c0, Λε(wε
0))<+∞and
aε(wε
0) = min {aε(wε)|(uε−vε)|Sε=c0}.
Here,
Λε(w) =εZ
Sεφ(v−u)dσ,
φ(x) =Zx
0g(ξ)dξ
and
aε(w) =aε(w,w).
6

Claudia Timofte
If we denote
bε(w) =bε(w,w),
one can prove (see, for details, [ 13]) that there exists a unique solution wε= (uε,vε)∈C0([0,T];Vε) of the above
problem and, moreover, we have the following estimates:
sup
t∈[0,T](bε(wε) + Λε(wε) +aε(wε))≤C(bε(wε
0) + Λε(wε
0) +aε(wε
0)),
ZT
0bε(∂t(vε−uε))dt≤C(bε(wε
0) + Λε(wε
0) +aε(wε
0)),
for a constant Cindependent of εand the initial data.
Let us take now Φ 1∈C∞
0(Ω) = D(Ω),Φ2∈C∞
0(Ω) andψ∈C∞
0((0,T)) =D((0,T)). We have:
ZT
0Z
ΩεAε
1∇uε· ∇Φ1ψdxdt +ZT
0Z
ΠεAε
2∇vε· ∇Φ2ψdxdt +ZT
0Z
Ωεβ(uε)Φ1ψdxdt +
αεZT
0Z
Sε(uε−vε)(Φ2−Φ1)dψ
dtdσdt+
εZT
0Z
Sεg(vε−uε)(Φ2−Φ1)ψdσdt =ZT
0Z
ΩεfΦ1ψdxdt +ZT
0Z
ΠεfΦ2ψdxdt. (6)
As already mentioned, we know that there exists a unique weak solution ( uε,vε) of (6). Denoting by Pεuε
andPεvεthe extensions of uεand, respectively, vεto the whole of Ω (see [ 8] and [ 13]), we can see that Pεuε
andPεvεare bounded in L2(0,T;H1
0(Ω)) (see, for details, [ 9], [13], [15] and [ 20]). So, by passing to a subse-
quence, we have Pεuε/arrowrighttophalf u weakly inL2(0,T;H1
0(Ω)). Also, Pεuε/arrowrighttophalf u weakly inL2(Ω×(0,T)) and strongly
inL1
loc(0,T;L1(Ω)) (see [ 2], Remark 2.1, Lemma 7.1 and Lemma 7.4). In a similar manner, we obtain that
Pεvε/arrowrighttophalf v weakly inL2(0,T;H1
0(Ω)), weakly in L2(Ω×(0,T)) and strongly in L1
loc(0,T;L1(Ω)). Moreover,
u, v∈C0([0,T];H1
0(Ω)) and∂
∂t(u−v)∈L2(0,T;L2(Ω)) (see [ 13]).
It is well-known by now how to pass to the limit, with ε→0, in the linear terms of (6) defined on Ωεand Πε
(see, for instance [ 9], [12], [15] and [ 20]). Also, recall that θis the weak- ⋆limit inL∞(Ω) of the characteristic
functionχεof the domain Ωε. Thus, we get:
ZT
0Z
ΩεAε
1∇uε· ∇Φ1ψdxdt →ZT
0Z
ΩA1∇u· ∇Φ1ψdxdt (7)
andZT
0Z
ΠεAε
2∇vε· ∇Φ2ψdxdt →ZT
0Z
ΩA2∇v· ∇Φ2ψdxdt, (8)
for any Φ 1,Φ2∈ D(Ω), ψ∈ D((0,T)).
Let us see now how we can pass to the limit in the nonlinear terms in (6).
7

Multiscale Analysis in Nonlinear Thermal Diffusion Problems in Composite Structures
For the third term in the left-hand side of (6), let us notice that, exactly like in [ 9], one can prove that for any
zε/arrowrighttophalfz weakly inH1
0(Ω), we get β(zε)→β(z) strongly in Lq(Ω), where q=2n
q(n−2) +n. Therefore, since for t
fixed,Pεuε(·,t) converges weakly in H1
0(Ω) (see [ 13]), it follows that
Z
Ωεβ(uε)Φ1dx→Z
Ωβ(u)θΦ1dx,
for any Φ 1∈ D(Ω). Using our hypotheses on the data, we see that, for any t,/bardblβ(uε)/bardbl2
L2(Ωε)≤C,/bardblβ(Pεuε)/bardbl2
L2(Ω)≤
C, withCdepending on Tand the data, but independent of ε. Here, byCwe have denoted different constants,
which can vary from line to line.
Using Lebesgue’s convergence theorem, we have
ZT
0Z
Ωεβ(uε)Φ1ψdxdt →ZT
0Z
Ωβ(u)θΦ1ψdxdt, (9)
for any Φ 1∈ D(Ω) and for any ψ∈ D((0,T)).
For passing to the limit in the other terms in (6), let us remember a result from [ 5].
Let us introduce, for any h∈Ls/prime(Γ), 1 ≤s/prime≤ ∞, the linear form µε
honW1,s
0(Ω) defined by
/angbracketleftµε
h,ψ/angbracketright=εZ
Sεh(x
ε)ψdσ ∀ψ∈W1,s
0(Ω),
with 1/s+ 1/s/prime= 1. It is proven in [ 5] that
µε
h→µhstrongly in ( W1,s
0(Ω))/prime, (10)
where
/angbracketleftµh,ψ/angbracketright=µhZ
Ωψdx,
with
µh=Z
Γh(y)dσ.
Ifh∈L∞(Γ) or even if his constant, we have (see [ 6])
µε
h→µhstrongly in W−1,∞(Ω). (11)
Also, for obtaining the limit behavior of our homogenization problem, let us recall another result from [ 9]. Let
Hbe a continuously differentiable function, monotonously non-decreasing and such that H(v) = 0 if and only if
v= 0. We shall suppose that there exist a positive constant Cand an exponent q, with 0 ≤q<n/ (n−2), such
8

Claudia Timofte
that|H| ≤C(1 +|v|q). If we denote by q=2n
q(n−2) +n, one can prove (see [ 9]) that for any zε/arrowrighttophalfz weakly in
H1
0(Ω), we get
H(zε)/arrowrighttophalfH (z) weakly in W1,q
0(Ω). (12)
Now, using (12) written for gand the convergence (11) written for h= 1, we obtain that
εZ
Sεg(vε−uε)(Φ2−Φ1)dσ→ |Γ|Z
Ωg(v−u)(Φ2−Φ1)dx,
for any Φ 1,Φ2∈ D(Ω). Integrating in time and using Lebesgue’s convergence theorem, it is not difficult to see
that
εZT
0Z
Sεg(vε−uε)(Φ2−Φ1)ψdσdt → |Γ|ZT
0Z
Ωg(v−u)(Φ2−Φ1)ψdxdt. (13)
Also, we have
αεZT
0Z
Sε(uε−vε)(Φ2−Φ1)dψ
dtdσdt→α|Γ|ZT
0Z
Ω(u−v)(Φ2−Φ1)dψ
dtdxdt. (14)
Since it is obvious how to pass to the limit in the other linear terms in (6), putting together (7)-(9) and (13)-(14),
we can pass to the limit in all the terms in (6) and, using standard density arguments, we obtain exactly the
variational formulation of the limit problem (3). Also, we can easily pass to the limit, with ε→0, in the initial
condition and we have
u(0,x)−v(0,x) =c0(x), x∈Ω.
Asuandvare uniquely determined (see [ 13]), the whole sequences PεuεandPεvεconverge to uand, respectively,
v, and this completes the proof of Theorem 2.1.
Acknowledgments This work was supported by the CNCSIS Grant Ideas 992, under contract 31/2007.
References
[1]M. Amar, D. Andreucci, P. Bisegna, R. Gianni, Nonlinear Analysis: Real Word Applications (6), 367-380
(2005)
[2]M. Amar, D. Andreucci, P. Bisegna, R. Gianni, Math. Mod. Meth. Appl. 14(9), 1261-1295 (2004)
[3]G. I. Barenblatt, Y. P. Zheltov, I. N. Kochina, Prikl. Mat.Mekh. 24, 852-864 (1960)
[4]H. Br´ ezis, J. Math. Pures et Appl. 51 (1), 1-168 (1972)
[5]D. Cioranescu, P. Donato, Asymptotic Analysis 1, 115-138 (1988)
[6]D. Cioranescu, P. Donato, H. Ene, Mathematical Methods in Applied Sciences 19, 857-881 (1996)
[7]D. Cioranescu, P. Donato, R. Zaki, Asymptotic Anal. 53 (4), 209-235 (2007)
9

Multiscale Analysis in Nonlinear Thermal Diffusion Problems in Composite Structures
[8]D. Cioranescu, J. Saint Jean Paulin, J. Math. Anal. Appl. 71, 590-607 (1979)
[9]C. Conca, J.I. D´ ıaz, C. Timofte, Math. Models Methods Appl. Sci. (M3AS) 13 (10), 1437-1462 (2003)
[10]C. Conca, P. Donato, RAIRO Mod´ el. Math. Anal. Num´ er. 22, 561-607 (1988)
[11]C. Conca, F. Murat, C. Timofte, ESAIM: Mod´ el Math Anal Num´ er (M2AN) 37 (5), 773-806 (2003)
[12]H. I. Ene, D. Polisevski, Z. angew. Math. Phys. 53 (6), 1052-1059 (2002)
[13]M. Pennacchio, G. Savar´ e, P.C. Franzone, SIAM J. Math. Anal. 37 (4), 1333-1370 (2005)
[14]L. Tartar, Probl` emes d’homog´ en´ eisation dans les ´ equations aux d´ eriv´ ees partielles (in Cours Peccot, Coll` ege
de France, 1977)
[15]C. Timofte, Acta Mathematica Scientia 29B (1), 74-82 (2009)
[16]C. Timofte, Z.Angew.Math.Mech. 87 (6), 406-412 (2007)
[17]C. Timofte, Acta Physica Polonica B 38 (8), 2433-2443 (2007)
[18]C. Timofte, Acta Physica Polonica B 39 (11), 2811-2822 (2008)
[19]M. Veneroni, Math. Meth. Appl. Sci. 29 (14), 1631-1661 (2006)
[20]W. Wang, J. Duan, Comm. Math. Phys. 275, 163-186 (2007)
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