HOMOGENIZATION OF THE LANDAU-LIFSHITZ-GILBERT EQUATION IN A CONTRASTED COMPOSITE MEDIUM

We study the Landau-Lifshitz-Gilbert equation in a composite ferromagnetic medium made of two different materials with highly contrasted properties. Over the so-called matrix domain, the effective field, the demagnetizing field and the bulk anisotropy field are scaled with regard to a parameter representing the size of the matrix blocks. This scaling preserves the physics of the magnetization as tends to zero. Using homogenization theory, we derive the corresponding effective model. To this aim we use the concept of twoscale convergence together with a new homogenization procedure for handling with the nonlinear terms. More precisely, an appropriate dilation operator is applied in a embedded cells network, the network being constrained by the microscopic geometry. We prove that the less magnetic part of the medium contributes through additional memory terms in the effective field. 1. Setting of the problem. Heterogeneous media are commonly adopted for electromagnetic applications in many branches of industry and science due to their ability to be tailored to meet specific requirements. For the reduction of eddy current loss, medium to high frequency components for electrical and electronic devices are frequently composed of heterogeneous soft magnets. Examples are the Mn-Zn ferrites, widely used in power electronics for transformers and inductor cores, or the soft magnetic composites, very promising for high speed electrical machines (see for instance part 1 in [3]). All these materials are designed for having both good magnetic properties and a quite high macroscopic resistivity. Nevertheless the interpretation of the experimental data is very difficult because of their sensitivity to many error sources. The development of analytical models for the determination of the 2010 Mathematics Subject Classification. Primary: 35Q60, 35B27; Secondary: 35K55, 82D40, 78A25.

1. Setting of the problem.Heterogeneous media are commonly adopted for electromagnetic applications in many branches of industry and science due to their ability to be tailored to meet specific requirements.For the reduction of eddy current loss, medium to high frequency components for electrical and electronic devices are frequently composed of heterogeneous soft magnets.Examples are the Mn-Zn ferrites, widely used in power electronics for transformers and inductor cores, or the soft magnetic composites, very promising for high speed electrical machines (see for instance part 1 in [3]).All these materials are designed for having both good magnetic properties and a quite high macroscopic resistivity.Nevertheless the interpretation of the experimental data is very difficult because of their sensitivity to many error sources.The development of analytical models for the determination of the effective properties of heterogeneous materials has a long tradition.More recently, thanks to the progress in computing power, the modeling of the electromagnetic behavior of heterogeneous media has been also faced by numerical approaches.As a drawback, the numerical implementation leads to exorbitant computational burden when fine spatial discretization have to coexist with the macroscopic sample size.
In the general context of problems described by differential equations in finely periodic structures, homogenization techniques have been widely applied to determine effective properties.Anyway, such asymptotic processes are performant and quite straightforward when dealing with global physical phenomena, while they show some limits when local effects are not negligible, even if local correctors are considered.
The main objective of this paper is to perform a rigorous derivation of the homogenized Landau-Lifshitz-Gilbert (LLG) equation associated to a highly contrasted composite ferromagnetic material.This is a typical example where a nonlinear and multiscale problem leads to difficulties for the justification of the effective model.We develop a new method, based on the use of an appropriate sequence of embedded cells together with dilation operators.The homogenization of the LLG equation is seldom addressed.A layered ferromagnetic medium was considered in [14].The effective behavior of the demagnetization field operator in periodically perforated domains is studied in [23] using the classical two-scale convergence method.Some nonlinear terms in the LLG model have the same structure than ones of the Ginzburg-Landau functional.We thus also mention [18], [21], [7] and the references therein.
The ferromagnetic medium is assumed to have two distinct components.The matrix part consists of disjoint blocks where the dynamics are slow, surrounded by a thin layer of another material with better magnetic properties.More precisely, the medium occupies the set Ω ⊂ R 3 which is assumed to be a bounded, two-connected domain with a periodic structure controlled by a parameter > 0 which represents the size of each block of the matrix (see also Figure 1).The standard period is a cell Q consisting of a two-connected matrix block Q m with external smooth boundary ∂Q m , surrounded by a two-connected domain Q f .The -composite medium consists of copies Q covering Ω.We denote by ∂Ω the external (Lipschitz) boundary of Ω, by γ the matrix boundary and by n and ν the corresponding exterior normals.The exterior normal to Q m will be denoted by ν.For any > 0, we denote by Ω m the matrix part of the domain and by Ω f the other part, so that where A is an appropriate infinite lattice.We denote by J = (0, T ) the time interval of interest, T > 0. For the sake of the simplicity we assume |Q| = 1, more precisely Let us now describe the PDEs system modeling the behavior of the magnetization in such a medium.The magnetization vector M ∈ R 3 is in the form where χ m (resp.χ f ) is the characteristic function of Ω m (resp.Ω f ).The magnetization is associated with the nonconvex constraint The time evolution of the magnetization vector may be described by the LLG equation ( [2], [15]): The term parameterized by a factor α describes Gilbert damping torque and the right-hand side accounts for torque by the effective field H e (M ) which is given by Tensor A satisfies where We also assume that A m is an admissible test function for the two-scale convergence (in the sense of [25]).The term φ va expresses the effects of the volume anisotropy energy.It reads where K v > 0 is a scalar bounded function and the constant vector u is the direction of the easy magnetization axis.In what follows we thus simply assume that function φ va,m being moreover periodic with regard to its second variable.In the magnetostatic approximation context ( [5]), the demagnetizing field H d (M ) satisfies, in J × R 3 , the equation curlH d = 0 and the stray field equation where µ is the permeability.Classical models keeps the latter equations or simply assume that H d is some potential depending on M .We consider both of these modelings by assuming where each continuous gradient function satisfies 0 , and where For the sake of the simplicity, we have assumed a constant permeability.We shall consider a potential formulation of this problem.Indeed, due to (5), there exists scalar potentials p and P such that In view of (6), (p , P ) is defined by We complete the model with initial, boundary and transfer conditions.The initial data for the magnetization is The stay field equation ( 7) is completed by an initial condition The external boundary condition is a no-flux type At the interface Γ between the two parts of the composite medium, we assume the continuity of the magnetization m = M on J × Γ (10) and the conservation of the scaled fluxes across Γ as follows Gathering all these elements, we finally get the following system The aim of the paper is to derive an effective (homogeneous) model for this composite microscopic problem, by letting → 0. We prove that it is still a Landau-Lifshitz-Gilbert equation, but with a new source term which is a memory term produced by the slow dynamics part of the microscopic model.On the contrary, the structure of the associated stray field equation is not modified by the matrix part of the microscopic model.
We use various tools of the homogenization theory.We begin by exploiting the periodic structure of the problem through two-scale convergence arguments ( [20,4]).The process let us exhibit the existence of memory terms due to the less conductive part of the domain in the effective model.But the -scaling in the matrix part of (13) clearly does not allow to get compactness results and to pass to the limit in the nonlinear terms.We thus adopt another approach.On the one hand, we introduce a dilation operator, in the spirit of the periodic unfolding method of e.g.[11].The -scaling disappears, at the expense of doubling the space dimension.Classical compactness results thus remain inaccessible.On the other hand, we thus exploit the periodic structure in a new way.It is based on the intuition that the lattice of matrix blocks tends to a set of points which is dense in Ω as tends to zero.Around any of these points, we succeed in constructing a sequence of embedded grids where we restrict the dimension and pass to the limit.We finally show that the obtained information is sufficient to identify the limit problem in the whole space.The method is original, even if a so-called density argument was already mentioned (but not detailed neither used) in [8].
The outline of this work is the following.The effective model is provided in the next section.The remaining part of the paper consists in its justification.For the sake of completeness, in Section 3, we begin by checking that the effective model may be computed through formal asymptotic expansions.Section 4 is devoted to the rigorous justification of the upscaling process, namely by proving that a subsequence of solutions of Problem (13) converges in some sense to a solution of the effective model.After stating uniform estimates, we apply two-scale convergence results to some extension of the solution in the most conductive part of the domain, Ω f .Next we introduce the dilation operator, the embedded grids approach and the 'density' arguments for the solution in Ω m .
2. Main result.Let (v j ) j=1...3 and (w j ) j=1...3 respectively the Q-periodic solutions of the following problems where the vector e j , 1 ≤ j ≤ 3, is the jth unit vector of the canonical orthonormal basis.We define A H and W H by The effective magnetization vector M and the effective demagnetizing field P satisfy Moreover the source terms involving m 0 and p 0 are computed thanks to the following problem: The problem is completed by the initial conditions: Remark 1. Inspection of the effective model reveals that the resulting homogenized problem is a LLG type model that contains a term representing memory effects which could be seen as a new magnetic excitation in the effective field.The memory term is induced by the slow dynamics part of the model, and it appears solely in the magnetization equation.The limiting stray field equation also depends on a new permeability, namely a kind of averaged permeability.
The effectiveness of the latter model is justified by a convergence result.Namely, we use the concept of two-scale convergence introduced by G. Nguetseng [20] and developed by G. Allaire [4].We refer to [22] (see Subsection 2.5.2) for the timedependent settings.Let Ω be an open subset in R 3 .A sequence of functions (v ) in L 2 (Ω ×J) is said to two-scale converge to a limit v 0 belonging to The convergence result is denoted by v 2 v 0 .
We have the following properties (see [4]).
Proposition 1. (i) From each bounded sequence (v ) in L 2 (Ω × J) we can extract a subsequence which two-scale converges.(ii) Let (v ) be a bounded sequence in H 1 (Ω × J) which converges weakly to v in H 1 (Ω ×J).Then (v ) two-scale converges to v and there exists a function Then for any sequence (w ) Remark 2. Choosing Ω = Ω (resp.Ω = R 3 ) in the definition and the properties above, we obtain the functional setting which is well suited for the study of the magnetization vector M (resp. of the demagnetizing field H ).
The main result of the paper is the following.
Theorem 2.1.Let (M , H ) be a solution of Problem (13) for > 0. There exists a subsequence of an appropriate extension of χ f (M , H ) on the one hand, and of (M , H ) on the other hand, which two-scale converges to a solution (M, P ) and (m 0 , p 0 ) of the effective model ( 18)-( 24).
3. Formal asymptotic expansions.In the present section, purely formal computations are developed for the guess of the effective model.These formal results are made rigorous by the limit process stated and proved in the next section.We now use formal asymptotic expansions.It means that, setting y = x/ for the fast space variable, we assume the following forms for the solutions: where we have denoted by χ f (resp.χ m ) the characteristic function of Q f (resp.Q m ).We insert these expansions in (13).Selecting the terms according to the powers of epsilon, we obtain the following cascade of equations.First, we consider the constraint (1).Whatever f = M or f = m , we infer from (1) that Next, terms of order −2 , −1 and 0 of (2) in Ω f × J give the following three equations: The same process in Ω m × J gives: The expansion of the boundary conditions for M and m on Ω × ∂Q m × J leads to: The same work on the equations characterizing the demagnetizing field gives: the first three equations being satisfied in Now we exploit the latter equations.First we infer from (27) completed with (32) that M 0 does not depend on the fast variable y.The same holds true for P 0 in view of ( 35) and (41): Then we characterize function M 1 .On the one hand, the variational formulation corresponding to (28) with (33) is On the other hand, in view of assertions (25), we also have where functions w j are defined in (14) and α is some function which does not depend on y.The first term on the right-hand side of equation (29) now reads Similarly, we infer from (36) and (42) that functions w j being defined by (15), and the two first terms in (37) read Next step consists in integrating over Q the equations characterizing the main order terms of the expansions, that is (29) and (37), in view of obtaining the effective model.In view of the latter computations, we get at first glance the following two equations in Ω × J: and (44) We also have, in view of (34), and, in view of (40), Now, setting M 0 = M and P 0 = P , we notice that the effective model corresponds to (43)-( 46) and (30), (38).
4. Rigorous derivation of the effective model.

4.1.
Uniform estimates on the microscopic model.The existence of a weak solution for (13) may be stated using classical arguments for this type of problem (see e.g.[5] and the references therein).Moreover we have the following estimates 2 with regard to the scaling parameter .
Proposition 2. Assume that M init ∈ H 1 (Ω).Then any weak solution of problem (13) satisfies the following estimates: 2 All along the paper, letter C denotes some generic constant.
Proof.The techniques used in the proof are similar to those developed for instance in [15].We rewrite the LLG equation in the form Multiplying ( 47) by (α∂ t M − (1 + α 2 )H e (M ) , we get Integrating with respect to space and then with respect to time, we obtain for any t ≥ 0, where the energy E(M ) is defined by The hypothesis M init ∈ H 1 (Ω) ensures that the initial energy E(M init ) is bounded (see for example [16]).Proposition 2 is proved.
In view of exploiting the a priori estimates obtained in Ω f which is an -dependent domain, we first need to extend the functions M and χ Ω P to the whole fixed domain Ω.To this aim, we first check that the structure of Ω = Ω f ∪ Γ ∪ Ω m satisfies the assumptions in [1].We then can claim that there exist three real numbers k i = k i (Q f ) > 0, i = 1, 2, 3, and a linear and continuous extension operator Π : Here Ω( k 1 ) = {x ∈ Ω : dist(x, ∂Ω) > k 1 }.To avoid dealing with boundary layers, we make the following additional assumption on the structure of the domain Ω : It means that we assume that the blocks are removed in an k 1 -neighborhood of ∂Ω.Therefore estimates in Proposition 2 lead to 4.2.Exploitation of two-scale arguments.We infer from the estimates listed in Proposition 2 and the estimates (49) the following convergences results.Proposition 3.There exist limit functions M ∈ H 1 (J; ) on the one hand, and ) on the other hand, such that, for some subsequence not relabelled for convenience, the following convergence results hold true: )), ∀r ≥ 1, and a.e. in Ω × J, χ R 3 \Ω P + Π (χ Ω P ) P weakly in L 2 (J; H 1 (R 3 )), We now aim using the latter convergence results to pass to the limit → 0 in (13).First of all, notice that ( 23) is a direct consequence of the definition of the two-scale limits (M, m 0 ) and (P, P 0 ).Now, we exploit the constraint equation ( 1) in the following auxiliary lemma.Proof.First we look for the derivative of the limit constraint.On the one hand, due to the a.e.convergence of Π M , we know that χ f |M | 2 χ f (y)|M | and thus On the other hand, since We conclude from (50)-(51 Deriving the latter relation with regard to x i , for any 1 ≤ i ≤ 3, we compute that in Ω × J. Thus the first result announced in the lemma.Now we look for the limit of the derivative of the constraint.Let 1 We now pass to the two-scale limit in the part of ( 13) that only contains linear operators.Let Ψ ∈ D(R 3 × J), Notice that these functions are test admissible for the two-scale convergence ( [25]).We have Letting → 0 in the latter relation we get Thanks to classical density arguments, the latter relation holds true for any Ψ ∈ L ∞ (J; ) such that ψ(x, y, t) = 0 if y ∈ Q f .Choosing Ψ 1 = 0 and ψ = 0 in (52), bearing in mind that Q ∇ y P 1 dy = 0 thanks to the Q-periodicity of P 1 , we recover the variational formulation of the following problem , where β is some function which does not depend on y and where w j , 1 ≤ j ≤ 3, is defined in (15).Therefore where W H is defined in (17).Equation ( 53) is thus actually (20) in the effective model.Finally, choosing Ψ = Ψ 1 = 0, we recover (22).
The same type of computations for the limit behavior of ( 2)-( 4), ( 8)- (11) give a more frustrating result because of the numerous nonlinearities.More precisely, we get Choosing Ψ = Ψ 1 = 0 in (54), we obtain the following "characterization" for m 0 in (55) Next, choosing Ψ 1 = ψ = 0 in (54) and using Lemma 4.1 for simplifying the vectorial product in the boundary condition, we assert that Using Ψ = ψ = 0 in (54) we get moreover for M 1 which leads to express M 1 using (v i ) 1≤i≤3 (see the characterization of M 1 in Section 3).Thus ( 56)-(57) actually reads (bearing also in mind (55) for the expression of the non-explicit term): In the next subsection, we introduce another strategy for computing the nonexplicit terms i , 1 ≤ i ≤ 3. Nevertheless, we already can prove that the effective problem, even with this partial formulation, is well-posed.Lemma 4.2.Problem (58)-( 59) associated with (55), ( 23) and ( 24) admits a weak solution.
Proof.For our purpose, we look for a regularity result for the term Qm div y ( 3 )dy.Since all the two-scale limits are defined in L 2 (Ω × J × Q m ), we know that equation ( 55) is satisfied in H −1 (Ω × J × Q m ).Then we can write, for any ϕ ∈ H 1 0 (J), We conclude that J div y ( 3 )dt has the same regularity than We are allowed to make the following computation The source term Qm div y ( 3 )dy in (58) thus belongs to L 2 (Ω × J).The existence of some weak solution to (58)-( 59), ( 23)-( 24) is then ensured by the classical parabolic theory.

4.3.
Exploitation of an appropriate dilation operator.It remains to pass to the limit in the nonlinear matrix terms of the problem for giving an explicit form to the terms i , 1 ≤ i ≤ 3, in (55).We thus have to use another technique than the two-scale convergence.A first idea consists in introducing a dilation operator for upscaling the fast variable x/ and thus removing the -weight in the H 1 estimates.Such an operator was formally used in [6].It is also behind the periodic unfolding method of Cioranescu et al [11].For each > 0, we define a dilation operator • mapping measurable functions on Ω m × J to measurable functions on where c (x) denotes the lattice translation point of the -cell domain containing x.This dilation annihilates the scaling distinction between the slow variable x and the fast variable y = x/ .Assume for instance a simple but not restrictive description of the periodic structure of Ω , more precisely A = Z 3 and The dilation operator has the following properties (see [6]).
This subsection is not completely disconnected from the latter one.Indeed, as emphasized in the following result, the limiting process based on two-scale convergence and the one based on weak convergence of dilated sequences are equivalent (see [8]).
and χ m v two-scale converges to v 0 , then we have ṽ = v 0 a.e. in Ω × J × Q m .
It means that for computing the non-explicit terms in (55), it is "sufficient" to fully characterize the weak limit of ( m , p ), m (resp.p ) being the dilated magnetization vector (resp.field potential).It is thus natural to write the equations satisfied by ( m , p ). Lemma 4.3.The dilated quantities ( m , p ) satisfy the following set of equations which are satisfied in L 2 (J; H −1 (Q m )) for almost every x ∈ Ω m .The boundary and initial conditions are Proof.We detail for instance the derivation of the equation (60) satisfied by m .The derivation of the one for p follows the same lines.For any given ψ ∈ L 2 (J; We multiply (2) by ψ such that ψ |t=T = 0. We integrate over Ω m .We recall that ).We thus get, for almost every x ∈ Ω m : We introduce the change of variable z → (y + k).We obtain The latter is the variational formulation of (60) with the initial condition m |t=0 = M init .We give some precisions about the boundary condition.Of course we can enlarge the definition of the dilation operator to a subset of Ω( k 1 ) strictly containing Ω m .This gives sense to the boundary condition m = M on ∂Q m × J.The result has been established for almost every x ∈ (Q m + k) and for all k ∈ Z 3 .Then it is valid almost everywhere in Ω m .
The good point in (60)-( 63) is clearly that the -scaling does not appear anymore.The uniform estimates leading to the following convergences, possibly for subsequences not relabeled for convenience, are thus straightforward: Notice that we have used Proposition 5 to ensure that the limit functions (m 0 , p 0 ) appearing here are actually the same than the ones already defined in Proposition 3. Moreover the equation satisfied by p 0 has already been derived in the latter subsection.Nevertheless, we still do not have any compactness result for m (x, y, t) because we have no information on the boundedness of its partial derivative with respect to x.This difficulty also appeared in [8] and [19].These authors solved it either by comparing the dilated problem with their formal guess for the limit problem ( [8]) or by proving that they actually deal with a Cauchy sequence ( [19]).
The complex structure of our equation does not allow such approaches.We thus adopt another method and we develop rigorously an idea already present in [10].Due to the definition of the dilation operator, one checks easily that the dilated functions restricted to a given matrix cell of Ω do not depend on x.Let k ∈ Z 3 .Let ( m k , p k ) be defined by and which let us define m k and p k .
For any > 0 such that (Q m + k) ∩ Ω = ∅, ( m k , p k ) is clearly a solution of ( 60)-(63) in Q m × J. On the other hand, any f k associated with some Thus, we have enough regularity properties to get with (60)-( 63) the same estimates for ( m k , p k ) than the ones obtained for ( m , p ).But here the estimate of ∇ y m k gives a uniform bound in H 1 (Q m ) for m k and thus enough compactness results to pass to the limit → 0 in (60).Furthermore, notice that k ., 3 (where |Ω| i denotes here the value of the measure of Ω in the i th direction).Then, for any k ∈ Z 3 , there exists (k) > 0 such that for any < (k), (Q m + k) ∩ Ω = ∅.Denoting by ( m k , p k ) the limit in L 2 (Q m × J) of ( m k , p k ), we get the following system: Another basic idea is that the subgrid defined in Ω by {k ∈ Z 3 ; (Q m +k)∩Ω = ∅} seems to become dense in Ω as → 0. Let us show that this point is sufficient to pass to the limit, at least in some part of the domain.Thanks to Section 2, we already know that our aim is to show that the limit (m 0 , p 0 ) also satisfies (64)-(65), that is ( 21)- (22).In brief, we are going to prove that m 0 (x, ) of problem (66) (see [17], [24], [12], [13]).
We define the set C ⊂ Ω by It means that C is the set of all points of Ω that are the center of an 0 -copy of Q (and thus of Q m ) at some step, 0 , of the convergence process → 0. We also define We have C = >0 C .We begin by restricting the limit process to the set C × J.To this aim, we develop our embedded grids approach.Let x 0 ∈ C.There exists some 0 > 0 such that x 0 ∈ Ω 0 m and x 0 is the center of an 0 -copy of Q.One checks easily3 that x 0 remains the center of an -copy of Q for any ≤ 0 .See also Figure 2.
We then can choose a particular numbering for the description of Ω , ≤ 0 : It means that for any ≤ 0 , x 0 is the center of the (0, 0, 0)th -copy of Q.We thus can exploit the latter remarks on the restricted functions m k for the value k = 0 = (0, 0, 0).We set for (y, t) ∈ Q m × J m 0x 0 (y, t) = m 0 (y, t) for the numbering (67).
Proof.The proof is a particular case of the derivation of ( 64)-(65), namely for k = 0. We thus know that completed by the initial and boundary conditions Indeed, as already mentioned, we can enlarge the definition of the dilation operator to a subset of Ω( k 1 ) strictly containing Ω m .This gives sense to the boundary condition m = M on ∂Q m × J.In particular The weak L 2 limit of M 0 is equal to the two-scale limit of M .Since Π M strongly converges in L 2 (Ω × J) to M , function M (which does not depend on y) is also the two-scale limit of the restriction M .Using the continuity of the trace operator, (69) gives at the limit → 0 the condition m 0 (y, t) = M (x 0 , t) on ∂Q m × J.We have proved that m 0 satisfies (66).The solution of (66) being unique for any fixed x = x 0 ∈ Ω, the whole sequence m 0x 0 converges to the solution of (66).This ends the proof of the lemma.

Conclusion.
Beyond the particular application considered in the present paper, we believe that the precise description of the embedded grids approach and of the density arguments coupled with the dilation method for the homogenization of nonlinear terms will be a very useful tool for many applications.This description is, as far as we know, completely original.

Figure 1 .
Figure 1.An example of periodic structure for the domain and the standard cell

Lemma 4 . 1 .
For any 1 ≤ i ≤ 3, the vectors ∂ xi M , ∂ xi M + ∂ yi M 1 and ∂ yi M 1 are perpendicular to the vector M almost everywhere in Ω × J × Q f .
M 1 is actually perpendicular to M .Due to the first part of the proof, the same holds true for ∂ yi M 1 .

L 2 #
(Q m )), lim →0 Qm×J m 0 ϕ dydt = lim →0 Qm×J m (x, y, t)χ |x0+ Qm (x) ϕ(y, t) dydt = Qm×J m 0 (x 0 , y, t) ϕ(y, t) dydt a.e.x 0 ∈ C.The limit behavior of ( m |x=x0 ) of course does not depend on the choice of the numbering of the -copies of Q in Ω m .Problem (66) thus characterizes the limit behavior of the restriction of m in C.This point is however not sufficient for our purpose.Indeed, on the one hand C is dense in Ω, but on the other hand the a.e.-convergence in C is not sufficiently meaningful since |C| = 0.The end of the paper consists in extending the result of the latter lemma from C to Ω. Let m * 0 be defined by m * 0 (x, y, t) = m * 0 (x)(y, t) where m * 0 (x) is defined by (66).Let us prove that we have actuallym 0 = m * 0 in L 2 (Ω × J; L 2 # (Q m )), that is lim →0 Ω×Qm×J ( m − m * 0 ) ϕ dxdydt = 0 (70) for any ϕ ∈ L 2 (Ω × J; L 2 # (Q m )), or, equivalently by density, for any compactly supported test function, ϕ ∈ C c (Ω × J; C # (Q m )).For the structuration of the paper, we announce this final result in the following lemma.Lemma 4.5.Let ϕ ∈ C c (Ω × J; C # (Q m )).Let η > 0. There exists > 0 such that for any < , Ω×Qm×J m − m * 0 ϕ dxdydt ≤ η.Proof.Let L = Ω×Qm×J m − m * 0 ϕ dxdydt.Function m being constant on each given -cell, we write L in the following form.