Increase of mass and nonlocal effects in the homogenization of magneto-elastodynamics problems

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Increase of mass and nonlocal effects in the

homogenization of magneto-elastodynamics problems

Marc Briane, Juan Casado-Diaz

To cite this version:

Marc Briane, Juan Casado-Diaz. Increase of mass and nonlocal effects in the homogenization of magneto-elastodynamics problems. Calculus of Variations and Partial Differential Equations, Springer Verlag, 2021, 60 (5), pp.article n°163. �10.1007/s00526-021-02027-0�. �hal-01825021v2�

Increase of mass and nonlocal effects in the

homogenization of magneto-elastodynamics problems

Marc BRIANE Juan CASADO-D´IAZ

Institut de Recherche Math´ematique de Rennes Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico Univ Rennes, INSA de Rennes, CNRS, IRMAR - UMR 6625 Universidad de Sevilla

[email protected] [email protected]

January 17, 2020

Abstract

The paper deals with the homogenization of a magneto-elastodynamics equation satis- fied by the displacementuεof an elastic body which is subjected to an oscillating magnetic fieldB_ε generating the Lorentz force∂_tu_ε×B_ε. When the magnetic fieldB_ε only depends on time or on space, the oscillations ofBεinduce an increase of mass in the homogenized equation. More generally, when the magnetic field is time-space dependent through a uni- formly bounded component G_ε(t, x) ofB_ε, besides the increase of mass the homogenized equation involves the more intricate limitgof∂_tu_ε×G_εwhich turns out to be decomposed in two terms. The first term of g can be regarded as a nonlocal Lorentz force the range of which is limited to a light cone at each point (t, x). The cone angle is determined by the maximal velocity defined as the square root of the ratio between the elasticity tensor spectral radius and the body mass. Otherwise, the second term of g is locally controlled inL²-norm by the compactness default measure of the oscillating initial energy.

Keywords: Magneto-elastodynamics, oscillating magnetic field, Lorentz force, homogeniza- tion, increase of mass, nonlocal effect

AMS subject classification: 74Q10, 74Q15, 35B27, 35L05

1 Introduction

In a insulating (vacuum-like) environment, an elastic three-dimensional body placed in an electric fieldE and a magnetic B is subjected to the Lorentz force (see, e.g., [2, Section 9.3])

f_L=ρ_e(E+v×B) +σ(E+v×B)×B, (1.1) where v is the velocity, σ the conductivity of the body and ρ_e is the density of free electrical charges, whileEandBsatisfy Maxwell’s system. In particular, the fieldsEandBare connected by the equation

curlE+∂_tB = 0.

In the present paper we focus on the magnetic Lorentz force∂tv×B rather than on the electrical force. We assume that

• the elastic body is a poor conductor, i.e. σ≈0,

• the electrical Lorentz force ρeE is negligible compared to the magnetic Lorentz force ρ_e(v×B),

which yields

f_L≈ρ_e(v×B). (1.2)

The second assumption holds in particular if E(t, x) =εe(t, x/ε) with 0 < ε1, since then O(ε) =E(t, x)B(t, x) = B(0, x)−

ˆ _t

0

(curle)(s, x/ε)ds=O(1).

Under these assumptions and setting ρ_e = 1, the displacement u of the body with velocity v =∂_tu, satisfies the “simplified” magneto-elastodynamics equation

ρ ∂_tt²u−Divx Ae(u)

+∂tu×B =f, (1.3)

where ρ is the mass density, A is the elasticity tensor of the body and e(u) is the symmetric strain tensor. The right-hand sidef encompasses all other body forces. Equation (1.3) can be extended to any dimension N ≥ 2, replacing the three-dimensional Lorentz force ∂_tu×B by B∂_tu, where B is now a N ×N skew-symmetric matrix-valued function.

In the framework of homogenization theory, our aim is to study the effect of a time-space oscillating magnetic field B_ε(t, x) on the magneto-elastodynamics equation (1.3).

LetT >0, let Ω be a bounded open set ofR^N andQ:= (0, T)×Ω. Consider the magneto- elastodynamics problem











ρ ∂_tt²u_ε−Div_x Ae(u_ε)

+B_ε∂_tu_ε =f_ε in Q u_ε = 0 on (0, T)×∂Ω

u_ε(0, .) = u⁰_ε, ∂_tu_ε(0, .) = u¹_ε in Ω,

(1.4)

whereB_ε is a skew-symmetric matrix-valued function in L^∞(Q)^N×N decomposed as

B_ε(t, x) = F_ε(x)+G_ε(t, x)+H_ε(t, x), with B_ε(0, x) =F_ε(x), G_ε(0, x) = H_ε(0, x) = 0, (1.5) f_ε ∈L¹(0, T;L²(Ω))^N, u⁰_ε ∈ H₀¹(Ω)^N, u¹_ε ∈L²(Ω)^N. Contrary to F_ε(x) the component G_ε(t, x) is assumed to be uniformly bounded with respect tot and x, but the time-space oscillations of G_ε(t, x) may produce a nonlocal effect. The component H_ε(t, x) is a compact perturbation of B_ε(t, x). Under suitable oscillations of the sequences B_ε,f_ε,u⁰_ε, u¹_ε, we can pass to the limit as ε tends to zero in (1.4) in order to derive the homogenized problem.

Homogenization of pde’s with variable coefficients has been the subject of numerous studies for bounded coefficients in the books [22, 12, 11, 1] and the references therein, [4, 18, 3] for periodically oscillating coefficients, as well as for non-uniformly bounded coefficients (from above or below) in the survey article [13]. Here, the asymptotic analysis of equation (1.4) involving the general sequence B_ε is in line with homogenization of pde’s with non-periodic coefficients which was initiated by Spagnolo [19] and Murat, Tartar [17]. More specifically, in the stationary case Tartar [20, 21] (see also [6] for an alternative approach) has studied the homogenization of the three-dimensional Stokes equation

−∆u_ε+b_ε×u_ε+∇p_ε=f in Ω, (1.6) perturbed by the oscillating drift term b_ε×u_ε representing the Coriolis force which plays an analogous role to the Lorentz force (1.2) in equation (1.3). To that end Tartar developed his

celebrated “oscillating test functions method” at the end of the Seventies, and he obtained a homogenized Brinkman [7] type equation

−∆u+b×u+∇p+M^∗u=f in Ω, (1.7)

where M^∗ is a non-negative symmetric matrix-valued function. If the magnetic field B_ε(x) is independent on time and T = ∞, a time Laplace transform of equation (1.4) leads us to an equation which is similar to (1.6). Therefore, Tartar’s homogenization result combined with an inverse Laplace transform should at the least modify the mass ρ in the homogenized equation of (1.4).

Alternatively, nonlocal effects without change of mass have been obtained in [8] for the homogenization of a scalar wave equation with a periodically oscillating matrix-valued function B_ε(t, x) = B(t, x, t/ε, x/ε), whereB(t, x, s, y) is bounded with respect to the variables (t, x) and periodically continuous with respect to the variables (s, y), using a two-scale analysis method.

In our non-periodic and vectorial setting we show that the time-space oscillations of the magnetic fieldB_ε(t, x) produce both an increase of mass and nonlocal effects through an abstract representation formula arising in the homogenized equation.

On the one hand, the first result of the paper is the derivation of an anisotropic effective mass%^∗ which is greater (in the sense of the quadratic forms) than the starting mass ρI_N. This increase of mass in the homogenization process is due to the oscillations of the magnetic field at the microscopic scale, which modify the linear momentum through the magnetic Lorentz force. At this point Milton and Willis [16] have explained the macroscopic change of mass obtained in composite elastic bodies at fixed frequency, by the existence of a hidden mass at the microscopic scale, which modifies Newton’s second law. From this observation, when the magnetic fieldB_ε is only time dependent, we can build an anisotropic internal massm_ε(t) such that in a multiplicative way

m_ε(t)∂_tu_ε ≈m^∗(t)∂_tu. (1.8) In contrast, when the magnetic field is independent of time,i.e. B_ε =F_ε which is assumed to converge weakly to zero inW^−1,p(Ω)^N×N for some p > N, we can build an anisotropic internal mass M_ε(x) =F_ε(x)W_ε(x) such that in an additive way

∂_tu_ε≈∂_tu+W_ε∂_tt²u. (1.9) The harmonic limit of m_ε(t) (due to the multiplicativity of (1.8)) or the arithmetic limit of M_ε(x) (due to the additivity of (1.9)) leads us to the anisotropic effective mass %^∗.

On the other hand, the second result of the paper shows that both time and space oscillations of the magnetic fieldB_ε(t, x) may also induce nonlocal effects which are absent if the magnetic field is only time dependent or only space dependent. Assuming that the componentG_ε of the magnetic field (see (1.5)) weakly converges to zero in L^∞(Q)^N×N and that H_ε is a compact perturbation, we prove (see Theorem 4.1, Theorem 5.9 and Theorem 5.14) that the limit g of the magnetic Lorentz forceG_ε∂_tu_ε admits the following decomposition

g = ˆ

Q

dΛ(s, y)∂_tu(s, y)−h₀, in Q. (1.10) First, the matrix-valued measure Λ in (1.10) can be regarded as the kernel of a nonlocal Lorentz force arising in the homogenized problem. The range of this nonlocal term is limited to each light cone ofQ, the angle of which is equal to 2 arctancwithc=p

kAk/ρ(kAkis the Frobenius norm of tensor A). A particular nonlocal term with some light cone range has been obtained

in the periodically oscillating case of [8]. The general nonlocal term (1.10) with the specific light cone range is thus substantial to the homogenization process in the present non-periodic setting. Next, we show (see Theorem 5.9, Corollary 5.11 and Corollary 5.12) that the second term h0 in (1.10) is locally controlled in L²-norm by the compactness default measure µ⁰ of the oscillating initial energy. The function h₀ acts as a new exterior force in the homogenized problem.

Therefore, collecting the two previous results we get that the homogenized problem of (1.4) can be written as











%^∗∂_tt²u−Divx Ae(u)

+H∂tu+g =f in Q u= 0 on (0, T)×∂Ω

u(0, .) =u⁰ in Ω,

(1.11)

and the initial velocity∂tu(0, .) actually depends on the effective mass %^∗. As a by-product of the energy estimate satisfied by the limitg, we obtain a corrector result for the homogenization problem (1.4) if the compactness default measureµ⁰ vanishes (see Remark 5.4). This holds in particular when the initial conditions are “well-prepared” (see Remark 5.1) in the spirit of the classical homogenization result [10] for the wave equation.

The paper is organized as follows:

In Section 2 we study the case where the magnetic fieldB_εonly depends on time. We derive (see Theorem 2.2) the homogenized problem (1.11) with the sole increase of mass (g = 0).

Section 3 is devoted to a stationary problem (see Theorem 3.1) which prepares the main homogenization result of the paper in Section 4. It is partly based on Tartar’s works [20, 21].

In Section 4 we consider a more general magnetic field B_ε satisfying (1.5). We prove (see Theorem 4.1) that the homogenized magneto-elastodynamics problem of (1.4) is (1.11).

Section 5 deals with several estimates of the limit g (see Theorem 5.3 and Theorem 5.9) and an abstract representation (see Theorem 5.14) which allow us to prove that the function g admits the decomposition (1.10). In some specific cases we get a complete representation of the functiong and the uniqueness of a solution to the limit problem (1.11) (see Corollary 5.11 and Corollary 5.12).

Notation

• Y¯ denotes the closure of a subset Y of a topological set X.

• A ∈ L(R^Ns ^×N;R^N×Ns ) is a positive definite symmetric fourth-order tensor, and kAk de- notes its Frobenius norm.

• |E| denotes the Lebesgue measure of a measurable set E of R^N.

• L(X;Y) denotes the space of continuous linear functions from the normed space X into the normed space Y.

• ·denotes the scalar product inR^N, : denotes the scalar product inR^N×N, and| · |denotes the associated norm in both cases.

• B(x, r) denotes the euclidien ball of center x ∈ R^N and of radius r > 0, and B_r simply denotes the ball B(0, r) centered at the origin.

• IN denotes the unit matrix of R^N×N, and M^t denotes the transposed of a matrix of M.

• R^Ns ^×N denotes the space of symmetric matrices of order N.

• Ω denotes a bounded open set of R^N forN ≥2, T >0, and Qthe cylinder (0, T)×Ω.

• Div denotes the vector-valued divergence operator taking the divergence of each row of a matrix-valued function.

• e(u) denotes the symmetrized gradient of a vector-valued function u.

• M(X) denotes the space of the Radon measures on a locally compact set X.

• C_c^∞(U) denotes the set of the smooth functions with compact support in an open subset U of R^N.

• D⁰(U) denotes the space of the distributions on an open subset U of R^N.

• →denotes a strong convergence, *a weak convergence, and *^∗ a weak-∗ convergence

• ,→ denotes a continuous embedding between two topological spaces.

• Oε denotes a sequence of ε which converges to zero as ε tends to zero, and which may vary from line to line.

• C denotes a positive constant which may vary from line to line.

2 Homogenization of an elastodynamics problem with a strong magnetic field only depending on time

Let Ω be a bounded open set of R^N with N ≥ 2, T > 0, Q = (0, T)×Ω, ρ > 0 and let A ∈ L(R^N×Ns ;R^N×Ns ) be a positive definite symmetric tensor. Let f ∈ L¹(0, T;L²(Ω))^N and let ß_ε∈C¹([0, T])^N×N be such that

ß^t_ε =−ß_ε and ß⁰_εß_ε = ß_εß⁰_ε in [0, T], ß_ε(0) = 0, (2.1) exp(−ρ⁻¹ß_ε)*^∗ M⁻¹ in L^∞(0, T)^N×N with M ∈C¹([0, T])^N×N (2.2) where ß⁰_ε denotes the time derivative of ß_ε and M is an invertible matrix-valued function. Here we need to assume that the weak limit of exp(−ρ⁻¹ß_ε) is an invertible matrix-valued function.

This is not automatically satisfied as shows the following example.

Example 2.1. LetN = 2 and ρ= 1. Consider the function b_ε defined by b_ε(t) =t/ε+b(t/ε) for t ∈ R, where b is a 2π-periodic function in C¹(R), and the skew-symmetric matrix-valued function

ßε :=

0 −b_ε b_ε 0

∈C¹(R)^2×2. We have

exp(−ρ⁻¹ß_ε) =

cos(b_ε) −sin(b_ε) sin(b_ε) cos(b_ε)

* 1

2π ˆ 2π

0

cos(t) cos(b(t))−sin(t) sin(b(t)) −cos(t) sin(b(t))−sin(t) cos(b(t)) cos(t) sin(b(t)) + sin(t) cos(b(t)) cos(t) cos(b(t))−sin(t) sin(b(t))

dt.

Hence, ifb= 0 then the weak limit of exp(−ρ⁻¹ßε) is the nul matrix. Otherwise, ifbis closed to the 2π-periodic function which agrees in [0,2π] with ^π₂ 1_[0,π], then the weak limit of exp(−ρ⁻¹ß_ε) is closed to the matrix

−1 π

1 −1

1 1

, and is thus invertible.

We consider the solution u_ε to the wave equation











ρ ∂_tt²u_ε−Div_x Ae(u_ε)

+ ß⁰_ε∂_tu_ε =f inQ u_ε = 0 on (0, T)×∂Ω

u_ε(0, .) =u⁰_ε, ∂_tu_ε(0, .) =u¹_ε in Ω,

(2.3)

where

u⁰_ε * u⁰ inH₀¹(Ω)^N, u¹_ε * u¹ inL²(Ω)^N. (2.4) We have the following homogenization result.

Theorem 2.2. Assume that conditions (2.1), (2.2), (2.4) hold true. Then, we have

u_ε* u^∗ in L^∞(0, T;H₀¹(Ω))^N ∩W^1,∞(0, T;L²(Ω)))^N, (2.5) where u is the solution to the equation











ρM^tM∂_tt²u−Div_x Ae(u)

+ρM^tM⁰∂_tu=f in Q u= 0 on (0, T)×∂Ω

u(0, .) = u⁰, ∂_tu(0, .) =M⁻¹(0)u¹ in Ω.

(2.6)

Remark 2.3. Since the matrix exp(ρ⁻¹ß_ε) is unitary, the lower semi-continuity of convex functionals yields for anyλ∈R^N and for any measurable set E ⊂Ω,

ˆ

E

M⁻¹λ·M⁻¹λ dx≤lim inf

ε→0

ˆ

E

exp(−ρ⁻¹ß_ε)λ·exp(−ρ⁻¹ß_ε)λ dx=|E| |λ|²,

which implies thatM⁻¹λ·M⁻¹λ ≤ |λ|² a.e. in Ω. Due to M= (M⁻¹)⁻¹, we equivalently get that for any λ ∈ R^N, Mλ·Mλ ≥ |λ|² a.e. in Ω. Hence, the homogenized equation (2.6) involves an effective anisotropic mass

ρM^tM≥ρ I_N a.e. in Ω,

which is greater than the initial oneρ. We will see in Section 4 that if we replace time oscillations by space oscillations, the homogenization process also induces a larger effective anisotropic mass but in quite a different way.

Proof of Theorem 2.2. First of all, since ß_ε is skew-symmetric, the classical estimates for the wave equation yield convergence (2.5) up to a subsequence.

By (2.1) we have exp(ρ⁻¹ßε)0

= ρ⁻¹ß⁰_εexp(ρ⁻¹ßε) = ρ⁻¹exp(ρ⁻¹ßε) ß⁰_ε. Hence, equa- tion (2.3) can be written as

ρ ∂_t exp(ρ⁻¹ß_ε)∂_tu_ε

−exp(ρ⁻¹ß_ε) Div_x Ae(u_ε)

= exp(ρ⁻¹ß_ε)f inQ, (2.7)

which implies that for any Φ∈C¹([0, T];C_c^∞(Ω))^N with Φ(T, .) = 0, recalling ßε(0) = 0, ˆ

Q

−ρexp(ρ⁻¹ß_ε)∂_tu_ε·∂_tΦ +Ae(u_ε) :e exp(−ρ⁻¹ß_ε)Φ dt dx

= ˆ

Ω

ρ u¹_ε·Φ(0, .)dx+ ˆ

Q

f· exp(−ρ⁻¹ß_ε)Φdt dx.

(2.8)

Forϕ∈C_c^∞(Ω)^N, define the function ξ_ε ∈L^∞(0, T)^N by ξ_ε(t) := ρexp ρ⁻¹ß_ε(t)

ˆ

Ω

∂_tu_ε(t, x)·ϕ(x)dx, for a.e. t∈(0, T). (2.9) By (2.7) and (2.5) we have

ξ_ε⁰ = ˆ

Ω

−Ae(u_ε) :e(exp(−ρ⁻¹ß_ε)ϕ) +f · exp(−ρ⁻¹ß_ε)ϕ

dx bounded in L^∞(0, T).

Hence, ξ_ε is bounded inW^1,∞(0, T), and up to a subsequence converges weakly-∗ to someξ in W^1,∞(0, T)^N. This combined with convergences (2.2) and (2.5) implies that

exp(−ρ⁻¹ß_ε)ξ_ε *^∗ M⁻¹ξ=ρ ˆ

Ω

∂_tu(t, x)·ϕ(x)dx in L^∞(0, T)^N. Due to the arbitrariness of ϕit follows that

exp(ρ⁻¹ß_ε)∂_tu_ε*^∗ M∂_tu in L^∞(0, T;L²(Ω))^N. (2.10) Moreover, integrating by parts with respect to x and noting that ßε is independent of x, the weak convergence (2.5) ofu_ε (see, e.g., [15, Chapter 3, Section 8]) and (2.2) yield

ˆ

Q

Ae(u_ε) :e exp(−ρ⁻¹ß_ε)Φ

dt dx=− ˆ

Q

u_ε : Div_x

Ae exp(−ρ⁻¹ß_ε)Φ dt dx

− ˆ

Q

u: Divx

Ae(M⁻¹Φ)

dt dx+Oε = ˆ

Q

Ae(u) :e(M⁻¹Φ)dt dx+Oε.

Therefore, passing to the limit in (2.8) with (2.10) and (2.5), we get that for any Φ∈C_c^∞(Q)^N, ˆ

Q

−ρM∂_tu·∂_tΦ +Ae(u) :e(M⁻¹Φ)

dt dx= ˆ

Q

f · M⁻¹Φdt dx.

which is equivalent to the first equation of (2.6).

Finally, let Φ∈C¹([0, T];C_c^∞(Ω))^N with Φ(T, .) = 0. Passing to the limit in (2.8) with the second convergences of (2.2) and (2.4) we get that

ˆ

Q

−ρM∂_tu·∂_tΦ +Ae(u) :e(M⁻¹Φ) dt dx

= ˆ

Ω

ρ u¹·Φ(0, .)dx+ ˆ

Q

f· M⁻¹Φdt dx,

which combined with the first equation of (2.6) gives the initial condition M(0)∂_tu(0, .) = u¹. The condition u(0, .) =u⁰ just follows from (2.5), which also implies that u_ε converges to u in

C⁰([0, T];L²(Ω)^N). The proof is now complete.

3 Homogenization of a stationary problem

Let Ω be a smooth bounded open set of R^N with N ≥ 2. Consider a sequence F_ε of matrix- valued functions in W^−1,p(Ω)^N×N with p > N, such that

F_ε*0 in W^−1,p(Ω)^N×N, (3.1)

and definew^j_ε, 1≤j ≤N, as the solution to ( −Div Ae(w_ε^j)

+F_εe_j = 0 in Ω

w_ε^j = 0 on ∂Ω, (3.2)

which satisfies (due to the regularity of Ω)

w^j_ε *0 in W₀^1,p(Ω)^N, ∀j ∈ {1. . . , N}. (3.3) Extracting a subsequence if necessary, we can assume the existence of a nonnegative symmetric matrix-valued functionM in L^p2(Ω)^N×N such that

Ae(w^j_ε) :e(w^k_ε)*(M e_j)·e_k in L^p2(Ω), ∀j, k ∈ {1, . . . , N}. (3.4) We have the following result which will be used in the next section withuεas a time average of the displacement and z_ε as a time average of the velocity in the elastodynamics problem.

Theorem 3.1. Consider two sequences z_ε∈H¹(Ω)^N and f_ε∈H⁻¹(Ω)^N such that

z_ε* z in H¹(Ω)^N, f_ε→f in H⁻¹(Ω)^N, (3.5) and define u_ε as the solution to

( −Div Ae(u_ε)

+F_εz_ε =f_ε in Ω

uε= 0 on ∂Ω. (3.6)

Then, up to a subsequence, we have

u_ε * u in H₀¹(Ω)^N (3.7)

uε−u−

N

X

j=1

w_ε^jzj →0 in H₀¹(Ω)^N, (3.8) Ae(u_ε) :e(u_ε)* Ae(u) :^∗ e(u) +M z·z in M( ¯Ω). (3.9) Proof. First of all, observe that thanks to Rellich-Kondrachov’s compactness theorem, we have v_ε¹, v_ε² *0 in H¹(Ω)^N ⇒ F_εv_ε¹·v_ε² *0 in D⁰(Ω). (3.10) Using that F_ε = div (Φ_ε) where Φ_ε is bounded in L^p(Ω)^N, and that by Sobolev’s imbedding

∇(z_εu_ε) belongs to L^N−1N (Ω)^N ⊂ L^p0(Ω)^N (due to p > N) where z_ε is bounded in H¹(Ω), we have by H¨older’s inequality

hF_ε, z_εu_εi

≤ kΦ_εk_L^p_(Ω)^Nk∇(z_εu_ε)k

L

N

N−1(Ω)^N ≤Cku_εk_H¹

0(Ω).

Hence, puttinguε as test function in (3.6) the former estimate combined with the boundedness off_ε inH⁻¹(Ω) implies thatu_ε is bounded in H₀¹(Ω). Therefore, convergence (3.7) holds up to a subsequence.

Now, given φ∈C_c^∞(Ω)^N, we put

u_ε−u−

N

X

j=1

w^j_εφ_j

as test function in (3.6). Thanks to (3.10) and (3.3), we get ˆ

Ω

Ae(u_ε) :

e(u_ε)−e(u)−

N

X

j=1

e(w_ε^j)φ_j dx+

* F_εz,

u_ε−u−

N

X

j=1

w_ε^jφ_j +

=O_ε. (3.11) On the other hand, putting

φ_j

u_ε−u−

N

X

i=1

w_εⁱφ_i as test function in (3.2), adding inj and using (3.3), we get

ˆ

Ω

AX^N

j=1

e(w_ε^j)φ_j :

e(u_ε)−e(u)−

N

X

j=1

e(w^j_ε)φ_j dx

+

*

Fεφ,(uε−u−

N

X

j=1

w_ε^jφj

+

=Oε.

(3.12)

Subtracting (3.11) and (3.12) we have ˆ

Ω

A

e(u_ε)−

N

X

j=1

e(w_ε^j)φ_j :

e(u_ε)−e(u)−

N

X

j=1

e(w^j_ε)φ_j dx

+

*

F_ε(z−φ),

u_ε−u−

N

X

j=1

w_ε^jφ_j +

=O_ε. This combined with the weak convergence

e(u_ε)−e(u)−

N

X

j=1

e(w^j_ε)φ_j *0 in L²(Ω)^N×N also yields

ˆ

Ω

A

e(u_ε)−e(u)−

N

X

j=1

e(w^j_ε)φ_j :

e(u_ε)−e(u)−

N

X

j=1

e(w_ε^j)φ_j dx

+

*

Fε(z−φ),

uε−u−

N

X

j=1

w_ε^jφj

+

=Oε.

From Rellich-Kondrachov’s compactness theorem and Cauchy-Schwarz’ inequality, we deduce lim sup

ε→0

ˆ

Ω

A

e(u_ε)−e(u)−

N

X

j=1

e(w^j_ε)φ_j :

e(u_ε)−e(u)−

N

X

j=1

e(w_ε^j)φ_j

dx≤Ckz−φk_H¹

0(Ω)^N.

Moreover, taking a sequenceφⁿ which converges strongly to z inH₀¹(Ω)^N and noting that

n→∞lim lim sup

ε→0

ˆ

Ω

AX^N

j=1

e(w^j_ε)(φⁿ_j −z_j)

:X^N

j=1

e(w_ε^j)(φⁿ_j −z_j)

dx= 0,

we conclude to limε→0

ˆ

Ω

A

e(u_ε)−e(u)−

N

X

j=1

e(w_ε^j)z_j :

e(u_ε)−e(u)−

N

X

j=1

e(w_ε^j)z_j

dx= 0, (3.13) which by Korn’s inequality proves (3.8).

It is immediate that (3.13) and (3.4) imply (3.9).

We also have the following lower semicontinuity result.

Lemma 3.2. Consider a sequence u_ε which satisfies the assumptions of Theorem 3.1. Then, up to subsequence, there exists a measurable function ζ : Ω→R^N, with

M ζ·ζ ∈L¹(Ω)^N, M ζ ∈L^p+22p (Ω)^N, (3.14) such that

F_ε^tu_ε * M ζ in H⁻¹(Ω)^N, (3.15) lim inf

n→∞

ˆ

Ω

Ae(u_ε) :e(u_ε)ϕ dx≥ ˆ

Ω

Ae(u) :e(u) +M ζ·ζ

ϕ dx, ∀ϕ∈C⁰( ¯Ω), ϕ≥0. (3.16) Proof. Forφ ∈C^∞( ¯Ω)^N, we have

F_ε^tu_ε, φ

=

F_εφ, u_ε

=−

N

X

j=1

ˆ

Ω

Ae(u_ε) :e(w_ε^j)φ_jdx+O_ε.

Therefore, definingZ ∈L^p+22p (Ω)^N by

Ae(uε) :e(w_ε^j)*−Zj inL^p+22p (Ω), (3.17) we get

F_ε^tu_ε* Z inH⁻¹(Ω)^N. (3.18)

Applying (3.18) tow^k_ε in place ofu_ε and recalling the definition ofM, we get

F_ε^tw_ε^k *−M e_k in H⁻¹(Ω)^N, (3.19) which implies

F_ε^tX^N

k=1

w^k_εφ_k

*−M φ in H⁻¹(Ω)^N, ∀φ ∈C^∞( ¯Ω)^N. (3.20) On the other hand, by (3.17), (3.4) and Cauchy-Schwarz’ inequality, we have for any function η∈L^p−22p (Ω)^N,

ˆ

Ω

Z·η dx

=

limε→0

ˆ

Ω

Ae(u_ε) :X^N

j=1

e(w_ε^j)η_j dx

≤

lim sup

ε→0

ˆ

Ω

Ae(uε) :e(uε)dx ¹₂ ˆ

Ω

M η·η dx ¹₂

.

Therefore,Z is orthogonal to any functionη∈L^p−22p (Ω)^N such that M η= 0 a.e. in Ω.Now, let us show the existence of a measurable function ζ : Ω → R^N such that Z =M ζ. To this end, consider the set

V :=n

M ξ:ξ∈L^p−12p (Ω)^No

which by H¨older’s inequality (recall that M ∈ L^p2(Ω)^N×N) is a linear subspace of L^p+12p (Ω)^N. Letη∈L^p−22p (Ω)^N. Due to the symmetry of M we have

η∈V^⊥ ⇔ ∀ξ ∈L^p+12p (Ω)^N, ˆ

Ω

M η·ξ dx= 0 ⇔ M η= 0 a.e. in Ω,

which implies that Z ∈ (V^⊥)^⊥ = V since L^p+12p (Ω)^N is a reflexive space (see, e.g., [5, Propo- sition 1.9]). Hence, there exists a sequence ζ_n in L^p−12p (Ω)^N such that M ζ_n converges strongly toZ in L^p+12p (Ω)^N. Up to replace ζn by its orthogonal projection on (kerM)^⊥, we may assume thatζ_n ∈(kerM)^⊥ a.e. in Ω. Next, consider the measurable pseudo-inverseM⁻¹ of the matrix- valued M defined for a.e. x ∈Ω by M(x)⁻¹(M(x)ξ) = ξ for any ξ ∈(kerM(x))^⊥. Then, we have for anyk >0,

1_{|M⁻¹_|≤k}ζ_n= 1_{|M⁻¹_|≤k}M⁻¹(M ζ_n)→1_{|M⁻¹_|≤k}M⁻¹Z strongly inL^p+12p (Ω)^N.

Since a strongly convergent sequence in L^q(Ω) converges up to a subsequence a.e. in Ω, using a diagonal procedure in the former convergences there exists a subsequenceζ_θ(n) such that for any k >0,

1{|M⁻¹|≤k}ζ_θ(n) →1{|M⁻¹|≤k}M⁻¹Z a.e. in Ω.

Therefore, the a.e. limitζ of ζ_θ(n) in Ω satisfies Z =M ζ a.e. in Ω.

It remains to prove (3.16) which in particular implies the first assertion of (3.14). Taking into account (3.3) and (3.7), for anyφ∈C⁰( ¯Ω)^N and ϕ∈C⁰( ¯Ω), ϕ≥0, we have

ˆ

Ω

A

e(u_ε)−e(u)−

N

X

j=1

e(w^j_ε)φ_j :

e(u_ε)−e(u)−

N

X

j=1

e(w_ε^j)φ_j ϕ dx

= ˆ

Ω

Ae(u_ε) :e(u_ε)ϕ dx− ˆ

Ω

Ae(u) :e(u)ϕ dx

+ ˆ

Ω

A X^N

j=1

e(w^j_ε)φj

:

X^N

j=1

e(w_ε^j)φj

ϕ dx−2

N

X

j=1

ˆ

Ω

Ae(uε) :e(w^j_ε)φjϕ dx+Oε

= ˆ

Ω

Ae(u_ε) :e(u_ε)ϕ dx− ˆ

Ω

Ae(u) :e(u)ϕ dx+ ˆ

Ω

M φ:φ ϕ dx+ 2 ˆ

Ω

M ζ·φ ϕ dx.

This proves

ε→0lim ˆ

Ω

Ae(u_ε) :e(u_ε)ϕ dx≥ ˆ

Ω

Ae(u) :e(u)ϕ dx− ˆ

Ω

M φ·φ ϕ dx−2 ˆ

Ω

M ζ·φ ϕ dx,

for any φ ∈ C⁰( ¯Ω)^N and any ϕ ∈ C⁰( ¯Ω), ϕ ≥ 0. Taking into account that M ζ belongs to L^p+22p (Ω)^N, we deduce by approximation that the above equality holds for any φ ∈ L^p−22p (Ω)^N. Thus we can choose in particularφ =−1B(0,R)∩{|ζ|<R}ζ. Then, passing to the limit asR tends to infinity thanks to the monotone convergence theorem we conclude to (3.16). As a by-product

we deduce from (3.16) withϕ= 1 that M ζ·ζ ∈L¹(Ω).

4 Homogenization of a general magneto-elastodynamics problem

Let Ω be a smooth bounded open set of R^N, N ≥ 2, T > 0, Q = (0, T)×Ω, ρ > 0 and let A∈L(R^Ns ^×N;R^N×Ns ) be a positive definite symmetric tensor.

Consider a sequence F_ε of skew-symmetric matrix-valued functions in L^∞(Ω)^N×N which satisfies (3.1) for some p > N, a sequence of skew-symmetric matrix-valued functions G_ε in L^∞(Q)^N×N such that

G_ε*^∗ 0 in L^∞(Q)^N×N, (4.1)

and a sequenceH_ε of skew-symmetric matrix-valued functions in L^∞(Q)^N×N such that

H_ε→H inH¹(0, T;W^−1,p(Ω))^N×N. (4.2) Define

B_ε(t, x) := F_ε(x) +G_ε(t, x) +H_ε(t, x) for (t, x)∈Q. (4.3) Recall that M is the non-negative symmetric matrix-valued function in L^p2(Ω)^N×N defined by (3.4).

The main result of the section is the following Theorem 4.1. Let fε∈L¹(0, T;L²(Ω))^N be such that

f_ε→f in L¹(0, T;L²(Ω))^N, (4.4) and u⁰_ε ∈H₀¹(Ω)^N, u¹_ε ∈L²(Ω)^N be such that

u⁰_ε * u⁰ in H₀¹(Ω)^N, u¹_ε * u¹ in L²(Ω)^N. (4.5) Then, there exist a measurable functionζ : Ω→R^N and a function g ∈L^∞(0, T;L²(Ω))^N such that the solution u_ε of











ρ ∂_tt²u_ε−Div_x Ae(u_ε)

+B_ε∂_tu_ε =f_ε in Q u_ε = 0 on (0, T)×∂Ω

u_ε(0, .) = u⁰_ε, ∂_tu_ε(0, .) =u¹_ε in Ω,

(4.6)

and u⁰_ε satisfy up to a subsequence

u_ε * u^∗ in L^∞(0, T;H₀¹(Ω))^N ∩W^1,∞(0, T;L²(Ω))^N, (4.7) G_ε∂_tu_ε * g^∗ in L^∞(0, T;L²(Ω))^N, (4.8) F_εu⁰_ε * M ζ in H⁻¹(Ω)^N, with M ζ ∈L^p+22p (Ω)^N, M ζ·ζ ∈L¹(Ω). (4.9) Moreover, the limit u is a solution to











(ρI_N +M)∂_tt²u−Div_x Ae(u)

+H∂_tu+g =f in Q u= 0 on (0, T)×∂Ω

u(0, .) = u⁰, ∂_tu(0, .) = (ρI_N +M)⁻¹(ρu¹+M ζ) in Ω,

(4.10)

with

M ∂_tu·∂_tu∈L^∞(0, T;L¹(Ω)). (4.11)

Remark 4.2.Actually, the functionggiven by convergence (4.8) is independent of the sequence H_ε (which is in some sense compact) but cannot be determined in terms of the limitsf,u⁰,u¹. In particular, we cannot prove an uniqueness result for the limit problem (4.10). In Section 5 we will give a specific representation about the functiong illuminating possible nonlocal effects in the homogenization process.

However, if ∂_tG_ε is assumed for instance to be bounded in L¹(0, T;L^∞(Ω))^N×N which cor- responds to the absence of time oscillations, then the functiong is zero. In this case the limit problem (4.10) is completely determined and has a unique solution. The limit elastodynamics equation (4.10) is then characterized by a magnetic fieldH and an increase of mass M which only depends on the space oscillations of Fε(x) through (3.4). This completes the picture of Section 2 where the magnetic field only depends on time. The general case with both space and time oscillations through G_ε(t, x) is much more intricate and leads to the undetermined functiong.

Note that the strong convergence (4.2) makes H_ε a compact perturbation of the magnetic field which simply gives the limitH in the homogenized equation (4.10).

Proof of Theorem 4.1. First of all (see, e.g. [14, Chapter 1]), it is classical that the limit problem (4.6) has one solution inC⁰([0, T];H₀¹(Ω))^N∩C¹([0, T];L²(Ω))^N and that, taking into account thatF_ε, H_ε are skew-symmetric, we have the energy identity

1 2

d dt

ˆ

Ω

ρ|∂tuε|² +Ae(uε) :e(uε)

dx

= ˆ

Ω

fε·∂tuεdx.

This implies that up to a subsequenceu_ε satisfies (4.7), (4.8). In particular, we have

uε→u inC⁰([0, T];L²(Ω))^N. (4.12) Moreover, we recall thatu∈L^∞(0, T;H₀¹(Ω))^N ∩W^1,∞(0, T;L²(Ω))^N gives

( u(t, .)∈H₀¹(Ω)^N, ∀t ∈[0, T]

t_n →t ⇒ u(t_n, .)* u(t, .) in H₀¹(Ω)^N, and that (4.7) implies

uε(t, .)* u(t, .) in H₀¹(Ω)^N, ∀t∈[0, T]. (4.13) Now, the idea is to take time-average values of u_ε and to apply the results of Section 3.

Integrating (4.6) with respect tot in (t₁, t₂) with 0≤t₁ < t₂ ≤T, we deduce that the function

¯ u_ε:=

ˆ _t2

t1

u_ε(s, .)ds in Ω satisfies

ρ ∂_tu_ε(t₂, x)−∂_tu_ε(t₁, x)

−Div_x Ae(¯u_ε)

+F_ε u_ε(t₂, x)−u_ε(t₁, x) +(H_εu_ε)(t₂, x)−(H_εu_ε)(t₁, x)−

ˆ t2

t1

∂_tH_εu_εdt+ ˆ t2

t1

G_ε∂_tu_εdt

= ˆ t2

t1

f_εdt in H⁻¹(Ω)^N.

(4.14)

First step. A corrector result for ¯u_ε.

By (4.7) we have

ρ ∂_tu_ε(t₂, .)−∂_tu_ε(t₁, .)

bounded in L²(Ω)^N. (4.15) By (4.2) we have

H_ε(t, .)→H(t, .) in C⁰([0, T];W^−1,p(Ω))^N×N, which combined with (4.13) and Rellich-Kondrachov’s theorem gives

(H_εu_ε)(t₂, .)−(H_εu_ε)(t₁, .)→(Hu)(t₂, .)−(Hu)(t₁, .) in H⁻¹(Ω)^N. (4.16) Similarly, we have ˆ t2

t1

∂_tH_εu_εdt → ˆ t2

t1

∂_tHu dt inH⁻¹(Ω)^N. (4.17) By (4.8) we also have ˆ _t2

t1

G_ε∂_tu_εdt * ˆ _t2

t1

g dt in L²(Ω)^N. (4.18)

The previous convergences (4.15), (4.16), (4.17), and (4.18) combined with (4.13) and (4.14) allow us to apply Theorem 3.1 to deduce the corrector result

¯

u_ε−u¯−

N

X

j=1

u_j(t₂, .)−u_j(t₁, .)

w_ε^j →0 in H¹(Ω)^N, (4.19) where we denote

¯ u:=

ˆ _t2

t1

u(s, .)ds in Ω.

Second step. Limit of (4.14).

We replace in (4.14), t₁, t₂ by t₁ +s, t₂ +s and we integrate with respect to s in (0, τ) with τ < T−t₂. Then, we can pass to the limit as ε tends to zero to deduce

ρ u(t₂ +τ, .)−u(t₂, .)−u(t₁+τ, .) +u(t₁, .)

−Div

Aeˆ τ 0

ˆ t2+s t1+s

u dt ds

+ lim

ε→0F_ε ˆ τ

0

u_ε(t₂ +s, .)−u_ε(t₁+s, .) ds

+

ˆ τ 0

(Hu)(t₂+s, .)−(Hu)(t₁+s, .) ds

+ ˆ τ

0

ˆ t2+s t1+s

−∂_tHu+g

dt ds= ˆ τ

0

ˆ t2+s t1+s

f dt ds inH⁻¹(Ω)^N,

(4.20) where the limit in the third term is taken in the weak topology of H⁻¹(Ω)^N. Moreover, by (3.1), (4.19), (3.19) and F_ε skew-symmetric we have

F_ε ˆ τ

0

u_ε(t₂+s, .)−u_ε(t₁+s, .) ds

=F_ε

ˆ t2+τ t2

u_ε(s, .)ds− ˆ t1+τ

t1

u_ε(s, .)ds

=

N

X

j=1

u_j(t₂+τ, .)−u_j(t₂, .)−u_j(t₁+τ, .) +u_j(t₁, .)

F_εw_ε^j+R_ε

=M u(t₂+τ, .)−u(t₂, .)−u(t₁+τ, .) +u(t₁, .) +R_ε,

(4.21)

where Rε denotes a sequence which converges weakly (strongly for the first one) to zero in H⁻¹(Ω)^N. Putting (4.21) in (4.20), dividing byτ and letting τ tend to zero, we get

ρ ∂_tu(t₂, .)−∂_tu(t₁, .)

−Div

Aeˆ _t2

t1

u dt +M ∂_tu(t₂, .)−∂_tu(t₁, .)

+ (Hu)(t₂, .)−(Hu)(t₁, .) +

ˆ t2

t1

−∂_tHu+g dt=

ˆ t2

t1

f dt inH⁻¹(Ω)^N. Finally, dividing by t₂−t₁ and letting t₂−t₁ tend to zero, we obtain

(ρI_N +M)∂_tt²u−Div_x Ae(u)

+H∂_tu+g =f inD⁰(Q)^N. (4.22) Third step. Limit of the initial conditions.

By (4.13) and (4.5) the limitu satisfies

u(0, .) = u⁰ in Ω. (4.23)

Now, it remains to find the initial velocity. Let us prove that (ρ∂_tu_ε+F_εu_ε)(t, .)* (ρI_N +M)∂_tu

(t, .) in H⁻¹(Ω)^N, ∀t∈[0, T). (4.24) By (4.6) we have

∂_t ρ∂_tu_ε+F_εu_ε+H_εu_ε

=f_ε+ Div_x Ae(u_ε)

+∂_tH_εu_ε−G_ε∂_tu_ε,

where the right-hand side is bounded inL¹(0, T;H⁻¹(Ω))^N by (4.1), (4.2), (4.4), (4.7). There- fore,

ρ∂_tu_ε+F_εu_ε+H_εu_ε is bounded in W^1,1(0, T;H⁻¹(Ω))^N,

Now, we fixt₀ ∈[0, T) and we observe that (3.1), u_ε∈C⁰([0, T];H₀¹(Ω))^N ∩C¹([0, T];L²(Ω))^N and (4.7) imply, up to a subsequence,

(ρ∂tuε+Fεuε)(t0, .)* L in H⁻¹(Ω)^N. (4.25) On the other hand, forτ ∈(0, T −t₀), we have

(ρ∂_tu_ε+F_εu_ε+H_εu_ε)(t₀, .)− 1 τ

ˆ _t0+τ

t0

(ρ∂_tu_ε+F_εu_ε+H_εu_ε)(t, .)dt

H⁻¹(Ω)^N

=

1 τ

ˆ τ 0

ˆ t0+t t0

∂_r(ρ∂_ru_ε+F_εu_ε+H_εu_ε)(r, .)dr

dt

H⁻¹(Ω)^N

≤ 1 τ

ˆ τ 0

ˆ t0+t t0

fε+ Divx Ae(uε)

+∂rHεuε−Gε∂ruε

H⁻¹(Ω)^Ndr

dt

≤ kf_ε(t, .)k_L¹_(t0,t0+τ;L²_(Ω))^N +τ

2 +C√ τ

∂_tH_ε

L²(t0,t0+τ;W^−1,p(Ω)^N×N)

ku_εk_L^∞_(0,T;H¹

0(Ω))^N

+CτkG_εk_L^∞_(Q)Nk∂_tu_εk_L^∞_(0,T;L2(Ω))^N. By (4.2), (4.7) and (4.13) we have

(H_εu_ε)(t₀, .)→(Hu)(t₀, .) in H⁻¹(Ω)^N,