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

Briane, Marc

Casado Díaz, Juan

idUS. Depósito de Investigación Universidad de Sevilla

SIAM J. MATH. ANAL. c© 2005 Society for Industrial and Applied MathematicsVol. 37, No. 2, pp. 343–346LACK OF COMPACTNESS IN TWO-SCALE CONVERGENCE∗MARC BRIANE† AND JUAN CASADO-DI´AZ‡Abstract. This article deals with the links between compensated compactness and two-scaleconvergence. More precisely, we ask the following question: Is the div-curl compactness assumptionsuﬃcient to pass to the limit in a product of two sequences which two-scale converge with respect tothe pair of variables (x, x/ε)? We reply in the negative. Indeed, the div-curl assumption allows usto control oscillations which are faster than 1/ε but not the slower ones.Key words. two-scale convergence, compensated compactness, counterexampleAMS subject classiﬁcations. 35B27, 35B40DOI. 10.1137/0406213511. Introduction. In order to study the asymptotic behavior of periodic prob-lems 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 := (− 12 , 12 )d, and let M be a positiveinteger. A bounded sequence uε in L1loc(Ω)M two-scale converges to a function uˆin L1loc(Ω × Rd)M and Y -periodic with respect to the last variable if, for any ψ ∈C∞c (Ω, C∞# (Y ))M , we havelimε→0∫Ωuε(x)ψ(x,xε)dx =∫Ω∫Yuˆ(x, y)ψ(x, y) dx dy.(1.1)A compactness theorem due to Nguetseng [7] establishes that if uε is bounded inLp(Ω)M , then there exists a subsequence of uε which two-scale converges to uˆ ∈Lp(Ω;Lp#(Y ))M .Taking in (1.1) ψ(x, y) independent of y, we deduce that if uε two-scale convergesto uˆ, then it converges weakly in Lp(Ω)M to u :=∫Yuˆ(x, y) dy. On the other hand, ifuε strongly converges to u in L1(Ω)M , then it also two-scale converges to u. Thereforetwo-scale convergence is stronger than weak convergence and weaker than the strongone. Moreover, it provides an expression of the limit of the product uε ψ(x,xε ) of (1.1)in which each term only weakly converges.In the periodic homogenization we usually deal with a sequence uε which is notonly bounded in Lp(Ω)M but whose some combinations of its derivatives are alsobounded. In this context, let us recall that if uε converges weakly in W1,p(Ω)M , for1 ≤ p < +∞, to a function u, then it converges strongly in Lploc(Ω)M (Lp(Ω)M if Ωsmooth) and so uε two-scale converges to u. Then we can conjecture that the classicalresults 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 replacethe weak convergence in Lp(Ω)M with two-scale convergence. In fact we have thefollowing result:∗Received by the editors December 13, 2004; accepted for publication (in revised form) December28, 2004; published electronically October 14, 2005.http://www.siam.org/journals/sima/37-2/62135.html†Centre de Mathe´matiques, I.N.S.A. de Rennes & I.R.M.A.R., 35043 Rennes cedex, France(mbriane@insa-rennes.fr).‡Dpto. de Ecuaciones Diferenciales y Ana´lisis Nume´rico, Universidad de Sevilla, 41012 Sevilla,Spain (jcasadod@us.es).343Downloaded 06/10/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php344 MARC BRIANE AND JUAN CASADO-DI´AZProposition 1.1. Let (Y, Y1, . . . , Yn) be (n+ 1) parallelotops of Rd of Lebesguemeasure 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 functionson (Rd)n+1 which are Y -periodic with respect to the variable y and Yk-periodic withrespect to the variable yk for any k = 1, . . . , n. Let εk = εk(ε) for k = 1, . . . , n be nwell-ordered scales such thatlimε→0ε1ε= limε→0εk+1εk= 0 for any k = 1, . . . , n− 1.(1.2)Consider the vector-valued sequences uε and vε deﬁned byuε(x) := U(x,xε,xε1, . . . ,xεn)and vε(x) := V(x,xε,xε1, . . . ,xεn),(1.3)and assume thatdivuε is compact in H−1(Ω) and curl vε is compact in H−1(Ω)d×d.(1.4)Then the two-scale limits uˆ of uε, vˆ of vε, and wˆ of uε·?vε exist and satisfywˆ = uˆ · vˆ.(1.5)Proposition 1.1 shows that the div-curl condition (1.4) implies some compactnessin the two-scale convergence process (as in the classical case) when the oscillationsof the sequences are faster than 1ε . Unfortunately, this is not the case for generalsequences, particularly when the oscillations are slower than 1ε . This assertion followsfrom 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 )d such that the sequence uε(x) := U(xε ) is divergence-free, the sequencevε(x) := V (xε ) is curl-free, but the two-scale limits of uε, vε, and uε · vε do notsatisfy (1.5).The key ingredient of this counterexample is that 2-periodic functions are consid-ered although the test functions are 1-periodic.In order to understand the lack of compactness in two-scale convergence, letus recall the equivalence between the two-scale convergence theory and the methodintroduced by Arbogast, Douglas, and Hornung [3] to study the oscillations of asequence uε in L1loc(Rd)M . Their method consists in introducing the function uˆε :Rd × Y → RM deﬁned byuˆε(x, y) =∑k∈Zd1εk+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 uε is bounded in Lp(Ω)M , with 1 < p < +∞. Thenuˆε converges weakly to uˆ in Lp (Ω;Lp(Y ))Mif and only if uε two-scale converges to uˆ.The functions uˆε(x, y) are not continuous with respect to the variable x. If acombination of derivatives of uε is bounded, we also get a bound for the same combi-nation of derivatives with respect to the variable y of uˆε but not with respect to thevariable 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.phpLACK OF COMPACTNESS IN TWO-SCALE CONVERGENCE 3452. Proof of the results. In this section we prove Proposition 1.1 and Theo-rem 1.2.Proof of Proposition 1.1. We follow the multiscale procedure of [2]. Thanks tothe separation of scales (1.2) the sequences uε, vε, and uε · vε, respectively, two-scale converge to uˆ :=∫Y1· · · ∫YnU , vˆ :=∫Y1· · · ∫YnV , and wˆ :=∫Y1· · · ∫YnU · V .Putting test functions of type εk Φ(x,xε ,xε1, . . . , xεk ) from k = n to 1 in the div-curlassumption (1.4) implies thatdivyk(∫Yk+1· · ·∫YnU)= 0 and curlyk(∫Yk+1· · ·∫YnV)= 0 for k = 1, . . . , n,whence, integrating by parts the product of∫Yk+1· · · ∫YnU and∫Yk+1· · · ∫YnV (whichis equal to the gradient in yk of a periodic function plus a function depending onlyon the other variables y1, . . . , yk−1) successively from k = n to 1, yieldswˆ =∫Y1· · ·∫YnU · V =(∫Y1· · ·∫YnU)·(∫Y1· · ·∫YnV)= uˆ · vˆ,which implies the desired equality (1.5).Proof of Theorem 1.2. Let us consider two vector-valued functions Φ,Ψ ∈ C∞c (Y )dsuch that div Φ = 0, curlΨ = 0, and Φ · Ψ = 0 (this is possible since d > 1),which we extend to Rd by Y -periodicity. Let η : R → R be the 1-periodic functionη :=∑i∈Z 1(i− 14 ,i+ 14 ) and let us deﬁne the following sequencesuε(x) := η( x12 ε)Φ(xε)and vε(x) := η( x12 ε)Ψ(xε).Since in each cube εk + εY , for k ∈ Zd, η( x12 ε ) is constant, and Φ(xε ), Ψ(xε ) vanishon the boundary of εk + εY , we have uε, vε ∈ C∞(RN ), divuε = 0, and curl vε = 0in Rd. Moreover, since η( x12 ε ) is constant in εk+ εY for any k ∈ Zd, it is invariant bythe transformation (1.6). So we getuˆε(x, y) = η( x12 ε)Φ(y), vˆε(x, y) = η( x12 ε)Ψ(y), ûε · vε(x, y) = η2( x12 ε)Φ(y) ·Ψ(y).By Theorem 1.3 the two-scale limits uˆ of uε, vˆ of vε, and wˆ of uε · vε are thus givenbyuˆ(x, y) =(∫ 12− 12η(s) ds)Φ(y) =12Φ(y), vˆ(x, y) =(∫ 12− 12η(s) ds)Ψ(y) =12Ψ(y),and wˆ(x, y) =(∫ 12− 12η2(s) ds)Φ(y) ·Ψ(y) = 12Φ(y) ·Ψ(y),whence wˆ = uˆ · vˆ.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.php346 MARC BRIANE AND JUAN CASADO-DI´AZ[3] T. Arbogast, J. Douglas, and U. Hornung, Derivation of the double porosity model of singlephase 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, Homoge´ne´isation d’un circuit e´lectrique, C. R. Acad. Sci. Paris, 324 (1997),pp. 537–542.[6] F. Murat, Compacite´ 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 homoge-nization, SIAM J. Math. Anal., 20 (1989), pp. 608–623.[8] L. Tartar, Compensated compactness and applications to partial diﬀerential equations, nonlin-ear 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

Lack of compactness in two-scale convergence

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Lack of compactness in two-scale convergence

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