Lack of compactness in two-scale convergence

SIAM J. MATH. ANAL.c 2005 Society for Industrial and Applied Mathematics Vol. 37, No. 2, pp. 343–346 LACK OF COMPACTNESS IN TWO-SCALE CONVERGENCE∗ MARC BRIANE†AND JUAN CASADO-D´ IAZ‡ Abstract. This article deals with the links between compensated compactness and two-scale convergence. More precisely, we ask the following question: Is the div-curl compactness assumption sufficient to pass to the limit in a product of two sequences which two-scale converge with respect to the pair of variables (x, x/ε)? We reply in the negative. Indeed, the div-curl assumption allows us to control oscillations which are faster than 1/ε but not the slower ones. Key words. two-scale convergence, compensated compactness, counterexample AMS subject classifications. 35B27, 35B40 DOI. 10.1137/040621351 1. Introduction. In order to study the asymptotic behavior of periodic problems arising in homogenization theory, Nguetseng introduced in [7] (see also Allaire [1]) the notion of two-scale convergence: Let Ω be a bounded open subset of Rd,Y:= (−1 2,1 2)d, and let Mbe a positive integer. A bounded sequence uεin L1 loc(Ω)Mtwo-scale converges to a function ˆu in L1 loc(Ω ×Rd)Mand Y-periodic with respect to the last variable if, for any ψ∈ C∞ c(Ω,C∞ #(Y))M,wehave lim ε→0Ω uε(x)ψx, x εdx =ΩY ˆu(x, y)ψ(x, y)dx dy.(1.1) A compactness theorem due to Nguetseng [7] establishes that if uεis bounded in Lp(Ω)M, then there exists a subsequence of uεwhich two-scale converges to ˆu∈ Lp(Ω; Lp #(Y))M. Taking in (1.1) ψ(x, y) independent of y, we deduce that if uεtwo-scale converges to ˆu, then it converges weakly in Lp(Ω)Mto u:= Yˆu(x, y)dy. On the other hand, if uεstrongly converges to uin L1(Ω)M, then it also two-scale converges to u. Therefore two-scale convergence is stronger than weak convergence and weaker than the strong one. Moreover, it provides an expression of the limit of the product uεψ(x, x ε) of (1.1) in which each term only weakly converges. In the periodic homogenization we usually deal with a sequence uεwhich is not only bounded in Lp(Ω)Mbut whose some combinations of its derivatives are also bounded. In this context, let us recall that if uεconverges weakly in W1,p(Ω)M, for 1≤p<+∞, to a function u, then it converges strongly in Lp loc(Ω)M(Lp(Ω)Mif Ω smooth) and so uεtwo-scale converges to u. Then we can conjecture that the classical results of the compensated compactness theory due to Murat and Tartar (see, e.g., [6] and [8]), and in particular the div-curl theorem, still hold true when we replace the weak convergence in Lp(Ω)Mwith two-scale convergence. In fact we have the following result: ∗Received by the editors December 13, 2004; accepted for publication (in revised form) December 28, 2004; published electronically October 14, 2005. http://www.siam.org/journals/sima/37-2/62135.html †Centre de Math´ematiques, I.N.S.A. de Rennes & I.R.M.A.R., 35043 Rennes cedex, France (m[email protected]). ‡Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, 41012 Sevilla, Spain (jcasado[email protected]). 343 Downloaded 06/10/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 344 MARC BRIANE AND JUAN CASADO-D´ IAZ Proposition 1.1. Let (Y,Y1,...,Y n)be (n+1)parallelotops of Rdof Lebesgue measure equal to 1, and let U, V be two vector-valued functions in L2(Ω; C#(Y×Y1× ···×Yn))d, where C#(Y×Y1×···×Yn)denotes the set of the continuous functions on (Rd)n+1 which are Y-periodic with respect to the variable yand Yk-periodic with respect to the variable ykfor any k=1,...,n.Letεk=εk(ε)for k=1,...,n be n well-ordered scales such that lim ε→0 ε1 ε= lim ε→0 εk+1 εk =0 for any k=1,...,n−1.(1.2) Consider the vector-valued sequences uεand vεdefined by uε(x):=Ux, x ε,x ε1 ,..., x εnand vε(x):=Vx, x ε,x ε1 ,..., x εn,(1.3) and assume that div uεis compact in H−1(Ω) and curl vεis compact in H−1(Ω)d×d.(1.4) Then the two-scale limits ˆuof uε,ˆvof vε,and ˆwof uε·?vεexist and satisfy ˆw=ˆu·ˆv.(1.5) Proposition 1.1 shows that the div-curl condition (1.4) implies some compactness in the two-scale convergence process (as in the classical case) when the oscillations of the sequences are faster than 1 ε. Unfortunately, this is not the case for general sequences, particularly when the oscillations are slower than 1 ε. This assertion follows from the following theorem, which is the main result of the present paper: Theorem 1.2. Assume that d≥2. Then there exist two functions U, V ∈ C∞ #(2Y)dsuch that the sequence uε(x):=U(x ε)is divergence-free, the sequence vε(x):=V(x ε)is curl-free, but the two-scale limits of uε,vε, and uε·vεdo not satisfy (1.5). The key ingredient of this counterexample is that 2-periodic functions are considered although the test functions are 1-periodic. In order to understand the lack of compactness in two-scale convergence, let us recall the equivalence between the two-scale convergence theory and the method introduced by Arbogast, Douglas, and Hornung [3] to study the oscillations of a sequence uεin L1 loc(Rd)M. Their method consists in introducing the function ˆuε: Rd×Y→RMdefined by ˆuε(x, y)=  k∈Zd 1εk+εY (x)uε(εk +εy).(1.6) The equivalence between the two approaches is then given by the following result (see, e.g., [5] and [4]): Theorem 1.3. Assume that uεis bounded in Lp(Ω)M, with 1<p<+∞. Then ˆuεconverges weakly to ˆuin Lp(Ω; Lp(Y))Mif and only if uεtwo-scale converges to ˆu. The functions ˆuε(x, y) are not continuous with respect to the variable x.Ifa combination of derivatives of uεis bounded, we also get a bound for the same combination of derivatives with respect to the variable yof ˆuεbut not with respect to the variable x. This explains the lack of compactness in two-scale convergence. Downloaded 06/10/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php LACK OF COMPACTNESS IN TWO-SCALE CONVERGENCE 345 2. Proof of the results. In this section we prove Proposition 1.1 and Theorem 1.2. Proof of Proposition 1.1. We follow the multiscale procedure of [2]. Thanks to the separation of scales (1.2) the sequences uε,vε, and uε·vε, respectively, twoscale converge to ˆu:= Y1···YnU,ˆv:= Y1···YnV, and ˆw:= Y1···YnU·V. Putting test functions of type εkΦ(x, x ε,x ε1,..., x εk) from k=nto 1 in the div-curl assumption (1.4) implies that divykYk+1 ···Yn U= 0 and curlykYk+1 ···Yn V= 0 for k=1,...,n, whence, integrating by parts the product of Yk+1 ···YnUand Yk+1 ···YnV(which is equal to the gradient in ykof a periodic function plus a function depending only on the other variables y1,...,y k−1) successively from k=nto 1, yields ˆw=Y1 ···Yn U·V=Y1 ···Yn U·Y1 ···Yn V=ˆu·ˆv, which implies the desired equality (1.5). Proof of Theorem 1.2. Let us consider two vector-valued functions Φ,Ψ∈C∞ c(Y)d such that div Φ = 0, curl Ψ = 0, and Φ ·Ψ= 0 (this is possible since d>1), which we extend to Rdby Y-periodicity. Let η:R→Rbe the 1-periodic function η:= i∈Z1(i−1 4,i+1 4)and let us define the following sequences uε(x):=ηx1 2εΦx εand vε(x):=ηx1 2εΨx ε. Since in each cube εk +εY , for k∈Zd,η(x1 2ε) is constant, and Φ(x ε), Ψ(x ε) vanish on the boundary of εk +εY , we have uε,v ε∈C∞(RN), div uε= 0, and curl vε=0 in Rd. Moreover, since η(x1 2ε) is constant in εk +εY for any k∈Zd, it is invariant by the transformation (1.6). So we get ˆuε(x, y)=ηx1 2εΦ(y),ˆvε(x, y)=ηx1 2εΨ(y),uε·vε(x, y)=η2x1 2εΦ(y)·Ψ(y). By Theorem 1.3 the two-scale limits ˆuof uε,ˆvof vε, and ˆwof uε·vεare thus given by ˆu(x, y)=1 2 −1 2 η(s)dsΦ(y)=1 2Φ(y),ˆv(x, y)=1 2 −1 2 η(s)dsΨ(y)=1 2Ψ(y), and ˆw(x, y)=1 2 −1 2 η2(s)dsΦ(y)·Ψ(y)=1 2Φ(y)·Ψ(y), whence ˆw=ˆu·ˆv. REFERENCES [1] G. Allaire,Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), pp. 1482–1518. [2] G. Allaire and M. Briane,Multiscale convergence and reiterated homogenization, Proc. Roy. Soc. Edinburgh A, 126 (1996), pp. 297–342. Downloaded 06/10/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 346 MARC BRIANE AND JUAN CASADO-D´ IAZ [3] T. Arbogast, J. Douglas, and U. Hornung,Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., 21 (1990), pp. 823–836. [4] D. Cioranescu, A. Damlamian, and G. Griso,Periodic unfolding and homogenization,C.R. Acad. Sci. Paris, 335 (2002), pp. 99–104. [5] M. Lenczner,Homog´en´eisation d’un circuit ´electrique, C. R. Acad. Sci. Paris, 324 (1997), pp. 537–542. [6] F. Murat,Compacit´e par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (1978), pp. 489– 507. [7] G. Nguetseng,A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), pp. 608–623. [8] L. Tartar,Compensated compactness and applications to partial differential equations, nonlinear analysis and mechanics, in Heriot-Watt Symposium IV, Pitman, San Francisco, 1979, pp. 136–212. Downloaded 06/10/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php