Unfolding-based corrector estimates for a reactiondiffusion system predicting concrete corrosion Fatima, T.; Muntean, A.; Ptashnyk, M.

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1 Unfoldng-based corrector estmates for a reactondffuson system predctng concrete corroson Fatma, T.; Muntean, A.; Ptashnyk, M. Publshed: //2 Document Verson Publsher s PDF, also known as Verson of Record ncludes fnal page, ssue and volume numbers Please check the document verson of ths publcaton: A submtted manuscrpt s the author's verson of the artcle upon submsson and before peer-revew. There can be mportant dfferences between the submtted verson and the offcal publshed verson of record. People nterested n the research are advsed to contact the author for the fnal verson of the publcaton, or vst the DOI to the publsher's webste. The fnal author verson and the galley proof are versons of the publcaton after peer revew. The fnal publshed verson features the fnal layout of the paper ncludng the volume, ssue and page numbers. Lnk to publcaton Ctaton for publshed verson APA: Fatma, T., Muntean, A., & Ptashnyk, M. 2. Unfoldng-based corrector estmates for a reacton-dffuson system predctng concrete corroson. CASA-report; Vol. 38. Endhoven: Technsche Unverstet Endhoven. General rghts Copyrght and moral rghts for the publcatons made accessble n the publc portal are retaned by the authors and/or other copyrght owners and t s a condton of accessng publcatons that users recognse and abde by the legal requrements assocated wth these rghts. Users may download and prnt one copy of any publcaton from the publc portal for the purpose of prvate study or research. You may not further dstrbute the materal or use t for any proft-makng actvty or commercal gan You may freely dstrbute the URL dentfyng the publcaton n the publc portal? Take down polcy If you beleve that ths document breaches copyrght please contact us provdng detals, and we wll remove access to the work mmedately and nvestgate your clam. Download date: 7. Oct. 27

2 EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematcs and Computer Scence CASA-Report -38 June 2 Unfoldng-based corrector estmates for a reacton-dffuson system predctng concrete corroson by T. Fatma, A. Muntean, M. Ptashnyk Centre for Analyss, Scentfc computng and Applcatons Department of Mathematcs and Computer Scence Endhoven Unversty of Technology P.O. Box MB Endhoven, The Netherlands ISSN:

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4 UNFOLDING-BASED CORRECTOR ESTIMATES FOR A REACTION-DIFFUSION SYSTEM PREDICTING CONCRETE CORROSION T. Fatma, A. Muntean and M. Ptashnyk Department of Mathematcs and Computer Scence, Insttute for Complex Molecular Systems Techncal Unversty Endhoven, Endhoven, The Netherlands, e-mal: t.fatma@tue.nl, a.muntean@tue.nl Department of Mathematcs I, RWTH Aachen, D-5256 Aachen, Germany, e-mal: ptashnyk@math.rwth-aachen.de abstract. We use the perodc unfoldng technque to derve corrector estmates for a reacton-dffuson system descrbng concrete corroson penetraton n the sewer ppes. The system, defned n a perodcally-perforated doman, s sem-lnear, partally dsspatve, and coupled va a non-lnear ordnary dfferental equaton posed on the sold-water nterface at the pore level. After dscussng the solvablty of the pore scale model, we apply the perodc unfoldng technques adapted to treat the presence of perforatons not only to get upscaled model equatons, but also to prepare a proper framework for gettng a convergence rate corrector estmates of the averagng procedure. Keywords: Corrector estmates, perodc unfoldng, homogenzaton, sulfate corroson of concrete, reacton-dffuson systems.. Introducton Concrete corroson s a slow natural process that leads to the deteroraton of concrete structures buldngs, brdges, hghways, etc. leadng yearly to huge fnancal losses everywhere n the world. In ths paper, we focus on one of the many mechansms of chemcal corroson, namely the sulfaton of concrete, and am to descrbe t macroscopcally by a system of averaged reacton-dffuson equatons whose effectve coeffcents depend on the partcular shape of the mcrostructure. The fnal am of our research s to become capable to predct quanttatvely the durablty of a well-understood cement-based materal under a controlled expermental setup well-defned boundary condtons. The strkng thng s that n spte of the fact that the basc physcal-chemstry of ths relatvely easy materal s known [], we have no control on how the mcrostructure changes n tme and space and to whch extent these spato-temporal changes affect the observable macroscopc behavor of the materal. The research reported here goes along the lne open n [], where a formal asymptotc expanson ansatz was used to derve macroscopc equatons for a corroson model, posed n a doman wth locally-perodc mcrostructure see

5 2 UNFOLDING AND CORRECTOR ESTIMATES [7] for a rgorous averagng approach of a reduced model defned n a doman wth locallyperodc mcrostructures. A two-scale convergence approach for perodc mcrostructures was studed n [], whle prelmnary multscale smulatons are reported n [3]. Wthn ths paper we consder a partally dsspatve reacton-dffuson system defned n a doman wth perodcally dstrbuted mcrostructure. Ths system was orgnally proposed n [2] as a free-boundary problem. The model equatons descrbe the corroson of sewer ppes made of concrete when sulfate ons penetrate the non-saturated porous matrx of the concrete vewed as a composte. The typcal concrete mcrostructure ncludes sold, water and ar parts, see Fg. 2.. One could argue that the mcrostructure of a concrete s nether unformly perodc nor locally perodc, and the randomness of the pores and of ther dstrbutons should be taken nto account. However, perodc representatons of concrete mcrostructures often provde good descrptons. For what the macroscopc corroson process s concerned, the dervaton of corrector estmates [for the perodc case] s crucal for the dentfcaton of convergence rates of mcroscopc solutons. The stochastc geometry of the concrete wll be studed as future work wth the hope to shed some lght on eventual connectons between the role played by a locally-perodc dstrbuted mcrostructure vs. statonary random-dstrbuted pores. In ths sprt, we thnk that there s much to be learnt from [8]. The man novelty of the paper s twofold: on one hand, we obtan corrector estmates under optmal regularty assumptons on solutons of the mcroscopc model and obtan the desred convergence rate hence, we have now a confdence measure of our averagng results; on the other hand, we apply for the frst tme an unfoldng technque to derve corrector estmates n perforated meda. The man deas of the methodology were presented n [2, 3] and appled to lnear ellptc equatons wth oscllatng coeffcents, posed n a fxed doman. Our approach strongly reles on these results. However, novel aspects of the method, related to the presence of perforatons n the consdered mcroscopc doman, are treated here for the frst tme; see secton 3. The man advantage of usng the unfoldng technque to prove corrector estmates s that only H -regularty of solutons of mcroscopc equatons and of unt cell problems s requred, compared to standard methods mostly based on energy-type estmates used n the dervaton of corrector estmates. As a natural consequence of ths fact, the set of choces of mcrostructures s now much larger. The paper s structured n the followng fashon: After ntroducng model equatons and the assumed mcroscopc geometry of the concrete materal, the secton 2 goes on wth the man assumptons and basc estmates ensurng both the solvablty of the mcroscopc problem and the convergence of mcroscopc solutons to a soluton of the macroscopc equatons, as. In secton 3 we state and prove the corrector estmates for the concrete corroson model, Theorem 3.6, determnng the range of valdty of the upscaled model.

6 UNFOLDING AND CORRECTOR ESTIMATES 3 Note that the technque developed n ths artcle can be appled n a straghtforward way to derve convergence rates for solutons of other classes of partal dfferental equatons, posed n domans wth perodcally-dstrbuted mcrostructures. 2. Problem descrpton 2.. Geometry. We assume that concrete pece conssts of a system of pores perodcally dstrbuted nsde the three-dmensonal cube Ω = [a, b] 3 wth a, b R and b > a. Snce usually the concrete n sewer ppes s not completely dry, we consder a partally saturated porous materal. We assume that every pore has three dstnct non-overlappng parts: a sold part, the water flm whch surrounds the sold part, and an ar layer boundng the water flm and fllng the space of Y as shown n Fg. 2.. Note that the dark black parts ndcate the water-flled parts n the materal where most of our model equatons are defned. The reference pore, Y = [, ] 3, has three par-wse dsjont domans Y, Y and Y 2 wth smooth boundares Γ and Γ 2 as shown n Fg. 2.. Moreover, Y = Ȳ Ȳ Ȳ2. Fgure. Left: Perodc approxmaton of the concrete pece. choce of the mcrostructure. Rght: Our Let be a small factor denotng the rato between the characterstc length of the pore Y and the characterstc length of the doman Ω. Let χ and χ 2 be the characterstc functons of the sets Y and Y 2, respectvely. The shfted set Y k s defned by Y k := Y + Σ 3 j=k j e j for k = k, k 2, k 3 Z 3, where e j s the j th unt vector. The unon of all Y k multpled by that are contaned wthn Ω defnes the perforated doman Ω, namely Ω := k Z 3{Y k Y k Ω}. Smlarly, Ω 2, Γ, and Γ 2 denote the unon of Y2 k, Γ k, and Γ k 2, contaned n Ω Mcroscopc equatons. We consder a mcroscopc model t u Du u = fu, v n, T Ω, t v Dv v = fu, v n, T Ω, t w Dw w = n, T Ω 2, t r = ηu, r on, T Γ, wth the ntal condtons u, x = u x, v, x = v x n Ω, w, x = w x n Ω 2, r, x = r x on Γ 2

7 4 UNFOLDING AND CORRECTOR ESTIMATES and the boundary condtons u =, v = on, T Ω Ω, w = on, T Ω Ω 2, 3 together wth Du u ν = ηu, r on, T Γ, Dv v ν = on, T Γ, Du u ν = on, T Γ 2, Dv v ν = a xw b xv on, T Γ 2, Dw w ν = a xw b xv on, T Γ 2. 4 We consder the space H Ω Ω = {u H Ω : u = on Ω Ω }, =, 2. Assumpton 2.. A D, t D L, T ; L pery 3 3, {u, v, w}, D t, xξ, ξ D ξ 2 for D >, for every ξ R 3 and a.a. t, x, T Y. A2 Reacton rate k 3 L perγ s nonnegatve and ηα, β = k 3 yrαqβ, where R : R R +, Q : R R + are sublnear and locally Lpschtz contnuous. Furthermore, Rα = for α < and Qβ = for β β max, wth some β max >. A3 f C R 2 s sublnear and globally Lpschtz contnuous n both varables,.e. fα, β C f + α + β, fα, β fα 2, β 2 C L α α 2 + β β 2 and fα, β = for α < or β <. A4 The mass transfer functons at the boundary a, b L perγ 2, ay and by are postve for a.a. y Γ 2 and there exsts A v, A w, M v, M w such that bye Avt M v = aye Awt M w for a.a. y Γ 2 and t, T. A5 Intal data u, v, w, r [H 2 Ω HΩ L Ω] 3 L perγ and u x, v x, w x a.e. n Ω, r x a.e. on Γ. We defne the oscllatng coeffcents: D t, x := D t, x, {u, v, w}, a x := a x, b x := b x, k x := k x. Defnton 2.2. We call u, v, w, r a weak soluton of 4 f u, v L 2, T ; H Ω Ω H, T ; L 2 Ω, w L 2, T ; H Ω Ω 2 H, T ; L 2 Ω 2, r H, T ; L 2 Γ and

8 UNFOLDING AND CORRECTOR ESTIMATES 5 satsfes the followng equatons T t u φ + Du u φ + fu, v φ T dxdt = ηu, r φdγdt, 5 Ω T t v φ + Dv v φ fu, v φ T dxdt = Γ a w b v φdγdt, 6 Ω T t w ϕ + Dw w ϕ T dxdt = Γ 2 a w b v ϕdγdt, 7 Ω 2 T T t r ψdγdt = Γ 2 ηu, r ψdγdt 8 Γ Γ for all φ L 2, T ; H Ω Ω, ϕ L 2, T ; H Ω Ω 2, ψ L 2, T Γ and u t u, v t v n L 2 Ω, w t w n L 2 Ω 2, r t r n L 2 Γ as t. Lemma 2.3. Under the Assumpton 2., solutons of the problem 4 satsfy the followng a pror estmates: u L,T ;L 2 Ω + u L 2,T Ω C v L,T ;L 2 Ω + v L 2,T Ω C, w L,T ;L 2 Ω 2 + w L 2,T Ω 2 C, /2 r L,T ;L 2 Γ + /2 t r L 2,T Γ C, 9 where the constant C s ndependent of. Proof. Frst, we consder as test functons φ = u n 5, φ = v n 6, ψ = w n 7 and use Assumpton 2., Young s nequalty, and the trace nequalty,.e. t Γ 2 w v dγdτ C t Ω 2 w w 2 dγdτ + C t Ω v v 2 dγdτ. Then, addng the obtaned nequaltes, choosng convenently and applyng Gronwall s nequalty mply the frst three estmates n Lemma. Takng ψ = r as a test functon n 8 and usng A2 from Assumpton 2. and the estmates for u, yeld the estmate for r. The test functon ψ = t r n 8, the sublnearty of R, the boundedness of Q and the estmates for u mply the boundedness of /2 t r L 2,T Γ. Lemma 2.4. Postvty and boundedness Let Assumpton 2. be fulflled. followng estmates hold: Then the u t, v t a.e. n Ω, w t a.e. n Ω 2 and u t, r t a.e. on Γ, for a.a. t, T.

9 6 UNFOLDING AND CORRECTOR ESTIMATES u t M u e Aut, v t M v e Avt a.e. n Ω, w t M w e Awt a.e. n Ω 2 and u t M u e Aut, r t M r e Art a.e. on Γ, for a.a. t, T. Proof. To show the postvty of a weak soluton we consder u as test functon n 5, v n 6, w n 7, and r n 8, where φ = mn{, φ} wth φ + φ =. The ntegrals nvolvng fu, v u, fu, v v and ηu, r u are zero, snce by Assumpton 2. fu, v s zero for negatve u or v and ηu, r s zero for negatve u. In the ntegrals over Γ 2 we use the postvty of a and b and the estmate v w = v + + v w v w. Due to the postvty of η, the rght hand sde n the equaton for r, wth the test functon ψ = r, s nonpostve. Addng the obtaned nequaltes, applyng both Young s and the trace nequaltes, consderng suffcently small, we obtan, due to postvty of the ntal data and usng Gronwall s nequalty, that u t L 2 Ω + v t L 2 Ω + w t L 2 Ω 2 + r t L 2 Γ, for a.a. t, T. Thus, negatve parts of the nvolved concentratons are equal zero a.e. n, T Ω, =, 2, or n, T Γ, respectvely. To show the boundedness of solutons, we consder u e Aut M u + as a test functon n 5, v e Avt M v + n 6 and w e Awt M w + n 7, where φ M + = max{, φ M} and A, M, = u, v, w are postve numbers, such that u x M u, v x M v, w x M w a.e n Ω, and A, M for = v, w are gven by A4 n Assumpton 2.. Addng the equatons for u, v, w and usng Assumpton 2. yeld, wth U M = u e Aut M u +, V M = v e Avt M v +, and W M = w e Awt M w +, τ t UM 2 + VM 2 + UM 2 + VM 2 dx + t WM 2 + WM 2 dx dt C Ω τ [ Ω Cf e Aut M u + e Avt M v A u e Aut M u U M + U M 2 + V M V M 2 + C f e Aut M u + e Avt M v A v e Avt M v V M dx + Ω 2 W M W M ] 2 dx dt. Ω 2 Choosng A u, M u such that C f e Aut M u + C f e Avt M v A u e Aut M u and C f e Aut M u + C f e Avt M v A v e Avt M v, and suffcently small, Gronwall s nequalty mples the estmates for u, v, w, stated n Lemma. Lemma 5. n Appendx and H -estmates for u n Lemma 2.3 mply u t and u t e Aut M u a.e on Γ for a.a. t, T. The assumpton on η and equaton 8 wth the test functon r e Art M r +, where r x M r a.e. on Γ, yeld τ 2 t r e Art M r A r e Art M r r e Art M r + dγdt = τ Γ Γ τ ηu, r r e Art M r + dγdt C η A u, M u r e Art M r + dγdt. Γ

10 UNFOLDING AND CORRECTOR ESTIMATES 7 Ths, for A r and M r, such that C η A r M r e ArT, mples the boundedness of r on Γ for a.a. t, T. Lemma 2.5. Under Assumpton 2., we have the followng estmates, ndependent of : t u L 2,T Ω + t v L 2,T ;H Ω + t w L 2,T ;H Ω 2 C. Proof. We test 5 wth φ = t u, and usng the structure of η, the regularty assumptons on R and Q and the boundedness of u and r on Γ, we estmate the boundary ntegral by t C Γ Ω ηu, r t u dγdτ = t u u 2 + u u 2 dx + C Γ k t Ru Qr Ru Q r t r dγdτ t Γ + t r 2 dγdτ, where Rα = α Rξdξ. Then, Assumpton 2., estmates n Lemma 2.3 and the fact that Du/2 2 for approprate, mply the estmate for t u. In order to estmate t v and t w, we dfferentate the correspondng equatons wth respect to the tme varable and then test the result wth t v and t w, respectvely. Due to assumptons on f and usng the trace nequalty, we obtan t t t v 2 dx + C t v 2 dxdτ C t w t w 2 dxdτ Ω t +C Ω 2 + Ω t w 2 dx + C Ω 2 Ω t u 2 + t v 2 + v 2 dxdτ + t t w 2 dx + C Ω 2 t t w 2 dxdτ C Ω Ω 2 Ω t t v 2 dx, Ω 2 t w 2 + w 2 dxdτ t v t v 2 dxdτ. The regularty assumptons mply that t v L 2 Ω and t w L 2 Ω 2 can be estmated by the H 2 -norm of v and w. Addng and, makng use of estmates for t u, v and w, and applyng Gronwall s Lemma, gve the desred estmates. Lemma 2.6. Exstence & Unqueness Let Assumpton 2. be fulflled. Then there exsts a unque global-n-tme weak soluton n the sense of Defnton 2.2. Proof. The Lpschtz contnuty of f, local Lpschtz contnuty of η and the boundedness of u and r on Γ ensure the unqueness result. The exstence of weak solutons follows by a standard Galerkn approach, [4], usng the a pror estmates n Lemmata 2.3, 2.4 and 2.5.

11 8 UNFOLDING AND CORRECTOR ESTIMATES 2.3. Unfolded lmt equatons. We defne Ω nt = Int k Z 3{Y k, Y k Ω}, Γ,nt = k Z 3{Γ k, Y k Ω}, R n = R n {Y + ξ, ξ Z n,l }, Ω = {x R n : dstx, Ω < l n}, l =, 2. Defnton 2.7. [4, 5, 7]. For any functon φ Lebesgue-measurable on perforated doman Ω, the unfoldng operator TY : Ω Ω Y, =, 2, s defned by φ [ ] x TY φx, y = + y a.e. for y Y Y, x Ω nt, a.e. for y Y, x Ω \ Ω nt, where k := [ x] denotes the unque nteger combnaton Σ3 k j e j of the perods such that x [ x] belongs to Y, 2. For any functon φ Lebesgue-measurable on oscllatng boundary Γ, the boundary unfoldng operator TΓ : Γ Ω Γ, =, 2 s defned by ] T Γ φx, y = φ x + y Y a.e. for y Γ, x Ω nt, a.e. for y Γ, x Ω \ Ω nt. We note that for w H Ω t holds that T Y w Ω = T Y w Ω Y. Lemma 2.8. Under the Assumpton 2., there exst u, v, w L 2, T ; H Ω H, T ; L 2 Ω, ũ, ṽ L 2, T Ω; H pery, w L 2, T Ω; H pery 2, and r H, T, L 2 Ω Γ such that up to a subsequence for and TY u u, TY v v n L 2, T Ω; H Y, t TY u t u, t TY v t v n L 2, T Ω Y, TY 2 w w, t TY 2 w t w n L 2, T Ω; H Y 2, TY u u + y ũ n L 2, T Ω Y, TY v v + y ṽ n L 2, T Ω Y, TY 2 w w + y w n L 2, T Ω Y 2, T T T Γ r r, t TΓ r t r n L 2, T Ω Γ, Γ u u n L 2, T Ω Γ, Γ 2 v v, TΓ 2 w w n L 2, T Ω Γ Proof. Applyng estmates n Lemmata 2.3, 2.5 and Convergence Theorem [7, 8], see Theorem 5.3 n Appendx, mples the convergences for u, v, w n 2. The strong convergence of r s acheved by showng that TΓ r s a Cauchy sequence n L 2, T Ω Γ, for the proof see [, 6]. A pror estmate for t r and the convergence propertes of TΓ, [7], mply the convergences of TΓ t r. To show the convergences 3, we make use of the trace theorem, [9], and of the strong convergence of TY u as,.e. TΓ u u L 2,T Ω Γ C TY u u L 2,T Ω;H Y as.

12 UNFOLDING AND CORRECTOR ESTIMATES 9 Theorem 2.9. Under the Assumpton 2., the sequences of weak solutons of the problem -4 converges as to a weak soluton u, v, w, r of a macroscopc model,.e. u, v, w L 2, T ; HΩ H, T ; L 2 Ω, r H, T ; L 2 Ω Γ and u, v, w, r satsfy the macroscopc equatons T u t uφ + D u t, y u + y ωu j φ + y φ + fu, vφ dydxdt Ω Y x j T = ηu, rφ dγ y dxdt, Ω Γ T v t vφ + D v t, y v + y ωv j φ + y φ fu, vφ dydxdt Ω Y x j T = ayw byvφ dγ y dxdt, 4 Ω Γ 2 T w t wφ 2 + D w t, y w + y ωw j φ 2 + y φ2 dydxdt Ω Y 2 x j T = ayw byvφ 2 dγ y dxdt, Ω Γ 2 T T t rψdγ y dxdt = ηu, rψdγ y dxdt, Ω Γ Ω Γ for φ, φ 2 L 2, T ; HΩ, φ L 2, T Ω; HperY, φ 2 L 2, T Ω; HperY 2 and ψ L 2, T Ω Γ, where ωu, j ωv j and ωw j are solutons of the correspondent unt cell problems y D ζ t, y y ω j ζ = D ζ t, y ω j ζ ν = 3 k= y D w t, y y ω j w = D w t, y ω j w ν = 3 k= yk D kj ζ t, y n Y, ζ = u, v, 5 D kj ζ t, yν k on Γ Γ 2, ω j ζ s Y -perodc, ω j ζ ydy =, Y 3 k= 3 k= yk D kj w t, y n Y 2, 6 D kj w t, yν k on Γ 2, ω j w s Y -perodc, Y 2 ω j wydy =. Proof. Due to consdered geometry of Ω and Ω 2 we have T T u φdxdt = TY u TY φdydxdt, =, 2. Ω Y Ω Applyng the unfoldng operator to 5-8, usng TY D t, x = D t, y, {u, v} and TY 2 D w t, x = D wt, y, consderng the lmt as and the convergences stated n Theorem 2.8, we obtan the unfolded lmt problem. Smlarly as for mcroscopc problem, usng local Lpschtz contnuty of η and f and boundedness of macroscopc solutons, whch follows drectly from the boundedness of mcroscopc solutons, we can show the

13 UNFOLDING AND CORRECTOR ESTIMATES unqueness of a soluton of the macroscopc model. Thus the whole sequence of mcroscopc solutons converge to a soluton of the unfolded lmt problem. The functons ũ, ṽ, w are defned n terms of u, v, w and solutons ω j u, ω j v, ω j w of unt cell problems 5 and 6, see [, 6]. 3. Corrector estmates Frst of all, we ntroduce the defnton of local average and averagng operators. After that, we show some techncal estmates needed n the followng. Defnton 3.. [2, 4]. For any φ L p Ω, p [, ] and =, 2, we defne the local average operator mean n the cells M Y : L p Ω L p Ω M Y φx = TY Y φx, ydy = φydy, x Ω. Y n Y [ x ]+Y 2. The operator Q : L p Ω,2 W, Ω, =, 2 s defned as Q nterpolaton of MY φ,.e. Q φξ = M Y φξ for ξ Z n and Q φx = k {,} n Q φξ + k x k... x kn n for x Y + ξ, ξ Z n, where for x Y + ξ and k = k,..., k n {, } n ponts x k l l x l ξ l x k, f k l l =, l = x l ξ l, f k l =. are gven by 3. The operator Q : W,p Ω W, Ω s defned by Q φ = Q Pφ Ω, where Q s gven n 2. and P : W,p Ω W,p R n s an extenson operator, n the case there exsts P, such that Pφ W,p R n C φ W,p Ω. Note TY M Y φ = M Y φ for φ L p Ω and M Y φx = M Y TY φx, addtonally x k... x kn n =. k {,} n Defnton 3.2. [7, 8]. For p [ + ] and =, 2, the averagng operator UY : L p Ω Y L p Ω s defned as Φ [ ] x + z, { } x dz for a.a. x Ω UY Y Y Y,nt, Φx = Y for a.a. x Ω \ Ω,nt. 2. UΓ : L p Ω Γ L p Γ s defned as Φ [ ] x + z, { } x dz for a.a. x Γ UΓ Y Y Y,nt, Φx = Y for a.a. x Γ \ Γ,nt. For ω HperY, due to y ω y = y T Y ω x = T Y x ω x and U Y y ω y = UY T Y x ω x = x ω x = yω x, we have that U Y y ω y = y ω x.

14 UNFOLDING AND CORRECTOR ESTIMATES 3.. Basc estmates. In ths subsecton, we prove some techncal estmates, used n the dervaton of corrector estmates. Proposton 3.3. For φ L 2, T ; H Ω and φ 2 L 2, T ; H Ω we have φ M Y φ L 2,T Ω C φ L 2,T Ω, φ 2 M Y φ 2 L 2,T Ω C φ 2 L 2,T Ω. 7 Proof. Ths proof s smlar to [2]. For φ L 2, T ; H Ω we can wrte x φ ξ+y x M Y φ ξ L 2, T ; H ξ + Y wth ξ + Y Ω. Usng Y Y and applyng Poncaré nequalty, we obtan T ξ+y T C n φ M Y φ ξ 2 dxdt = T y φ y 2 dydt = C 2 ξ+y φ y Y ξ+y x φ x 2 dxdt. φ zdz 2 n dydt ξ+y ξ+y Then, we add all nequaltes for ξ Z n, such that ξ + Y Ω, and obtan the frst estmate n 7. The second estmate follows from the decomposton of Ω nto ξ Z nξ + Y and Poncaré s nequalty as n the prevous estmate. Lemma 3.4. For φ L 2, T ; H 2 Ω, φ 2 L 2, T ; H Ω and ω H pery, we have the followng estmates φ M Y φ L 2,T Ω C φ L 2,T ;H 2 Ω, M Y x φ Q Y x φ y ω L 2,T Ω C φ L 2,T ;H 2 Ω ω L 2 Y, Q Y φ 2 M Y φ 2 L 2,T Ω C φ 2 L 2,T Ω, Q Y φ φ L 2,T Ω C φ L 2,T Ω, Q Y φ 2 φ 2 L 2,T Ω C φ 2 L 2,T Ω, 8 φ TΓ φ L 2,T Ω Γ C φ L 2,T Ω + C φ L 2,T Ω, Q Y φ 2 L 2,T Ω C φ 2 L 2,T Ω, Q Y ωy ωy L 2 Y C y ω L 2 Y, T Y Q Y φ 2 Q Y φ 2 L 2 Ω Y C φ 2 L 2,T Ω. Proof. The frst nequalty follows drectly from the frst estmate n 7 appled to φ. To show the second nequalty, we use the defnton of the operator Q, the equalty k {,} x k n... x kn n =, and obtan Q Y φx M Y φx = Q Y φξ + k M Y φξ x k... x kn n. k {,} n

15 2 UNFOLDING AND CORRECTOR ESTIMATES Then, t follows ξ+y 2 n Q Y φx M Y φx 2 x 2 y ω dx Q Y φξ + k Q Y φξ 2 Y n y ωy 2 dy. k {,} n For any φ W,p IntY Y + e j, the followng estmate holds M Y +e j φ M Y φ = M Y φ + e j φ φ + e j φ L p Y C φ L p Y Y +e j. Thus, by the defnton of Q Y φx and by a scalng argument ths mples Q Y φξ + k Q Y φξ C φ L 2 ξ+y ξ+k+y. 9 We sum over ξ Z n wth ξ + Y Ω and obtan the desred estmate. Usng 9 we obtan also that Q Y φ M Y φ 2 dx Ω 2 C n φ 2 L 2 ξ+y ξ+k+y 2 C φ 2 dx. k {,} n Ω ξ+y Ω In the same way, usng the estmates stated n Proposton 3.3, the fourth and ffth estmates n 8 follows from: Ω Γ Q Y φ 2 φ 2 L 2,T Ω Q Y φ 2 M Y φ 2 L 2,T Ω + M Y φ 2 φ 2 L 2,T Ω C φ 2 L 2,T Ω. For φ H Ω applyng the trace theorem to a functon n L 2 Γ yelds φ TΓ φ 2 dγdx φ M Y φ 2 + M Y φ TΓ φ dγdx 2 C 2 Γ Ω C 2 Γ 2 C φ 2 dx + C Ω Ω Γ Ω Y φ 2 dx + C φ 2 dx + Ω M Y φ T Y φ 2 + y M Y φ T Y φ 2 dydx M Y φ φ 2 dx + φ 2 dx. Ω Y y T Y φ 2 dydx Ω Ω To obtan an estmate for the gradent of Q Y φ 2, wth φ 2 L 2, T ; H Ω, we defne k j = k,..., k j, k j+,..., k n, k j = k,..., k j,, k j+,..., k n, k j = k,..., k j,, k j+,..., k n and calculate Q Y φ 2 x j = kj Q Y φ 2 ξ + k j Q Y φ 2 ξ + k j x k... x k j j... xk j+ j+ xkn n. Now, applyng 9 we obtan the estmates for Q Y φ 2 n L 2, T Ω.

16 UNFOLDING AND CORRECTOR ESTIMATES 3 For y Y we have Q Y ωyy ωy = Q Y ωk ωyȳ k... ȳn kn, where k {,} n ȳ k y l l ξ l, f k l =, l =. The Poncaré s nequalty and the perodcty of ω y l ξ l, f k l = mply the estmate for Q Y ωy ωy. To derve the last estmate, we consder T Y Q Y φ 2 Q Y φ 2 L 2 Ω Y T Y Q Y φ 2 M Y Q Y φ 2 L 2 Ω Y + M Y Q Y φ 2 Q Y φ 2 L 2 Ω Y C Q Y φ 2 M Y Q Y φ 2 L 2 Ω +C M Y Q Y φ 2 Q Y φ 2 L 2 Ω C Q Y φ 2 L 2 Ω C φ 2 L 2 Ω Perodcty defect. In the dervaton of error estmates we use a generalzaton of the Theorem 3.4 proved n [2] for functons defned n a perforated doman: Theorem 3.5. For any φ H Ω, =, 2, there exsts ˆψ L 2 Ω; H pery : Here φ = Q Y φ. ˆψ L 2 Ω;H Y C φ L 2 Ω n, T Y φ φ y ˆψ L 2 Y ;H Ω C φ L 2 Ω n. The proofs of Theorem 3.5 go the same lnes as n [2, Theorem 3.4], usng the estmates T Y φ L 2 Ω Y C φ L 2 Ω, Q Y φ L 2 Ω C φ L 2 Ω Error estmates. Under addtonal regularty assumptons on the soluton of the macroscopc problem, we obtan a set of error estmates. We emphasze here agan that the most mportant pont s that only H -regularty for the solutons of the mcroscopc model and of the cell problems s requred. Theorem 3.6. Suppose u, v, w, r are solutons of the mcroscopc problem -4 and u, v, w L 2, T ; H 2 Ω H, T Ω, r H, T ; L 2 Ω Γ are solutons of the macroscopc equatons 4. Then we have the followng corrector estmates: u u L 2,T Ω + u u v v L 2,T Ω + v v w w L 2,T Ω 2 + w w Q Y xj u y ωu j 2 L 2,T Ω C 2, Q Y xj v y ωv j 2 L 2,T Ω C 2, Q Y 2 xj w y ωw j 2 L 2,T Ω 2 C 2, 2 r U Γ rt, x, y L 2,T Γ C 2.

17 4 UNFOLDING AND CORRECTOR ESTIMATES 4. Proof of Theorem 3.6. We defne dstance functon ρx = dstx, Ω, domans ˆΩ ρ,n = {x Ω, ρx < } and ˆΩ,ρ,n = {x Ω, ρx < }, where ρ = nf{ ρ, }. Defnton of ρ yelds x ρ L Ω n = xρ L ˆΩ ρ,n n =. 2 Then, for Φ H 2 Ω and ω j estmates, [2], H Y, =, 2, j = u, v, w, we obtan the followng Φ L 2 ˆΩ + Q ρ,n n Y Φ L 2 ˆΩ + M ρ,n n Y Φ L 2 ˆΩ C ρ,n n 2 Φ H 2 Ω, ω j + ω j C L 2 ˆΩ 2 y ω j L 2,ρ,n L 2 ˆΩ Y n,,ρ,n n ρ x Φ L 2 Ω n xφ L 2 ˆΩ ρ,n n C 2 Φ H 2 Ω, 2 x ρ xj Φ L 2 Ω n C 2 + Φ H 2 Ω, x ρ Q Y xj Φω j C 2 Φ H 2 Ω ω j L 2 L 2 Ω Y, ρ x Q Y xj Φω j C Φ H 2 Ω ω j L 2 L 2 Ω Y. Now, for φ L 2, T ; H Ω gven by φ x = u x ux ρ x x Q Y xj uxωu j we consder an extenson φ from, T Ω nto, T Ω such that φ L 2,T Ω C φ L 2,T Ω and φ L 2,T Ω C φ L 2,T Ω. Due to zero boundary condtons such extenson can be defned for whole Ω. Notce that Q Y xj u and u are n L 2, T ; H Ω, but not n L 2, T ; HΩ. We consder φ L 2, T ; HΩ and ˆψ L 2, T Ω, HperY, gven by Theorem 3.5, as test functons n the macroscopc equaton 4 for u: τ t u φ u + D u y u + y ωu j φ + y ˆψ Ω Y x dydxdt j τ + fu, v φ τ dydxdt + ηu, r φ dγdxdt =. Ω Y Ω Γ In the frst term and n the last two ntegrals we replace φ by M Y φ, φ by TΓ φ, and u by TY u. As next step, we ntroduce ρ n front of u and xj u and replace φ by Q Y φ. Now, usng Theorem 3.5, we replace φ + y ˆψ, by TY φ, where φ = Q Y φ and obtan τ + τ t TY um Y φ + D u yρ u + Ω Y Ω Y T Y fu, vm Y φ dydxdt + τ u y ωu j TY x φ dydxdt j Ω Γ ηu, rt Γ φ dγdxdt = R u,

18 where R u = τ Ω Y +ρ D u u + UNFOLDING AND CORRECTOR ESTIMATES 5 [ t u T Y um Y φ + t u φ M Y φ u y ωu j Q Y x φ φ + TY φ φ y ˆψ j +ρ u D u u + y ωu j φ x + y ˆψ + fu, v φ M Y φ j ] τ +f TY fm Y φ dydxdt + ηu, rtγ φ φ dγdxdt. Ω Γ Then we remove ρ, replace u by M Y u, xj u by M Y xj u and, usng M Y φ = T Y M Y φ, we apply the nverse unfoldng τ + Ω τ t um Y φ + Du M Y u + Ω fu, vm Y φ dxdt + τ x M Y xj u y ωu φ j dxdt Ω Γ ηu, rt Γ φ dγdxdt = R u + R 2 u, where R 2 u = τ Ω Y [ ρ D u y u + +D u y M Y u u + xj u y ωuy j TY φ M Y xj u xj u ] y ωuyt j Y φ dydxdt. Introducng ρ n front of M Y xj u and replacng M Y φ by φ, M Y u by u, M Y xj u by Q Y xj u yeld where τ = R 3 u = +D u Ω τ τ Ω [ t uφ + Du u + x ] ρ Q Y xj u y ωu j φ + fu, vφ dxdt Ω Γ ηu, rt Y φ dγdxdt + R u + R 2 u + R 3 u, 22 [ t u + f φ M Y φ + ρ D u u M Y u + x M Y xj u y ωu j φ ρ Q Y xj u M Y xj u x ] y ωu φ j dxdt.

19 6 UNFOLDING AND CORRECTOR ESTIMATES Now, we subtract from the equaton for u the equaton 22 and obtan for the test functon φ = u u ρ τ D u Ω n Q Y xj uω j u the equalty [ t u uu u ρ Q Y xj uωu j + u u ρ Q Y xj u y ωu j u u + fu, v fu, v u u ρ Q Y xj uωu ] j dxdt + τ Ω Γ ηt u, T r ηu, r T Γ u u ρ x ρ Q Y xj u ωu j Q Y xj uωu j dγdxdt = Ru, where R u = Ru + Ru 2 + Ru. 3 We consder ψ = TΓ r r as a test functon n the equatons for TΓ r and r and, usng local Lpschtz contnuty of η and boundedness of u, u, r, r, obtan τ τ t TΓ r r 2 dγdxdt C T Γ r r 2 + TΓ u u 2 dγdxdt. Ω Γ Ω Γ Applyng Gronwall s nequalty and consderng T Γ r x, y = r y yeld TΓ r r 2 L 2 Ω Γ C T Γ u u 2 L 2,τ Ω Γ + T Γ r r 2 L 2 Ω Γ C TΓ u u 2 L 2,τ Ω Γ + T Γ u u 2 L 2,τ Ω Γ. Then, for the boundary ntegral usng the estmate n Lemma 3.4 we obtan τ ηtγ r, TΓ u ηr, utγ φ dγdxdt Ω Γ C TΓ r r L 2,τ Ω Γ + TΓ u u L 2,τ Ω Γ φ L 2,τ Γ C u u L 2,τ Ω + u u L 2,τ Ω + u L 2,τ Ω φ L 2,τ Ω + φ L 2,τ Ω. 23 Therefore, the ellptcty assumpton, the Lpschtz contnuty of f and Young nequalty, appled to the estmate for the boundary ntegral 23, mply τ C Ω τ t û ρ Q Y xj uωu j 2 + û ρ τ + 2 Ω Ω û ρ Q Y xj uωu j 2 + ˆv ρ u 2 dxdt + R u + C u, Q Y xj u y ωu j 2 dxdt Q Y xj vωv j 2 dxdt

20 where û = u u, ˆv = v v and τ Cu := C 2 Ω UNFOLDING AND CORRECTOR ESTIMATES 7 Q Y t xj uω j u Q Y xj uω j u 2 + Q Y xj uω j u 2 + Q Y xj vω j v 2 + Q Y xj u y ω j u 2 dxdt + C τ ˆΩ,ρ,n Q Y xj uωu j 2 dxdt C 2 u 2 L 2,T ;H 2 Ω + 2 u 2 H,T Ω + u 2 L 2,T ;H 2 Ω ωu 2 H Y n +C 2 v 2 L 2,T ;H Ω ω v 2 L 2 Y n. Here we used that 2 ρ Q Y xj u ω j u 2 dx 2 Q Y xj uω j u 2 dx + Q Y xj uω j u 2 dx. Ω Ω ˆΩ,ρ,n The estmates of the error terms n the subsecton 4. mply R u = R u + R 2 u + R 3 u /2 C + u H,T Ω + u L 2,T ;H 2 Ω + v L 2,T ;H Ω + r L 2,T Ω Γ φ L 2,T ;H Ω. Then, applyng Young s nequalty, we obtan τ C Ω τ t û ρ Q xj uωu j 2 + û ρ Ω Q Y xj u y ωu j 2 dxdt û ρ Q Y xj uωu j 2 + ˆv ρ Q Y xj uωv j 2 dxdt +C u 2 H,T Ω + u 2 L 2,T ;H 2 Ω + ω u 2 H Y n +C 2 v 2 L 2,T ;H Ω + ωv 2 H Y n + 2 r 2 L,T Ω Γ. Smlarly, estmates for v v n Q Y xj vωv j and w w n Q Y 2 xj wωw j are obtaned. The only dfference s the boundary term. Applyng the trace theorem and estmates n Lemma 3.4, the boundary term can be estmated by Ω Γ 2 C ayw byv φ aytγ 2 w bytγ 2 vtγ 2 φ dγdx w T Γ2 w + v TΓ 2 v TΓ 2 φ + w + v φ M Y φ Ω Γ 2 +w + v M Y φ TΓ 2 φ dγdx C v H Ω + w H Ω φ H Ω.

21 8 UNFOLDING AND CORRECTOR ESTIMATES Thus, we obtan for ˆv = v v and ŵ = w w τ C C τ Ω τ t ˆv ρ Q Y xj vωv j 2 + ˆv ρ Q Y xj v y ωv j 2 dxdt û ρ Q Y xj uωu j 2 + ˆv ρ Ω ŵ ρ Q Y 2 xj wωw j ŵ ρ Ω 2 Q Y xj vωv dxdt j 2 + Q Y 2 xj w y ωw j 2 dxdt +C v 2 L 2,T ;H 2 Ω + v 2 H,T Ω + ωv 2 H Y n +C 2 u 2 L 2,T ;H Ω + w 2 L 2,T ;H Ω + Cv, τ t ŵ ρ Q Y 2 xj wωw j 2 + ŵ ρ Q Y 2 xj w y ωw dxdt j 2 C C τ Ω 2 τ Ω =j ˆv ρ Q Y xj vωv j ˆv ρ Ω 2 = ŵ ρ Q Y 2 xj wωw j ŵ ρ +C w 2 L 2,T ;H 2 Ω + w 2 H,T Ω +C 2 v 2 L 2,T ;H Ω + C w, Q Y xj v y ωv dxdt j 2 + Q Y 2 xj w y ωw j 2 dxdt + ωv 2 H Y n where τ C v := C 2 Ω Q Y t xj vω j v Q xj vω j v 2 + Q Y xj uω j u 2 + Q Y xj vω j v 2 + Q Y xj v y ω j v 2 dxdt + τ +C 2 Ω 2 = τ ˆΩ,ρ,n Q Y2 xj wω jw 2 + Q Y2 xj w y ω jw 2 dxdt Q Y xj vω j v 2 dxdt C 2 v 2 L 2,T ;H 2 Ω + 2 v 2 H,T Ω + v 2 L 2,T ;H 2 Ω ω v 2 H Y n + 2 C u 2 L 2,T ;H Ω ω u 2 L 2 Y n + 2 C w 2 L 2,T ;H Ω ω w 2 H Y 2 n

22 UNFOLDING AND CORRECTOR ESTIMATES 9 and C w := 2 C τ Ω 2 Q Y 2 t x wωw j 2 + Q Y2 x wωw j 2 + Q Y2 xj wωw j 2 + Q Y2 x w y ωw j 2 dxdt + 2 C τ + ˆΩ 2,ρ,n τ Ω Q Y xj vωv j 2 + Q Y xj v y ωv j 2 dxdt Q Y2 xj wωw j 2 dxdt 2 v 2 L 2,T ;H Ω ω v 2 H Y n +C w 2 L 2,T ;H 2 Ω + 2 w 2 L 2,T ;H 2 Ω + 2 w 2 H,T Ω ωw 2 H Y 2 n. For suffcently small, addng the all estmates, removng ρ by usng the estmates 2, applyng Gronwall s nequalty and consderng that u = u, v = v, v = v we obtan the estmates for u, v, w, stated n the theorem. To obtan the estmate for r UΓ r, we consder the equatons for TΓ r and r wth the test functon TΓ r r. Usng the propertes of UΓ, the local Lpschtz contnuty of η, and Gronwall s nequalty, yelds Γ r U Γ r 2 dγ C TΓ r r 2 dγ y dx Ω Γ t [ û +C + 2 C Ω TΓ r r 2 dγ C Ω Γ t Q Y xj uω j u t Ω Γ T Γ u u 2 dγdτ + TΓ u M Y u 2 + M Y u u 2 dγdτ Ω Γ û Q Y xj u y ωu j 2 Q Y xj uωu j Q Y xj uωu j 2 + Q Y xj u y ωu j 2] dxdτ C + 2 u 2 L 2,T ;H 2 Ω + u 2 H,T Ω + v 2 L 2,T ;H 2 Ω + v 2 H,T Ω + w 2 L 2,T ;H 2 Ω + w 2 H,T Ω + r 2 L,T Ω Γ. 4.. Estmates of the error terms. Now, we proceed to estmatng the error terms R u, R 2 u, and R 3 u. Usng the defnton of ρ, the extenson propertes of φ, Theorem 3.5, and the estmates 2 we obtan Ω Y D u yρ u + C u L 2 ˆΩ,ρ,n + u y ωu j φ x + ˆψ dydx j y ωu j L 2 Y φ L 2 Ω + ˆψ L 2 Ω Y C /2 u H 2 Ω + y ωu j L 2 Y φ L 2 Ω.

23 2 UNFOLDING AND CORRECTOR ESTIMATES The Theorem 3.5 and the estmates 2 and 2 mply τ ρ D u y u + Ω Y xj u y ωu j T Y φ φ y ˆψ dydxdt C /2 + u L 2,T ;H 2 Ω + y ωu j L 2,T Y φ L 2,T Ω. We notce M Y φ = M Y φ and usng estmates 2 and 2, Lemma 3.4, the fact that φ s an extenson of φ from Ω nto Ω and φ = φ a.e n, T Ω, mples τ Ω Y ρ D u u + ρ D u u + u y ωu j Q x Y φ φ dydxdt j u y ωu j L x 2,τ Ω Y Q Y φ φ L 2,τ Ω j C u L 2,T ˆΩ,ρ,n + 2 u L 2 + ωu j L 2 Y φ L 2,τ Ω C /2 + u L 2,T ;H 2 Ω + ωu j L 2 Y φ L 2,τ Ω. Applyng the estmates n Lemma 3.4, yelds τ Ω Y t u T Y u M Y φ + t u φ M Y φ dydxdt C t u L 2,T Ω φ L 2 Ω + t u L 2 Ω φ L 2 Ω. Due to Lpschtz contnuty of f, we can estmate τ Ω Y fu, v TY fu, vm Y φ + fu, v φ M Y φ dydxdt C u L 2,T Ω + v L 2,T Ω φ L 2,τ Ω +C + u L 2,T Ω + v L 2,T Ω φ L 2,τ Ω. For the boundary ntegral we have τ ηu, rtγ φ φ dγdxdt ηu, r L 2,τ Ω Γ Ω Γ T Γ φ M Y φ L 2,τ Ω Γ + M Y φ φ L 2,τ Ω Γ C + u L 2,T Ω + r L,T Ω Γ T Y φ M Y φ L 2,τ Ω;H Y + M Y φ φ L 2,τ Ω C + u L 2,T Ω + r L,T Ω Γ φ L 2,τ Ω.

24 UNFOLDING AND CORRECTOR ESTIMATES 2 Thus, collectng all estmates from above we obtan the estmate for R u: Ru C /2 + u L 2,T ;H 2 Ω + ωu j L 2 Y φ L 2,τ Ω Usng the estmates 2 mples τ +C u H,T Ω + v L 2,T ;H Ω φ L 2,τ;H Ω. Ω ρ D u x M Y xj u y ωu j φ dxdt x M Y xj u L 2,τ ˆΩ yωu j,ρ,n L 2 ˆΩ φ L 2,τ Ω,ρ,n C u L 2,T ;H 2 Ω y ωu j L 2 Y φ L 2,τ Ω. Thus, the last estmate and applyng the estmates 8 and 2 yelds R 2 u u L 2,τ ˆΩ nt,ρ, + yω u L 2 Y n n T Y φ L 2,τ Ω Y +C u L 2,τ;H 2 Ω + y ω u L 2 Y n n T Y φ L 2,τ Ω Y /2 + C u L 2,T ;H 2 Ω + y ω u L 2 Y n n φ L 2,τ Ω. Due to estmates n 2 and n Lemma 3.4 we obtan also Ru 3 C t u L 2,T Ω + f L 2,T Ω + u L 2,T ;H 2 Ω y ω u L 2 Y n n + 2 u L 2,T Ω + 2 u L 2,T Ω y ω u L 2 Y n n φ L 2,τ Ω. In the smlar way we show the estmates for the error terms n the equatons for v and w: R v C 2 + v L 2,T ;H 2 Ω + v H,T Ω + u L 2,T ;H Ω φ 2 L 2,τ;H Ω, + w L 2,T ;H Ω R w C 2 + w L 2,T ;H 2 Ω + w H,T Ω + v L 2,T ;H Ω φ3 L 2,τ;H Ω 2. References [] R.E. Beddoe, H.W. Dorner, Modellng acd attack on concrete: Part. The essental mechansms. Cement and Concrete Research 3525, [2] M. Böhm, F. Jahan, J. Devnny, G. Rosen, A movng-boundary system modelng corroson of sewer ppes. Appl. Math. Comput ,

25 22 UNFOLDING AND CORRECTOR ESTIMATES [3] V. Chalupecký, T. Fatma, A. Muntean, Multscale sulfate attack on sewer ppes: Numercal study of a fast mcro-macro mass transfer lmt. Journal of Math-for-Industry 22, 2B-7, 7 8. [4] D. Coranescu, A. Damlaman, G. Grso, Perodc unfoldng and homogenzaton, SIAM J. Math. Anal., 428, 4, [5] D. Coranescu, P. Donato, R. Zak, Perodc unfoldng and Robn problems n perforated domans, C. R. Acad. Sc. Pars, 34226, [6] D. Coranescu, P. Donato, R. Zak, Asymptotc behavor of ellptc problems n perforated domans wth nonlnear boundary condtons, Asymptotc Analyss, 5327, [7] D. Coranescu, P. Donato, R. Zak, The perodc unfoldng method n perforated domans, Portugalae Mathematca, 6326, [8] D. Coranescu, A. Damlaman, P. Donato, G. Grso, R. Zak, The perodc unfoldng method n domans wth holes, 2, preprnt. [9] L.C. Evans, Partal Dfferental Equatons, AMS, Phladelpha, NY, 26. [] T. Fatma, A. Muntean, Sulfate attack n sewer ppes : dervaton of a concrete corroson model va two-scale convergence. CASA Report No. -7, Technsche Unverstet Endhoven 2. [] T. Fatma, N. Arab, E.P. Zemskov, A. Muntean, Homogenzaton of a reacton-dffuson system modelng sulfate corroson n locally-perodc perforated domans. J. Engng. Math. 692, 2, [2] G. Grso, Error estmate and unfoldng for perodc homogenzaton, Asymptotc Analyss, 424, [3] G. Grso, Interor error estmate for perodc homogenzaton, Comptes Rendus Mathematque, 34 25, 3, [4] J.L. Lons, Quelques méthodes de résoluton des problèmes aux lmtes non lnéares. Edtons Dunod, Pars, 969. [5] G.M. Leberman, Second Order Parabolc Dfferental Equatons, World Scentfc, Sngapore, 996. [6] A. Marcnak-Czochra, M. Ptashnyk, Dervaton of a macroscopc receptor-based model usng homogenzaton technques, SIAM J. Math. Anal. 428,,

26 UNFOLDING AND CORRECTOR ESTIMATES 23 [7] T.L. van Noorden, A. Muntean, Homogenzaton of a locally-perodc medum wth areas of low and hgh dffusvty. European Journal of Appled Mathematcs, 2, to appear. [8] V.V. Zhkov, A.L. Pyatnsk, Homogenzaton of random sngular structures and random measures, Izv. Math., 726,, Appendx Lemma 5.. Let Ω R n be a bounded doman wth Lpschtz boundary. If z H Ω L Ω, then z L Ω. Proof. Let z H Ω L Ω. Snce C Ω s dense n H Ω, we consder a sequence of smooth functons {f n } C Ω, such that f n z n H Ω and f n L Ω z L Ω. Applyng the trace theorem, see [9], we obtan f n z n L 2 Ω. Thus, there exsts a subsequence {f n } {f n } convergng pontwse,.e., f n x zx a.e. x Ω, and, due to f n x L Ω z L Ω, follows that z L Ω z L Ω a.e. x Ω. Lemma 5.2. [4, 5]. For w L p Ω, p [,, we have T Y w L p Ω Y = Y /p w L 2 Ω,nt Y /p w L 2 Ω. 2. For u L p Γ, p [,, we have T Γ u L p Ω Γ = /p Y /p u L 2 Γ,nt /p Y /p u L 2 Γ. 3. If w L p Ω, p [, then T Y w w strongly n L p Ω Y as. 4. For w W,p Ω, < p < +, T Γ w L p Ω Γ C w L p Ω + w L p Ω n. 5. For w W,p Ω holds TY w L p Ω, W,p Y and y TY w = TY w. 6. Let v L p pery and v x = v x, then T Y v x, y = vy. 7. For v, w L p Ω and φ, ψ L p Γ holds T Y v w = T Y vt Y w and T Γ φ ψ = T Γ φt Γ ψ. Theorem 5.3. [7, 8] Let p, and =, 2.. For {φ } W,p Ω satsfes φ W,p Ω C, there exsts a subsequence of {φ } stll denoted by φ, and φ W,p Ω, ˆφ L p Ω; W,p pery, such that TY φ φ strongly n L p,p loc Ω; WperY, T Y φ φ weakly n L p Ω; W,p pery, T Y φ φ + y ˆφ weakly n L p Ω Y. 2. For {φ } W,p Ω such that φ W,p Ω C there exsts a subsequence of {φ } stll denoted by φ and φ W,p Ω, φ L p Ω; WperY,p such that T Y φ φ strongly n L p Ω; W,p Y, T Y φ φ + y φ weakly n L p Ω Y.

27 24 UNFOLDING AND CORRECTOR ESTIMATES 3. For {ψ } L p Γ such that /p ψ L p Γ C there exsts a subsequence of {ψ } and ψ L p Ω Γ such that T Γ ψ ψ weakly n L p Ω Γ. Proposton 5.4. [7, 8]. The operator UY s formal adjont and left nverse of TY,.e for φ L p Ω φx a.e. for Ω,nt, UY TY φx = a.e. for Ω \ Ω,nt. 2. For φ L p Ω Y holds U Y φ L p Ω Y /p φ L p Ω Y. Theorem 5.5. [2] For any φ H Ω, there exsts ˆφ H pery ; L 2 Ω: ˆφ H Y ;L 2 Ω C φ L 2 Ω n, T x φ φ y ˆφ L 2 Y ;H Ω n C φ L 2 Ω n Theorem 5.6. [3] For any φ H Ω there exsts ˆφ H pery ; L 2 Ω: ˆφ H Y ;L 2 Ω C φ L 2 Ω n, T x φ φ y ˆφ L 2 Y ;H Ω n C φ L 2 Ω n + C φ L 2 ˆΩ,3 n, where ˆΩ,l = {x R n : dstx, Ω < l n}. The proofs of Theorems 5.5, 5.6 and 3.5 are based on the followng fundamental results: Theorem 5.7. [2] For any φ H Y, X and X separable Hlbert space, there exsts a unque ˆφ HperY, X, =, 2, such that φ ˆφ HperY, X and ˆφ H Y,X φ H Y,X, φ ˆφ H Y,X C φ ej +Y j φ Y j H /2 Y j,x. Theorem 5.8. [2] For any Φ W,p Y and for any k, k {,..., n}, there exsts ˆΦ k W k = {φ W,p Y, φ = φ + e j, j {,..., k}}, such that Φ ˆΦ k W,p Y C k Φ ej +Y j where the constant C s ndependent on n, Y j Φ Y j W /p Y j, =, 2, = {y Y, y j = }, j {,..., n}.

28 PREVIOUS PUBLICATIONS IN THIS SERIES: Number Authors Ttle Month -34 M.E. Hochstenbach L. Rechel Combnng approxmate solutons for lnear dscrete ll-posed problems May -35 E.J. Brambley M. Darau S.W. Renstra The crtcal layer n sheared flow May -36 M. Oppeneer W.M.J. Lazeroms S.W. Renstra R.M.M. Matthej P. Sjtsma Acoustc modes n a duct wth slowly varyng mpedance and nonunform mean flow and temperature May -37 M.E. Hochstenbach N. Mcnnch L. Rechel Dscrete ll-posed leastsquares problems wth a soluton norm constrant June -38 T. Fatma A. Muntean M. Ptashnyk Unfoldng-based corrector estmates for a reactondffuson system predctng concrete corroson June Ontwerp: de Tantes, Tobas Baanders, CWI