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

Calc. Var. (2021) 60:163 https://doi.org/10.1007/s00526-021-02027-0 Calculus of Variations Increase of mass and nonlocal effects in the homogenization of magneto-elastodynamics problems Marc Briane1 ·Juan Casado-díaz2 Received: 17 January 2020 / Accepted: 30 April 2021 / Published online: 24 July 2021 © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021 Abstract The paper deals with the homogenization of a magneto-elastodynamics equation satisfied by the displacement uεof an elastic body which is subjected to an oscillating magnetic field Bε generating the Lorentz force ∂tuε×Bε. When the magnetic field Bεonly depends on time or on space, the oscillations of Bεinduce an increase of mass in the homogenized equation. More generally, when the magnetic field is time-space dependent through a uniformly bounded component Gε(t,x)of Bε, besides the increase of mass the homogenized equation involves the more intricate limit gof ∂tuε×Gεwhich turns out to be decomposed in two terms. The first term of gcan 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 gis locally controlled in L2-norm by the compactness default measure of the oscillating initial energy. Mathematics Subject Classification 74Q10 ·74Q15 ·35B27 ·35L05 1 Introduction In a insulating (vacuum-like) environment, an elastic three-dimensional body placed in an electric field Eand a magnetic Bis subjected to the Lorentz force (see, e.g.,[2, Sect. 9.3]) fL=ρe(E+v×B)+σ(E+v×B)×B,(1.1) Communicated by A. Malchiodi. BMarc Briane [email protected] Juan Casado-díaz [email protected] 1Institut de Recherche Mathématique de Rennes, Univ Rennes, INSA de Rennes, CNRS, IRMAR - UMR 6625, Rennes, France 2Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Sevilla, Spain 123 163 Page 2 of 36 M. Briane, J. Casado-díaz where vis the velocity, σthe conductivity of the body and ρeis the density of free electrical charges, while Eand Bsatisfy Maxwell’s system. In particular, the fields Eand Bare connected by the equation curl E+∂tB=0. In the present paper we focus on the magnetic Lorentz force ∂tv×Brather than on the electrical force. We assume that •the elastic body is a poor conductor, i.e. σ≈0, •the electrical Lorentz force ρeEis negligible compared to the magnetic Lorentz force ρe(v ×B), which yields fL≈ρe(v ×B). (1.2) The second assumption holds in particular if E(t,x)=εe(t,x/ε) with 0 <ε1, since then O(ε) =E(t,x)B(t,x)=B(0,x)−t 0 (curl e)(s,x/ε) ds =O(1). Under these assumptions and setting ρe=1, the displacement uof the body with velocity v=∂tu, satisfies the “simplified” magneto-elastodynamics equation ρ∂ 2 ttu−DivxAe(u)+∂tu×B=f,(1.3) where ρis the mass density, Ais the elasticity tensor of the body and e(u)is the symmetric strain tensor. The right-hand side fencompasses 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,whereBis now a N×Nskew-symmetric matrix-valued function. In the framework of homogenization theory, our aim is to study the effect of a time-space oscillating magnetic field Bε(t,x)on the magneto-elastodynamics equation (1.3). Let T>0, let be a bounded open set of RNand Q:= (0,T)×. Consider the magneto-elastodynamics problem ⎧ ⎨ ⎩ ρ∂ 2 ttuε−DivxAe(uε)+Bε∂tuε=fεin Q uε=0on(0,T)×∂ uε(0,.)=u0 ε,∂ tuε(0,.)=u1 εin , (1.4) where Bεis a skew-symmetric matrix-valued function in L∞(Q)N×Ndecomposed as Bε(t,x)=Fε(x)+Gε(t,x)+Hε(t,x), with Bε(0,x)=Fε(x), Gε(0,x)=Hε(0,x)=0,(1.5) fε∈L1(0,T;L2())N,u0 ε∈H1 0()N,u1 ε∈L2()N. Contrary to Fε(x)the component Gε(t,x)is assumed to be uniformly bounded with respect to tand x, but the time-space oscillations of Gε(t,x)may produce a nonlocal effect. The component Hε(t,x)is a compact perturbation of Bε(t,x). Under suitable oscillations of the sequences Bε,fε,u0 ε,u1 ε, we can pass to the limit as εtends 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 [5,10,12,23] and the references therein, [1,3,19]for periodically oscillating coefficients, as well as for non-uniformly bounded coefficients (from above or below) in the survey article [14]. Here, the asymptotic analysis of equation (1.4) involving the general sequence Bεis in line with homogenization of pde’s with non-periodic 123 Increase of mass and nonlocal effects in the homogenization... Page 3 of 36 163 coefficients which was initiated by Spagnolo [20] and Murat, Tartar [18]. More specifically, in the stationary case Tartar [21,22](seealso[7] for an alternative approach) has studied the homogenization of the three-dimensional Stokes equation −uε+bε×uε+∇pε=fin , (1.6) perturbed by the oscillating drift term bε×uεrepresenting 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 [8] type equation −u+b×u+∇p+M∗u=fin , (1.7) where M∗is a non-negative symmetric matrix-valued function. If the magnetic field Bε(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 ρin the homogenized equation of (1.4). Considering the dynamical system of thermoelasticity with space varying coefficients Brahim-Otsmane, Francfort and Murat [4] obtained a similar homogenized dynamical system of thermoelasticity involving an increase of the heat capacity. Alternatively, nonlocal effects without change of mass have been obtained in [9]for the homogenization of a scalar wave equation with a periodically oscillating matrix-valued function Bε(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),usingatwoscale analysis method. In our non-periodic and vectorial setting we show that the time-space oscillations of the magnetic field Bε(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 ρIN.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 [17] 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 field Bεis only time dependent, we can build an anisotropic internal mass mε(t) such that in a multiplicative way mε(t)∂ tuε≈m∗(t)∂ tu.(1.8) In contrast, when the magnetic field is independent of time, i.e. Bε=Fεwhich is assumed to converge weakly to zero in W−1,p()N×Nfor some p>N∨3, we can build an anisotropic internal mass Mε(x)=Fε(x)Wε(x)such that in an additive way ∂tuε≈∂tu+Wε∂2 ttu.(1.9) The harmonic limit of mε(t)(due to the multiplicativity of (1.8)) or the arithmetic limit of Mε(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 field Bε(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 component Gεof the magnetic field (see (1.5)) weakly converges to zero in L∞(Q)N×Nand that Hεis 123 163 Page 4 of 36 M. Briane, J. Casado-díaz a compact perturbation, we prove (see Theorem 4.1, Theorem 5.10 and Theorem 5.15)that the limit gof the magnetic Lorentz force Gε∂tuεadmits the following decomposition g=Q d(s,y)∂ tu(s,y)−h0,in Q.(1.10) First, the matrix-valued measure 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 of Q, the angle of which is equal to 2 arctan cwith c=√A/ρ (Ais the Frobenius norm of tensor A). A particular nonlocal term with some light cone range has been obtained in the periodically oscillating case of [9]. 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 5.10, Corollary 5.12 and Corollary 5.13) that the second term h0in (1.10) is locally controlled in L2-norm by the compactness default measure μ0of the oscillating initial energy. The function h0acts 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 ⎧ ⎨ ⎩ ∗∂2 ttu−DivxAe(u)+H∂tu+g=fin Q u=0on(0,T)×∂ u(0,.)=u0in , (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 limit g, we obtain a corrector result for the homogenization problem (1.4) if the compactness default measure μ0vanishes (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 [13] for the wave equation. The paper is organized as follows: In Sect. 2we study the case where the magnetic field Bεonly depends on time. We derive (see Theorem 2.3) the homogenized problem (1.11) with the sole increase of mass (g=0). Section 3is devoted to a stationary problem (see Theorem 3.1) which prepares the main homogenization result of the paper in Sect. 4. It is partly based on Tartar’s works [21,22]. In Sect. 4we consider a more general magnetic field Bεsatisfying (1.5). We prove (see Theorem 4.1) that the homogenized magneto-elastodynamics problem of (1.4)is(1.11). Section 5deals with several estimates of the limit g(see Theorem 5.3 and Theorem 5.10) and an abstract representation (see Theorem 5.15) 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 function gand the uniqueness of a solution to the limit problem (1.11) (see Corollary 5.12 and Corollary 5.13). Notation •(e1,...,eN)denotes the canonical basis of RN. •For any ξ,η ∈RN,ξηdenotes the symmetric matrix in RN×Nthe entries of which are 1/2(ξiηj+ξjηi),i,j∈{1,...,N}. •¯ Ydenotes the closure of a subset Yof a topological set X. •A∈L(RN×N s;RN×N s)is a positive definite symmetric fourth-order tensor, and A denotes its Frobenius norm. •|E|denotes the Lebesgue measure of a measurable set Eof RN. •L(X;Y)denotes the space of continuous linear functions from the normed space Xinto the normed space Y. 123 Increase of mass and nonlocal effects in the homogenization... Page 5 of 36 163 •·denotes the scalar product in RN,:denotes the scalar product in RN×N,and|·|denotes the associated norm in both cases. •B(x,r)denotes the euclidean ball of center x∈RNand of radius r>0, and Brsimply denotes the ball B(0,r)centered at the origin. •INdenotes the unit matrix of RN×N,andMtdenotes the transposed of a matrix of M. •RN×N sdenotes the space of symmetric matrices of order N. •denotes a bounded open set of RNfor N≥2, T>0, and Qthe cylinder (0,T)×. •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 Uof RN. •D(U)denotes the space of the distributions on an open subset Uof RN. •→denotes a strong convergence, a weak convergence, and ∗ a weak-∗convergence •→denotes a continuous embedding between two topological spaces. •Oεdenotes a sequence of εwhich converges to zero as εtends to zero, and which may vary from line to line. •Cdenotes 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 be a bounded open set of RNwith N≥2, T>0, Q=(0,T)×,ρ>0andlet A∈L(RN×N s;RN×N s)be a positive definite symmetric tensor. Let f∈L1(0,T;L2())N. For a given function bε∈C1([0,T])and for a constant skew-symmetric matrix ß ∈RN×N, let ßε∈C1([0,T])N×Nbe the skew-symmetric matrix-valued function defined by ßε(t)=bε(t)ßfort∈[0,T],bε(0)=0,(2.1) and satisfying exp(−ρ−1ßε)∗ M−1in L∞(0,T)N×N,(2.2) where Mis an invertible matrix-valued function in C1([0,T])N×N. Remark 2.1 Actually we can extend condition (2.1)to ßt ε=−ßεand ß εßε=ßεß εin [0,T],ßε(0)=0,(2.3) where ß εdenotes the time derivative of ßε. It is clear that condition (2.1) is a particular case of condition (2.3). In dimension 2 conditions (2.1)and(2.3) are clearly equivalent, since any skew-symmetric matrix in R2×2is proportional to the 90◦rotation matrix. In dimension 3 the situation is more intricate. Assume that condition (2.3) holds. Then, there exists a vector-valued function xε∈C1([0,T])Nsuch that ßεy=xε×yfor y∈R3,(2.4) which implies that condition (2.3) is equivalent to the system of ODE’s ∀y∈R3,(xε·y)x ε=(x ε·y)xεin (0,T), xε(0)=0. 123 163 Page 6 of 36 M. Briane, J. Casado-díaz On the one hand, if xεdoes not vanish in (0,T), then using the previous equation with y=xε we get that xε |xε|=x ε |xε|−(x ε·xε)xε |xε|3=0in(0,T), xε(0)=0. Hence, there exist a function bε=|xε|∈C1([0,T])and a constant unit vector ξ∈R3such that xε(t)=bε(t)ξ for t∈[0,T],bε(0)=0, which combined with (2.4) yields condition (2.1) with ß y=ξ×y. On the other hand, consider two functions αε,β ε∈C1([0,T])such that for some t0∈(0,T), αε(t)>0andβε(t)=0ift∈(0,t0) αε(t)=0andβε(t)>0ift∈(t0,T). Then, for any independent vectors ξ,η ∈R3, the skew-symmetric matrix-valued function ßε defined by ßεy:= (αεξ+βεη) ×yfor y∈R3, satisfies condition (2.3) but not condition (2.1). Therefore, conditions (2.1)and(2.3) are not generally equivalent in dimension 3. Under condition (2.1)or(2.3) we need to assume that the weak limit of exp(−ρ−1ßε)is an invertible matrix-valued function. This is not automatically fulfilled as shows the following example. Example 2.2 Let N=2andρ=1. Consider the function bεdefined by bε(t)=t/ε +b(t/ε) for t∈R,wherebis a 2π-periodic function in C1(R), and the skew-symmetric matrix-valued function ßε:= 0−bε bε0∈C1(R)2×2. We have exp(−ρ−1ßε)=cos(bε)−sin(bε) sin(bε)cos(bε) 1 2π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, if b=0 then the weak limit of exp(−ρ−1ßε)is the nul matrix. Otherwise, if bis closed to the 2π-periodic function which agrees in [0,2π]with π 21[0,π], then the weak limit of exp(−ρ−1ßε)is closed to the matrix −1 π1−1 11 , and is thus invertible. We consider the solution uεto the wave equation ⎧ ⎨ ⎩ ρ∂ 2 ttuε−DivxAe(uε)+ß ε∂tuε=fin Q uε=0on(0,T)×∂ uε(0,.)=u0 ε,∂ tuε(0,.)=u1 εin , (2.5) 123 Increase of mass and nonlocal effects in the homogenization... Page 7 of 36 163 where u0 εu0in H1 0()N,u1 εu1in L2()N.(2.6) We have the following homogenization result. Theorem 2.3 Assume that conditions (2.3),(2.2),(2.6)hold true. Then, we have uε∗ uinL ∞(0,T;H1 0())N∩W1,∞(0,T;L2()))N,(2.7) where u is the solution to the equation ⎧ ⎨ ⎩ ρMtM∂2 ttu−DivxAe(u)+ρMtM∂tu=finQ u=0on (0,T)×∂ u(0,.)=u0,∂ tu(0,.)=M−1(0)u1in . (2.8) Remark 2.4 Since the matrix exp(ρ−1ßε)is unitary, the lower semi-continuity of convex functionals yields for any λ∈RNand for any measurable set E⊂, E M−1λ·M−1λdx ≤lim inf ε→0E exp(−ρ−1ßε)λ ·exp(−ρ−1ßε)λ dx =|E||λ|2, which implies that M−1λ·M−1λ≤|λ|2a.e. in .DuetoM=(M−1)−1, we equivalently get that for any λ∈RN,Mλ·Mλ≥|λ|2a.e. in . Hence, the homogenized equation (2.8) involves an effective anisotropic mass ρMtM≥ρINa.e. in , which is greater than the initial one ρ. We will see in Sect. 4that 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 2.3 The proof is performed under the general assumption (2.3). First of all, since ßεis skew-symmetric, the classical estimates for the wave equation yield convergence (2.7) up to a subsequence. By (2.3)wehaveexp(ρ−1ßε)=ρ−1ß εexp(ρ−1ßε)=ρ−1exp(ρ−1ßε)ß ε. Hence, equation (2.5) can be written as ρ∂ texp(ρ−1ßε)∂ tuε−exp(ρ−1ßε)DivxAe(uε)=exp(ρ−1ßε)fin Q,(2.9) which implies that for any ∈C1([0,T];C∞ c())Nwith (T,.)=0, recalling ßε(0)=0, Q−ρexp(ρ−1ßε)∂ tuε·∂t+Ae(uε):eexp(−ρ−1ßε)dt dx = ρu1 ε·(0,.)dx +Q f·exp(−ρ−1ßε) dt dx. (2.10) For ϕ∈C∞ c()N, define the function ξε∈L∞(0,T)Nby ξε(t):= ρexp ρ−1ßε(t) ∂tuε(t,x)·ϕ(x)dx,for a.e. t∈(0,T). (2.11) By (2.9)and(2.7)wehave ξ ε=−Ae(uε):e(exp(−ρ−1ßε)ϕ) +f·exp(−ρ−1ßε)ϕdx bounded in L∞(0,T). 123 163 Page 8 of 36 M. Briane, J. Casado-díaz Hence, ξεis bounded in W1,∞(0,T), and up to a subsequence converges weakly-∗to some ξin W1,∞(0,T)N. This combined with convergences (2.2)and(2.7) implies that exp(−ρ−1ßε)ξ ε∗ M−1ξ=ρ ∂tu(t,x)·ϕ(x)dx in L∞(0,T)N. Due to the arbitrariness of ϕit follows that exp(ρ−1ßε)∂ tuε∗ M∂tuin L∞(0,T;L2())N.(2.12) Moreover, integrating by parts with respect to xand noting that ßεis independent of x,the weak convergence (2.7)ofuε(see, e.g.,[16, Chapter 3, Sect. 8]) and (2.2) yield Q Ae(uε):eexp(−ρ−1ßε)dt dx =−Q uε:DivxAeexp(−ρ−1ßε)dt dx −Q u:DivxAe(M−1)dt dx +Oε=Q Ae(u):e(M−1) dt dx +Oε. Therefore, passing to the limit in (2.10) with (2.12)and(2.7), we get that for any ∈ C∞ c(Q)N, Q−ρM∂tu·∂t+Ae(u):e(M−1)dt dx =Q f·M−1dt dx. which is equivalent to the first equation of (2.8). Finally, let ∈C1([0,T];C∞ c())Nwith (T,.) =0. Passing to the limit in (2.10) with the second convergences of (2.2)and(2.6) we get that Q−ρM∂tu·∂t+Ae(u):e(M−1)dt dx = ρu1·(0,.)dx +Q f·M−1dt dx, which combined with the first equation of (2.8) gives the initial condition M(0)∂ tu(0,.)=u1. The condition u(0,.) =u0just follows from (2.7), which also implies that uεconverges to uin C0([0,T];L2()N). The proof is now complete.  3 Homogenization of a stationary problem This section follows the spirit of [21]and[4]. Let be a smooth bounded open set of RNwith N≥2. Consider a sequence Fεof matrix-valued functions in W−1,p()N×Nwith p>N∨3, such that Fε0inW−1,p()N×N,(3.1) Each entry Fi,j ε,1 ≤i,j≤N, of the matrix-valued distribution Fεread as Fi,j ε= −div(i,j ε),wherei,j εis a bounded sequence in Lp()N. In the sequel Fεzfor z∈ H1()N, denotes the vector-valued distribution defined by Fεz·ei= N  j=1−div(zji,j ε)+i,j ε·∇zjfor i∈{1...,N}.(3.2) 123 Increase of mass and nonlocal effects in the homogenization... Page 9 of 36 163 Define wj ε,1≤j≤N, as the solution to −DivAe(w j ε)+Fεej=0in wj ε=0on∂, (3.3) which satisfies (due to the regularity of ) wj ε0inW1,p 0()N,∀j∈{1...,N}.(3.4) Extracting a subsequence if necessary, we can assume the existence of a nonnegative symmetric matrix-valued function Min Lp 2()N×Nsuch that Ae(w j ε):e(wk ε)(Mej)·ekin Lp 2(), ∀j,k∈{1,...,N}.(3.5) We have the following result which will be used in the next section with uεas a time average of the displacement and zεas a time average of the velocity in the elastodynamics problem. Theorem 3.1 Consider two sequences zε∈H1()Nand fε∈H−1()Nsuch that zεzinH 1()N,fε→finH −1()N,(3.6) and recalling (3.2)define uε∈H1 0()Nas the solution to −DivAe(uε)+Fεzε=fεin  uε=0on ∂. (3.7) Then, up to a subsequence, we have uεuinH 1 0()N(3.8) uε−u− N  j=1 wj εzj→0in H1 0()N,(3.9) Ae(uε):e(uε)∗ Ae(u):e(u)+Mz ·zin M(¯ ). (3.10) Proof First of all, observe that by (3.2), the compact imbedding of H1()Ninto L 2p p−2()N, and Hölder’s inequality involving exponents p,2, 2p p−2,wehaveforanysequencesvε,v ε, which converge weakly to zero in H1 0()N,andanyϕ∈C∞ c(), Fεvε·v ε,ϕ= N  i,j=1i,j ε·∇(ϕ vε, j)v ε,i+i,j ε·∇v ε,iϕv ε, jdx →0. This proves the compactness result vε,v ε0inH1()N⇒Fεvε·v ε0inD(). (3.11) 123 163 Page 16 of 36 M. Briane, J. Casado-díaz Second step. Limit of (4.14). We replace in (4.14), t1,t2by t1+s,t2+sand we integrate with respect to sin (0,τ)with τ<T−t2. Then, we can pass to the limit as εtends to zero to deduce ρu(t2+τ,.) −u(t2,.)−u(t1+τ,.) +u(t1,.) −Div Aeτ 0t2+s t1+s udtds  +lim ε→0Fετ 0uε(t2+s,.)−uε(t1+s,.) ds +τ 0(Hu)(t2+s,.)−(Hu)(t1+s,.) ds +τ 0t2+s t1+s−∂tHu +gdt ds =τ 0t2+s t1+s fdtds in H−1()N,(4.20) where the limit in the third term is taken in the weak topology of H−1()N. Moreover, by (3.1), (4.19), (3.20)andFεskew-symmetric we have Fετ 0uε(t2+s,.)−uε(t1+s,.) ds=Fεt2+τ t2 uε(s,.)ds −t1+τ t1 uε(s,.)ds = N  j=1uj(t2+τ,.) −uj(t2,.)−uj(t1+τ,.) +uj(t1,.) Fεwj ε+Rε =Mu(t2+τ,.) −u(t2,.)−u(t1+τ,.) +u(t1,.) +Rε,(4.21) where Rεdenotes a sequence which converges weakly (strongly for the first one) to zero in H−1()N. Putting (4.21)in(4.20), dividing by τand letting τtend to zero, we get ρ∂tu(t2,.)−∂tu(t1,.) −Div Aet2 t1 udt +M∂tu(t2,.)−∂tu(t1,.) +(Hu)(t2,.)−(Hu)(t1,.) +t2 t1−∂tHu +gdt =t2 t1 fdt in H−1()N. Finally, dividing by t2−t1and letting t2−t1tend to zero, we obtain (ρIN+M)∂2 ttu−DivxAe(u)+H∂tu+g=fin D(Q)N.(4.22) Third step. Limit of the initial conditions. By (4.13)and(4.5) the limit usatisfies u(0,.)=u0in . (4.23) Now, it remains to find the initial velocity. Let us prove that (ρ∂tuε+Fεuε)(t, .)(ρIN+M)∂tu(t,.) in H−1()N,∀t∈[0,T). (4.24) By (4.6)wehave ∂tρ∂tuε+Fεuε+Hεuε=fε+DivxAe(uε)+∂tHεuε−Gε∂tuε, where the right-hand side is bounded in L1(0,T;H−1())Nby (4.1), (4.2), (4.4), (4.7). Therefore, ρ∂tuε+Fεuε+Hεuεis bounded in W1,1(0,T;H−1())N, 123 Increase of mass and nonlocal effects in the homogenization... Page 17 of 36 163 Now, we fix t0∈[0,T)and we observe that (3.1), uε∈C0([0,T];H1 0())N∩ C1([0,T];L2())Nand (4.7) imply, up to a subsequence, (ρ∂tuε+Fεuε)(t0, .)Lin H−1()N.(4.25) On the other hand, for τ∈(0,T−t0),wehave     (ρ∂tuε+Fεuε+Hεuε)(t0,.)−1 τt0+τ t0 (ρ∂tuε+Fεuε+Hεuε)(t,.)dt    H−1()N =    1 ττ 0t0+t t0 ∂r(ρ∂ruε+Fεuε+Hεuε)(r,.)drdt    H−1()N ≤1 ττ 0t0+t t0 fε+DivxAe(uε)+∂rHεuε−Gε∂ruε H−1()Ndrdt ≤fε(t,.)L1(t0,t0+τ;L2())N +τ 2A+C√τ ∂tHε L2(t0,t0+τ;W−1,p()N×N)uεL∞(0,T;H1 0())N +CτGεL∞(Q)N∂tuεL∞(0,T;L2())N. By (4.2), (4.7)and(4.13)wehave (Hεuε)(t0,.)→(Hu)(t0,.) in H−1()N, 1 τt0+τ t0 (ρ∂tuε+Hεuε)(t,.)dt →1 τt0+τ t0 (ρ∂tu+Hu)(t,.)dt in H−1()N. By (4.21)wealsohave 1 τt0+τ t0 Fεuεdt1 τt0+τ t0 M∂tudt in H−1(). Therefore, we deduce     L+(Hu)(t0,.)−1 τt0+τ t0(ρIN+M)∂tu+Hu(t,.)dt    H−1()N ≤f(t,.)L1(t0,t0+τ;L2())N+C√τ+τ.(4.26) Next, equation (4.22), combined with u∈L∞(0,T;H1 0())N∩W1,∞(0,T;L2())N implies as above (ρIN+M)∂tu+Hu ∈W1,1(0,T;H−1())N→C0([0,T];H−1())N. Hence, passing to the limit in (4.26)asτtends to zero, we get L=(ρIN+M)∂tu(t0, .), which implies (4.24). Convergence (4.24) combined with (4.5) yields ρu1 ε+Fεu0 ε(ρIN+M)∂tu(0,.) in H−1()N.(4.27) Therefore, by (3.16)andFεskew-symmetric there exists a measurable function ζsatisfying (4.9), which yields the second initial condition of (4.10). Finally, the proof of estimate (4.11) is given in Lemma 5.9 below. This concludes the proof of Theorem 4.1. 123 163 Page 18 of 36 M. Briane, J. Casado-díaz 5 Energy estimates and nonlocal effects The aim of this section is to estimate more precisely the function garising in the homogenized problem (4.10). 5.1 Energy estimate First of all, observe that the following inequality holds (ρIN+M)−1(ρξ +Mη) ·(ρξ +Mη) ≤ρ|ξ|2+Mη·η, ∀ξ,η ∈RN.(5.1) In order to show it, set ϒ:= (ρ IN+M)−1(ρξ +Mη). Then, using successively the Cauchy- Schwarz inequality with the non-negative symmetric matrix Mand the Cauchy-Schwarz inequality in R2,wehave (ρIN+M)ϒ ·ϒ=(ρξ +Mη) ·ϒ≤(ρ|ξ|2+Mη·η)1 2(ρIN+M)ϒ ·ϒ1 2, which gives (5.1). From (4.5), applying the lower semicontinuity (3.17) and convergence (3.16) with uε= u0 ε, and applying the inequality (5.1) with ξ=u1and η=ζ, we can assume, up to extract a subsequence, that there exists a non-negative Radon measure μ0defined on ¯ such that ρ|u1 ε|2+Ae(u0 ε):e(u0 ε) ∗ μ0+Ae(u0):e(u0)+(ρIN+M)−1(ρu1+Mζ)·(ρu1+Mζ) in M(¯ ).(5.2) Remark 5.1 The measure μ0represents the compactness default with respect to the initial conditions u0 ε,u1 ε. Now, assume that the initial conditions are well-prepared (see [13]for the classical homogenization of the wave equation without Lorentz force) in the following sense: u1 εu1in H1()N,−divAe(u0 ε)+Fεu1 εis compact in H−1()N.(5.3) Then, using the convergence (3.9) with uε=u0 εand zε=u1 ε, combined with convergences (3.1), (3.20), we get that Mζ=Mu1. Moreover, by (3.10) and Rellich-Kondrachov’s compactness theorem we have ρ|u1 ε|2+Ae(u0 ε):e(u0 ε)∗ ρ|u1|2+Ae(u0):e(u0)+Mu1·u1in M(¯ ), which proves that the measure μ0vanishes. Let us introduce the following notations. Definition 5.2 Set c:= A ρ.(5.4) For ¯x∈¯ ,S∈(0,T)and t∈(0,S),wedenote B(¯x,S,t):= B¯x,c(S−t)∩. (5.5) K(¯x,S,t):= ∂B¯x,c(S−t)∩, (5.6) and recall that Bδis the ball centered at the origin of radius δ>0. 123 Increase of mass and nonlocal effects in the homogenization... Page 19 of 36 163 We have the following result. Theorem 5.3 Under the assumptions and the notations of Theorem 4.1, for any ¯x∈¯ , 0<S1<S2,s∈(0,S1),δ>0, and ψ∈L2(0,T;L 2p p−2())N, the solution uεof (4.10) satisfies lim sup ε→0 1 2S2 S1BδB(¯x+z,S,s)ρ|∂t(uε−u)|2 +Ae(uε)−e(u)− N  j=1 e(w j ε)ψ j:e(uε)−e(u)− N  j=1 e(w j ε)ψ jdxdzdS ≤1 2S2 S1Bδ μ0¯ B(¯x+z,S,0)dzdS +S2 S1BδB(¯x+z,S,t) M(∂tu−ψ)·(∂tu−ψ)dxdzdS +S2 S1Bδs 0B(¯x+z,S,t) g·∂tudxdtdzdS.(5.7) Remark 5.4 Assuming that the measure μ0and the function gvanish, estimate (5.7)gives the corrector result lim sup ε→0 1 2S2 S1BδB(¯x+z,S,s)ρ|∂t(uε−u)|2 +Ae(uε)−e(u)− N  j=1 e(w j ε)ψ j:e(uε)−e(u)− N  j=1 e(w j ε)ψ jdxdzdS ≤S2 S1BδB(¯x+z,S,t) M(∂tu−ψ)·(∂tu−ψ)dxdzdS.(5.8) By virtue of Remark 4.2 and Remark 5.1 a sufficient condition for estimate (5.8) to be satisfied is that the sequence ∂tGεis bounded in L1(0,T;L∞())N×Nand that the initial conditions are well-prepared in the sense of (5.3). From Theorem 5.3 we deduce the following estimate for the function gwhich will be improvedinSect.5.2. Corollary 5.5 Under the same assumptions of Theorem 5.3, there exists a constant C >0 which only depends on supε>0GεN×N L∞(Q)such that the function g of (4.10)satisfies B(¯x,S,s)|g|2dx ≤Cμ0¯ B(¯x,S,0)+Cs 0B(¯x,S,t)|∂tu|2dx1 2 dt2 ,(5.9) 0≤μ0¯ B(¯x,S,0)+s 0B(¯x,S,t) g·∂tudxdt,(5.10) for any ¯x∈¯ , any S ∈(0,T)and a.e. s ∈(0,S). Remark 5.6 For ¯x∈¯ and S∈(0,T), define the cone of vertex (¯x,S)and angle equal to 2arctanc C(S,¯x):= (t,x):0<t<S,x∈B¯x,c(S−t),(5.11) where cis the wave propagation velocity defined by (5.4). Then, estimate (5.9) means that the norm of gover the cone section at time t=sis bounded by the measure μ0of the cone 123 163 Page 20 of 36 M. Briane, J. Casado-díaz section at time t=0 plus the norm of the velocity ∂tuover the truncated cone in the time interval (0,s). Proof of Corollary 5.5 By (4.8)and(5.7) there exists a constant C>0 which only depends on sup GεN×N L∞(Q)such that for any S1,S2with 0 <S1<S2<T,s∈(0,S1),δ>0, and ψ∈L2(0,T;L 2p p−2())N, S2 S1BδB(¯x+z,S,s)|g|2dxdzdS ≤CS2 S1Bδ μ0¯ B(¯x+z,S,0)dzdS +CS2 S1BδB(¯x+z,S,t) M(∂tu−ψ)·(∂tu−ψ)dxdzdS +CS2 S1Bδs 0B(¯x+z,S,t) g·∂tudxdtdzdS.(5.12) Moreover, by virtue of (4.11) and using an approximation by truncation in the space of the functions v∈L∞(0,T;L2())Nwith Mv·v∈L∞(0,T;L1()), the sequence ψn:= ∂tu1{|∂tu|≤n}in L∞(Q)Nsatisfies lim n→∞ M(∂tu−ψn)·(∂tu−ψn) L1(Q)=0. Using this approximation in (5.12) it follows that S2 S1BδB(¯x+z,S,s)|g|2dxdzdS ≤CS2 S1Bδ μ0¯ B(¯x+z,S,0)dzdS +CS2 S1Bδs 0B(¯x+z,S,t) g·∂tudxdtdzdS.(5.13) Making S1,S2tend to S,thenδtend to zero, this implies that for any S∈(0,T)and a.e. s∈(0,S), B(¯x,S,s)|g|2dx ≤Cμ0¯ B(¯x,S,0)+Cs 0B(¯x,S,t)|g·∂tu|dxdt.(5.14) Now, defining (s):= B(¯x,S,s)|g|2dx,A:= Cμ0¯ B(¯x,S,0),K(s):= B(¯x,S,s)|∂tu|2dx1 2 , and using the Cauchy-Schwarz inequality in (5.14), it follows that (s)≤A+Cs 0 K(t)(t)1 2dt. By a Gronwall’s type argument this provides (5.9) for another constant C, which concludes the proof of (5.9). The proof of (5.10) easily follows from (5.13)bytakingS1=S, dividing by (S2−S)δN, then letting this quantity tend to zero.  To prove Theorem 5.3 we need the following results. 123 Increase of mass and nonlocal effects in the homogenization... Page 21 of 36 163 Lemma 5.7 Let be a smooth (C1-regular) open set in RN,δ>0and U ∈ W1,1(0,T;L1()).Forx 0∈RNand R ∈C1(0,T),R>0, we define (t):= BδB(x0+z,R(t))∩ U(t,x)dxdz,for t ∈[0,T]. Then, ∈W1,1(0,T)and (t)=BδB(x0+z,R(t))∩ ∂tU(t,x)dxdz +R(t)Bδ∂B(x0+z,R(t))∩ U(t,x)ds(x)dz,for a.e. t ∈(0,T). (5.15) Remark 5.8 Thanks to Fubini’s theorem the function U(t,·)in Lemma 5.7 above is integrable on the boundary ∂B(x0+z,R(t)) for a.e. z∈Bδ, and a.e. t∈(0,T). Therefore, this yields a sense to the last integral of (5.15) under the sole assumption that U(t,·)∈L1(). We will use this additional integration over the ball Bδwith the (total) energy density U(t,x):= ρ|∂tuε|2+Ae(uε):e(uε)(5.16) in the proof of Theorem 5.3.below. Lemma 5.9 The limit u of the solution uεof (4.6)satisfies (4.11). Moreover, for any ν∈ C0(¯ Q)Nwith |ν|≤1, and for any ϕ∈C0(¯ Q)with ϕ≥0,wehave lim inf ε→0T 0c 2ρ|∂tuε|2+c 2Ae(uε):e(uε)−Ae(uε):(∂tuεν)ϕdxdt ≥T 0c 2ρIN+M)∂tu·∂tu+c 2Ae(u):e(u)−Ae(u):(∂tuν)ϕdxdt. (5.17) We also have Ae(uε):e(w j ε)∗ M∂tu·ejin L∞0,T;L 2p p+2(),∀j∈{1,...,N}.(5.18) Proof of Theorem 5.3 Let ¯x∈¯ ,S∈(0,T),t∈(0,S)and δ>0. First, assume that ∂tuε∈L∞(0,T;H1 0())N. So, we may put ∂tuεas test function in (4.6), which due to the skew-symmetry of the matrix-valued function Bεyields 1 2 d dt ρ|∂tuε|2+Ae(uε):e(uε)dx= fε(t,x)·∂tuεdx. In the general case, the former equality remains true using an approximation argument. Hence, we deduce that the energy density (5.16) belongs to W1,1(0,T;L1()). Then, integrating with respect to xover B(¯x+z,S,t)and to zover Bδ, by virtue of Lemma 5.7 we get that 123 163 Page 22 of 36 M. Briane, J. Casado-díaz (here νdenotes the unit exterior normal to B(¯x+z,S,t)) 1 2 d dt BδB(¯x+z,S,t)ρ|∂tuε|2+Ae(uε):e(uε)dxdz =BδB(¯x+z,S,t)ρ∂ 2 ttuε·∂tuε+Ae(uε):e(∂tuε)dxdz −c 2BδK(¯x+z,S,t)ρ|∂tuε|2+Ae(uε):e(uε)ds(x)dz =BδB(¯x+z,S,t)ρ∂ 2 ttuε−Divx(Ae(uε))·∂tuεdxdz +BδK(¯x+z,S,t)Ae(uε):(∂tuεν) −c 2ρ|∂tuε|2−c 2Ae(uε):e(uε)ds(x)dz =BδB(¯x+z,S,t) fε·∂tuεdxdz +BδK(¯x+z,S,t) Ae(uε):(∂tuεν) −c 2ρ|∂tuε|2−c 2Ae(uε):e(uε)ds(x)dz.(5.19) Now, integrating with respect to tin (0,s)with 0 <s<S,weobtain 1 2BδB(¯x+z,S,s)ρ|∂tuε|2+Ae(uε):e(uε)dxdz −1 2BδB(¯x+z,S,0)ρ|u1 ε|2+Ae(u0 ε):e(u0 ε)dxdz =Bδs 0B(¯x+z,S,t) fε·∂tuεdxdtdz −Bδs 0K(¯x+z,S,t) c 2ρ|∂tuε|2+c 2Ae(uε):e(uε)−Ae(uε):(∂tuεν)ds(x)dtdz.(5.20) Using estimate (5.17)in(5.20) and recalling (5.2) we then deduce lim sup ε→01 2BδB(¯x+z,S,s)ρ|∂tuε|2+Ae(uε):e(uε)dxdz ≤Bδs 0B(¯x+z,S,t) f·∂tudxdtdz+1 2Bδ μ0¯ B(¯x+z,S,0)dz +1 2B(¯x+z,S,0)Ae(u0):e(u0)+(ρIN+M)−1(ρu1+Mζ) :(ρu1+Mζ)dxdz −Bδs 0K(¯x+z,S,t)c 2(ρIN+M)∂tu·∂tu +c 2Ae(u):e(u)−Ae(u):(∂tuν)ds(x)dtdz.(5.21) Moreover, the non-negativity of the last integral of (5.20), convergences (4.4), (4.5), (4.7), and the inclusion B(¯x+z,S,s)⊂imply that there exists a constant Cδsuch that BδB(¯x+z,S,s)ρ|∂tuε|2+Ae(uε):e(uε)dxdz ≤Cδ.(5.22) 123 Increase of mass and nonlocal effects in the homogenization... Page 23 of 36 163 Next, similarly to (5.20) with equation (4.10)wehave 1 2BδB(¯x+z,S,s)(ρIN+M)∂tu·∂tu+Ae(u):e(u)dxdz =1 2BδB(¯x+z,S,0)Ae(u0):e(u0)+(ρIN+M)−1(ρu1+Mζ)·(ρu1+Mζ)dxdz +Bδs 0B(¯x+z,S,t) (f−g)·∂tudxdtdz −Bδs 0K(¯x+z,S,t)c 2(ρIN+M)∂tu·∂tu +c 2Ae(u):e(u)−Ae(u):(∂tuν)ds(x)dtdz.(5.23) On the other hand, for any ψ∈C∞ c(Q)N,wehave 1 2S2 S1BδB(¯x+z,S,s)ρ|∂t(uε−u)|2+ +Ae(uε)−e(u)− N  j=1 e(w j ε)ψ j:e(uε)−e(u)− N  j=1 e(w j ε)ψ jdxdzdS =1 2S2 S1BδB(¯x+z,S,s)ρ|∂tuε|2+Ae(uε):e(uε)dxdzdS −S2 S1BδB(¯x+z,S,s)⎛ ⎝ρ∂tuε·∂tu+Ae(uε):e(u)+ N  j=1 e(w j ε)ψ j⎞ ⎠dxdzdS +1 2S2 S1BδB(¯x+z,S,s)ρ|∂tu|2+Ae(u):e(u)dxdzdS +S2 S1BδB(¯x+z,S,s) Ae(u):N  j=1 e(w j ε)ψ jdxdzdS +1 2S2 S1BδB(¯x+z,S,s) AN  j=1 e(w j ε)ψ j:N  j=1 e(w j ε)ψ jdxdzdS. Passing to the limit as εtends to zero thanks to (5.18)and(3.5)weget lim sup ε→0 1 2S2 S1BδB(¯x+z,S,s)ρ|∂t(uε−u)|2 +Ae(uε)−e(u)− N  j=1 e(w j ε)ψ j:e(uε)−e(u)− N  j=1 e(w j ε)ψ jdxdzdS ≤lim sup ε→01 2S2 S1BδB(¯x+z,S,s)ρ|∂tuε|2+Ae(uε):e(uε)dxdzdS −1 2S2 S1BδB(¯x+z,S,s)ρ|∂tu|2+Ae(u):e(u)+2M∂tu·ψ−Mψ·ψdxdzdS, 123 163 Page 24 of 36 M. Briane, J. Casado-díaz which, by the Lebesgue dominated convergence theorem together with estimate (5.22), yields lim sup ε→0 1 2S2 S1BδB(¯x+z,S,s)ρ|∂t(uε−u)|2 +Ae(uε)−e(u)− N  j=1 e(w j ε)ψ j:e(uε)−e(u)− N  j=1 e(w j ε)ψ jdxdzdS ≤S2 S1 lim sup ε→01 2BδB(¯x+z,S,s)ρ|∂tuε|2+Ae(uε):e(uε)dxdzdS −1 2S2 S1BδB(¯x+z,S,s)ρ|∂tu|2+Ae(u):e(u)+2M∂tu·ψ−Mψ·ψdxdzdS. (5.24) Estimate (5.24) combined with (5.21)and(5.23) finally yields (5.7)forψ∈C∞ c(Q)N.The case where ψ∈L2(0,T;L 2p p−2())Neasily follows by approximating ψby a sequence in C∞ c(Q)N. 5.2 Fine estimate of the function g Corollary 5.5 can be improved by the following result. Theorem 5.10 Under the assumptions of Theorem 4.1 there exist a subsequence of εstill denoted by ε, a constant C >0which only depends on supε>0GεL∞(Q)N×Nand a continuous linear operator G:L1(0,T;L2())N→L∞(0,T;L2())Nsuch that for any w∈L1(0,T;L2())N, any ¯x∈¯ , any S ∈(0,T)and a.e. s ∈(0,S), B(¯x,S,s)Gw 2dx ≤Cs 0B(¯x,S,t)|w|2dx1 2 dt2 ,(5.25) 0≤s 0B(¯x,S,t)Gw·wdxdt (5.26) and such that the functions g and u in the limit problem (4.10)defined up to a subsequence of ε, satisfy B(¯x,S,s)g−G(∂tu) 2dx ≤Cμ0¯ B(¯x,S,0),(5.27) where μ0is the measure defined by (5.2)up to a subsequence of ε. Remark 5.11 Theorem 5.10 shows that the function gof problem (4.10) is the difference of G(∂tu)and a function h0which only depends on the initial conditions u0 ε,u1 εof problem (4.6) through the measure μ0. The additional term h0acts as a new exterior force in the limit equation (4.10). As a consequence of Theorem 5.26 we can now get a full representation of the limit problem (4.10) for some particular choices of the initial conditions. Our first result refers to the case of well-prepared initial conditions in the sense of Remark 5.1. 123 Increase of mass and nonlocal effects in the homogenization... Page 25 of 36 163 Corollary 5.12 Consider the subsequence of εdefined by Theorem 5.10. Assume that the initial conditions u0 ε,u 1 εin (4.6)satisfy (5.3). Then, the solution uεof (4.6)satisfies (4.7), where u is the unique solution to ⎧ ⎨ ⎩ (ρIN+M)∂2 ttu−DivxAe(u)+H∂tu+G(∂tu)=finQ u=0on (0,T)×∂ u(0,.)=u0,∂ tu(0,.)=u1in , (5.28) As an example of not well-prepared initial data consider the case where the initial conditions do not depend on ε. Corollary 5.13 There exists a subsequence of εsuch that Theorem 5.10 holds and such that there exists a constant C >0, which only depends on supε>0GεL∞(Q)Nand a continuous linear operator F:L2()N→L∞(0,T;L2())Nsuch that for any v∈L2()N, any ¯x∈¯ , any S ∈(0,T)and a.e. s ∈(0,S), B(¯x,S,s)|F(v)|2dx ≤CB(¯x,S,0) (ρI+M)−1Mv·vdx,(5.29) and such for any u0∈H1 0() and u1∈L2(), the solution uεof (4.6)with u0 ε=u0, u1 ε=u1satisfies (4.7), where u is the unique solution to ⎧ ⎨ ⎩ (ρIN+M)∂2 ttu−DivxAe(u)+H∂tu+G(∂tu)=f+F(u1)in Q u=0on (0,T)×∂ u(0,.)=u0,∂ tu(0,.)=ρ(ρIN+M)−1u1in , (5.30) Moreover, for any ¯x∈¯ , any S ∈(0,T)and a.e. s ∈(0,S), s 0B(¯x,S,s) F(u1)·∂tudxdt ≤s 0B(¯x,S,s) G(∂tu)·∂tudxdt +ρ 2B(¯x,S,0) (ρI+M)−1Mu1·u1dx.(5.31) The proof of Theorem 5.10 is based on the following result. Lemma 5.14 Let w∈C∞ c(Q)Nand let vk εfor k ∈Nbe the solution to ⎧ ⎨ ⎩ ρ∂ 2 ttvk ε−DivxAe(vk ε)+(Fε+Gε)∂tvk ε+k∂tvk ε−w=0in Q vk ε=0on (0,T)×∂ vk ε(0,.)=0,∂ tvk ε(0,.)=0in . (5.32) Then, there exists a constant Cw>0such that for any k ∈N,     vk ε−t 0 wds    2 L∞(0,T;H1 0())N+ ∂tvk ε−w  2 L∞(0,T;L2())N+k ∂tvk ε−w  2 L2(Q)≤Cw. (5.33) Proof of Theorem 5.10 Let {wn,n∈N}be a subset of C∞ c(Q)Nwhich is dense in L1(0,T;L2())N.Letuk,n ε,k,n∈N, be the solution to ⎧ ⎨ ⎩ ρ∂ 2 ttuk,n ε−DivxAe(uk,n ε)+(Fε+Gε)∂tuk,n ε+k∂tuk,n ε−wn=0inQ uk,n ε=0on(0,T)×∂ uk,n ε(0,.)=0,∂ tuk,n ε(0,.)=0in. (5.34) 123 163 Page 32 of 36 M. Briane, J. Casado-díaz Now, consider z∈Bδand (¯ t,¯x)∈(0,T)×∂ such that R(¯ t)= 0,¯x∈∂B(z,R(¯ t)) ∩∂. Since is C1-regular, there exists a ball B(¯x,δ¯x), an open set O⊂RN−1with 0 ∈Oand a function φ=φ(ζ) ∈C1(O;RN)such that φ(0)=¯x,φis injective in O,Rank(Dφ)(ζ ) =N−1,∀ζ∈O,∂∩B(¯x,δ¯x)=φ(O). Since R(¯ t)= 0, applying the implicit function theorem to the function (t,ζ)∈(0,T)×O→|φ(ζ)|−R(t), we deduce that δ¯xand Ocan be chosen small enough to ensure the existence of ε>0anda function ψin C1(O;(¯ t−ε, ¯ t+ε)) such that ψ(0)=¯ t,R(ψ(ζ )) =|φ(ζ)|,∀ζ∈O,t∈(¯ t−ε, ¯ t+ε), ζ ∈O R(t)=|φ(ζ)|⇒t=ψ(ζ). (5.57) Therefore, we have (t,x):t∈(¯ t−ε, ¯ t+ε) :x∈B(¯x,δ¯x)∩∂B(z,R(t)) ∩∂}=ψ(ζ),ζ:ζ∈O, which thus has null N-dimensional measure. Since for any integer n≥1, the set (we may assume that Ris defined in an interval larger than [0,T]) (¯ t,¯x)∈[0,T]×∂ :¯x∈∂B(z,R(t)), |R(t)|≥1/n is a compact set, we deduce that this set has zero measure. Hence, the set (¯ t,¯x)∈[0,T]×∂ :¯x∈∂B(z,R(t)), R(t)= 0 also has null N-dimensional measure. This shows that in the last term of (5.56), we have R(t)∂B(z,R(t))∩∂ U(t,x)ds(x)=0, for any z∈Bδ, and a.e. t∈(0,T). Therefore, the derivative formula (5.15) holds.  Proof of Lemma 5.9 It is enough to consider the case where ϕ∈C1(¯ Q)and ν∈C1(¯ Q).For any integer n≥1andanyk∈{0,...,n−1},set ¯un,k ε(x):= n Tk+1 nT k nT uε(t,x)dt,¯vn,k ε(x):= n Tuεk+1 nT,x−uεk nT,x, (5.58) ¯un,k:= n Tk+1 nT k nT u(t,x)dt,¯vn,k:= n Tuk+1 nT,·−uk nT,·,(5.59) ¯ϕn,k(x):= n Tk+1 nT k nT ϕ(t,x)dt,¯νn,k(x):= n Tk+1 nT k nT ν(t,x)dt.(5.60) Taking into account that for any ξ∈RNwith |ξ|≤1, the function Qξdefined as Qξ:(v, V)∈RN×RN×N s−→ c 2ρ|v|2+c 2AV :V−AV :vξ∈R 123 Increase of mass and nonlocal effects in the homogenization... Page 33 of 36 163 is convex and (4.19), we have T 0 Qν∂tuε,e(uε)ϕdxdt = n−1  j=0k+1 nT k nT Q¯νn,k∂tuε,e(uε)¯ϕn,kdxdt −C n ≥T n n−1  k=0 Q¯νn,k¯vn,k ε,e(¯un,k ε)¯ϕn,kdxdt −C n =T n n−1  k=0 Q¯νn,k¯vn,k ε,e(¯un,k)+ N  j=1 e(w j ε)¯vn,k j,ε ¯ϕn,kdx +Oε−C n =T n n−1  k=0 Q¯νn,k¯vn,k+(¯vn,k ε−¯vn,k), e(¯un,k) + N  j=1 e(w j ε)(¯vn,k j+(¯vn,k ε−¯vn,k))¯ϕn,kdx +Oε−C n. Using the weak convergence to zero of e(w j ε)in Lp()N×Nwith p>N, Rellich- Kondrachov’s compactness theorem for ¯vn,k ε−¯vn,k, the non-negativity of the quadratic form Qνand the definition (3.5)ofM,wehave t 0 Qν∂tuε,e(uε)ϕdxdt ≥T n n−1  k=0 Q¯νn,k ¯vn,k,e(¯un,k)+ N  j=1 e(w j ε)¯vn,k j¯ϕn,kdx +Oε−C n =T n n−1  k=0Q¯νn,k¯vn,k,e(¯un,k)+c 2M¯vn,k·¯vn,k¯ϕn,kdx +Oε−C n. Therefore, we have just proved lim inf ε→0T 0 Qν∂tuε,e(uε)ϕdxdt ≥T n n−1  k=0Q¯νn,k¯vn,k,e(¯un,k)+c 2M¯vn,k·¯vn,k¯ϕn,kdx −C n. Passing to the limit as ntends to infinity, we finally obtain (4.11)and(5.17). Finally, let us prove (5.18). The sequence Ae(uε):e(w j ε)is bounded in L∞0,T;L 2p p+2(). Hence, it is enough that convergence (5.18) holds in the distributions sense in Q. Let ϕ∈C∞ c(Q). With notations (5.58)and(5.59) we have for any integer n≥1, Q Ae(uε):e(w j ε)ϕdxdt = n−1  k=0k+1 nT k nT Ae(uε):e(w j ε)¯ϕn,kdx +O1 n =T n n−1  k=0 Ae(¯un,k ε):e(w j ε)¯ϕn,kdx +O1 n. 123 163 Page 34 of 36 M. Briane, J. Casado-díaz Then, again using convergence (4.19), the weak convergence to zero of e(w j ε)in Lp()N×N and the definition (3.5)ofM,weget Q Ae(uε):e(w j ε)ϕdxdt =T n n−1  k=0 Ae¯un,k+ N  i=1 e(wi ε)¯vn,k i:e(w j ε)¯ϕn,kdx +Oε+O1 n. =T n n−1  k=0 M¯vn,k·ej¯ϕn,kdxdt +Oε+O1 n. Passing successively to the limit as εtends to zero for a fixed n, and to the limit as ntends to infinity, we obtain that lim ε→0Q Ae(uε):e(w j ε)ϕdxdt =Q M∂tu·ejϕdxdt, which concludes the proof of Lemma 5.9. Proof of Lemma 5.14 Set v(t,.)=t 0 w(s,.)ds,for t∈[0,T]. Putting ∂tvk ε−was test function in (5.32) and using that Fε,Gεare skew-symmetric, we get 1 2 d dt ρ∂tvk ε−w 2+Ae(vk ε−v) :e(vk ε−v)dx +k∂tvk ε−w 2dx =d dt  Fε(vk ε−v) ·wdx − Fε(vk ε−v) ·∂twdx − Gεw·∂tvk ε−wdx − ρ∂ tw·(∂tvk ε−w) dx − Ae(v) :e(∂tvk ε−w) dx.(5.61) Setting Ek ε(t):= 1 2ρ∂tvk ε−w 2+Ae(vk ε−v) :e(vk ε−v)dx, hk ε(t):=  Fε(vk ε−v) ·wdx, equality (5.61) implies that dEk ε dt +k∂tvk ε−w 2dx ≤CwEk ε+1+dhk ε dt .(5.62) Applying Gronwall’s lemma and noting that Ek ε(0)=0, we get Ek ε(t)≤Cw+Cwt 0 Ek ε(s)ds, which again using Gronwall’s lemma gives Ek ε(t)≤Cw,∀t∈[0,T].(5.63) 123 Increase of mass and nonlocal effects in the homogenization... Page 35 of 36 163 This combined with (5.62) proves that kQ∂tvk ε−w 2dxdt ≤Cw. Finally, the former estimate and (5.63) yield the desired estimate (5.33).  Acknowledgements The authors wish to thank the unknown referee for his/her very careful reading of the paper, and for several relevant comments which have clarified some proofs. The second author also thanks the Université Rennes 1 and the Institut de Recherche Mathématique de Rennes for their invitation during his stay between June 25th and July 8th 2017, for the starting of this work. This paper has been partially supported by Project MTM2017 of the Ministerio de Ciencia, Innovación y Universidades of Spain. References 1. Bakhvalov, N., Panasenko, G.: Homogenisation: averaging processes in periodic media. Mathematical problems in the mechanics of composite materials, translated from the Russian by D. Leˇıtes, Mathematics and its Applications (Soviet Series) 36, Kluwer Academic Publishers Group, Dordrecht, 1989, pp. 402 (1989) 2. 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