The Stokes equations with Fourier boundary conditions on a wall with asperities

THE STOKES EQUATIONS WITH FOURIER BOUNDARY CONDITIONS ON A WALL WITH ASPERITIES Youce AMIRAT∗ , Blanca CLIMENT† , En ique FERN ´ ANDEZ-CARA†and Jacques SIMON∗ Abs ac We s udy he e ec o he ugosi y o a wall on he solu ion o he S okes sys em complemen ed wi h Fou ie bounda y condi ions. We conside he case o small pe iodic aspe i ies o size ε. We p o e ha he eloci y ield, p essu e and d ag espec i ely con e ge o he eloci y ield, p essu e and d ag o a ho- mogenized S okes p oblem, whe e a di e en ic ion coe icien appea s. This shows ha , con a ily o he case o Di ichle bounda y condi ions, ugosi y is dominan he e. ∗Labo a oi e de Ma h´ema iques Appliqu´ees, Uni e si ´e Blaise Pascal (Cle mon -Fe and 2), 63177 Aubi`e e Cedex, F ance, E-mails: ami a @uc ma.uni -bpcle mon . , [email p o ec ed] bpcle mon . . †Dp o. de Ecuaciones Di e enciales y An´alisis Num´e ico, Uni e sidad de Se illa, Ap do. 1160, 41080 Se illa, Spain, E-mails: blanca@nume .us.es, ca [email p o ec ed]. 0 1 In oduc ion Le us conside a luid in a domain Oε, limi ed a he bo om by a plane wall P and a he op by a wall Rε. We assume ha Pmo es a a cons an eloci y while Rεis a es . The la e is assumed o consis o a smoo h wall Rco e ed wi h pe iodically dis ibu ed aspe i ies o small size ε. We a e hen conce ned wi h he asymp o ic beha iou , as ε→0, o he eloci y and he p essu e in he luid. The case in which he luid adhe es o he walls has been conside ed in [?], [?] and [?]. In [?], [?], he wall Ris a pla e. Using bounda y laye co ec o s, i is p o ed ha , ou side a neighbou hood o he ugose zone, he low beha es asymp o ically as a Coue e low, up o an exponen ially small e o . An accu a e app oxima ion o he d ag is gi en which shows ha he e is no palpable d ag educ ion. These esul s ha e been hen ex ended o he case o a low go e ned by Na ie -S okes equa ions, see [?]. Le us also men ion a ecen pape by W. J¨age and A. Mikeli´c [?] on he lamina iscous channel low, wi h he la e al su ace o he channel con aining su ace i egula i ies. The luid sa is ies a no-slip bounda y condi ion on he ugose su ace and i is supposed ha a uni o m p essu e g adien is main ened in he longi udinal di ec ion in he channel. So he limi low is a Hagen-Poiseuille low. Using he co eponding bounda y laye s, he au ho s de i e a wall law which gi es an app oxima ion o he angen ial d ag o ce a o de O(ε3/2). As appea s om he quo ed wo ks, i he luid is assumed o adhe e o he walls, ha is o say, i Di ichle condi ions a e imposed, hen he e ec o ugosi y is negligible. Mo e p ecisely, he d ag, he eloci y ield, and he p essu e ela ed o Rεcon e ge o hose ela ed o Ras ε→0. In his pape , we assume ha , on he walls, he luid sa is ies condi ions o Fou ie kind. In pa icula , on he ugose wall we impose σ·n+ku = 0 on Rε,(1) whe e σis he usual s ess enso and kis a ic ion coe icien . We p o e ha hese condi ions, con a ily o Di ichle (no-slip) condi ions, b ing a leading pa o ugosi y : as ε→0, he d ag expe ienced by he ugose wall Rεcon e ges o he d ag associa ed wi h he smoo h wall Rp o ided wi h an homogenized ic ion coe icien Kwhich is no cons an and depends on he p o ile o aspe i ies. We calcula e he limi low and we gi e es ima es o he de ia ions o he d ag, eloci y ield and p essu e, in e ms o he size εo he aspe i ies. In he pa icula case o a pla e, he limi d ag is la ge han he d ag o he smoo h wall, see Rema k 2.1. In he case o Laplace o Poisson equa ion wi h Fou ie o Neumann bound- a y condi ions, simila esul s o hose o he p esen pape ha e been ob ained by O.A. Oleinik, A.S. Shamae and G.A. Yosi ian [?] and by G.A. Chechkin, A. F ied- man and A.L. Pia ni ski [?]. See also E. Sanchez-Palencia [?]. 1 In ac , i is no comple ely ealis ic o assume ha a luid sa is ies (??), unless he wall Rεhas some kind o po osi y, see R.L. Pan on [?, p. 149–152]. I would be mo e app op ia e, ins ead, o impose he slip condi ions u·n= 0,(σ·n) ang +ku = 0 on Rε,(2) whe e (σ·n) ang deno es he angen ial componen o σ·n. Howe e , (??) may be used o la ge kas a o mal app oxima ion o he Di ichle condi ion u= 0. I would be e y in e es ing o ex end he p esen esul s o he case in which con- di ions like (??) a e conside ed, bu he e a e some echnical di icul ies o do his. Ou esul s may be conside ed as a i s s ep in his di ec ion. Le us men ion he e ha slip bounda y condi ions ha e been conside ed by G. Allai e [?], and D. Cio- anescu, P. Dona o and H. I. Ene [?] o homogeniza ion o S okes o Na ie -S okes equa ions in domains con aining pe iodically dis ibu ed obs acles. Fo condi ions o Fou ie kind, we e e o C. Conca [?], [?]. 2 The main esul The smoo h wall Ris assumed o be he g aph o a unc ion on IR2, wi h is Lipschi z-con inuous, posi i e and (l1, l2)-pe iodic (3) ( he la e means ha is pe iodic wi h espec o xiwi h pe iod li o i= 1 o 2). Fo each ε > 0, he ugose wall Rεis assumed o be he g aph o he unc ion ε de ined on IR2as ollows : o any x′= (x1, x2), ε(x′) = (x′)(1 + εη(x′,x′ ε)).(4) He e, η=η(x′, y′) is a unc ion on IR2×IR2sa is ying ηis Lipschi z-con inuous and l1, l2-pe iodic wi h espec o x′and y′. In o de o ensu e ha εsa is ies (??), we assume ha εkηkL∞(S2)≤1 2,1 εis an in ege numbe . (5) The luid occupies he unbounded domain Oε={x∈IR3:x′∈IR2,0< x3< ε(x′)}. Se ing S= (0, l1)×(0, l2), Oεcan be iewed as gene a ed by pe iodic ansla ions o he bounded domain Ωε={x∈IR3:x′∈S, 0< x3< ε(x′)}. 2 Obse e ha ∂Ωεconsis s o he ollowing pa s o he walls Rεand P Rε={x∈IR3:x′∈S, x3= ε(x′)}, P ={x∈IR3:x′∈S, x3= 0} and he la e al imma e ial bounda y L={x∈IR3:x′∈∂S, 0≤x3≤ ε(x′)}. Fo each m≥0, le us in oduce he space Hm pe (Ωε) = { ∈Hm loc(Oε) : ∈Hm(Ωε), (x+ (l1,0,0)) = (x+ (0, l2,0)) = (x) o a.e. x∈ Oε}. In o he wo ds, Hm pe (Ωε) is o med by all (l1, l2)-pe iodic unc ions on Oεwhich a e Hmin any bounded subse (and no only in compac subse s, as Hm loc means). As usual, o m= 0, we w i e L2 pe (Ωε). The eloci y ield and he p essu e a e assumed o sa is y                uε∈(H1 pe (Ωε))3, pε∈L2 pe (Ωε), −ν∆uε+∇pε= 0,∇·uε= 0 in Oε, σε·n+ku = 0 on Rε, σε·n+k(u−g) = 0 on P, (6) whe e ν > 0 is he iscosi y, σε=σ(uε, pε) = −pεId + ν(∇uε+ ∇uε),(7) nis he ou wa ds uni no mal ec o ield, k > 0 is a ic ion coe icien and g= (g′,0) is he eloci y o he wall P. The exis ence and uniqueness o a solu ion is p o ed in p oposi ion 3.1. Rema k ha he bounda y condi ions on Rεand Pa e meaning ul, since (??) gi es σε∈ (L2 pe (Ωε))3×3and ∇·σε=−∇pε+ν∆uε+ ∇(∇·uε) = 0.(8) This allows o de ine he no mal ace σε·nin (H−1/2 loc (∂Oε))3. The hyd odynamical d ag Tεassocia ed wi h he bounded pa Rεo he wall Rε( he d ag o he whole wall is in ini e) is by de ini ion he p ojec ion o he o ce exe ed on Rεby he luid, ha is Tε=−g·ZRε σε·n ds. (9) 3 Thanks o he bounda y condi ion on Rε, i eads as well Tε=g·ZRε uεds. Le us now de ine he homogenized ic ion coe icien K=K(x′, (x′)). Fi s , we in oduce he ollowing unc ion m∈L∞(IR2×IR2) : o any (x′, y′)∈IR2×IR2, m(x′, y′) = 1 + |∇ (x′) + (x′)∇y′η(x′, y′)|2 1 + |∇ (x′)|21 2.(10) Then we pu , o all x′∈IR2, hmi(x′) = 1 |S|ZSm(x′, y′)dy′. Ob iously, we ha e hmi ∈ L∞(IR2). The homogenized ic ion coe icien is gi en R as ollows : o all x′∈IR2, K(x′, (x′)) = khmi(x′).(11) I is he e o e a (l1, l2)-pe iodic unc ion which belongs o L∞(R). We will p o e ha (uε, pε) con e ges in an app op ia e sense o he unique solu- ion o he sys em                u0∈(H1 pe (Ω))3, p0∈L2 pe (Ω), −ν∆u0+∇p0= 0,∇·u0= 0 in O, σ0·n+Ku0= 0 on R, σ0·n+k(u0−g) = 0 on P, (12) whe e σ0=−p0Id + ν(∇u0+ ∇u0), Ris he g aph o ,Ois he domain bounded by Rand Pand Ω = {x∈IR3:x′∈S, 0< x3< (x′)}. Fu he mo e, we will p o e ha he limi d ag is T0=−g·ZRσ0·n ds =g·ZRKu0ds. No ice ha , in gene al, u0is no he eloci y ield ela ed o he smoo h wall R because K6=k, and T0is no he d ag expe ienced by R. Since he domain Ωε a ies wi h ε, he con e gence o uεand pεcanno hold in he whole domain Ω. We will ob ain con e gence ou side a neighbou hood o Ro a bi a y small size δ > 0, ha is, in all subdomains o he o m ωδ={x∈IR3:x′∈S, 0< x3< (x′)−δ}. Ou main esul is he ollowing. 4 Theo em 2.1 Assume ∈W3,∞(IR2). The e exis s C > 0such ha , o any ε > 0 sa is ying (??), we ha e : |Tε−T0| ≤ C√ε. (13) Mo eo e , o any δ > 0 he e exis s Cδ>0such ha , o any εsa is ying (??), we ha e : kuε−u0kH1(ωδ)≤Cδ√ε, (14) kpε−p0kL2(ωδ)≤Cδ√ε. (15) ⊔⊓ Rema k 2.1 The d ag Tεo a ugose pla e is s ic ly g ea e han he d ag To he co esponding (homogenized) smoo h pla e. Indeed, assume ha (x′)≡l3(a posi i e eal numbe ) and ηdepends only on y′. Then hmiis independen o x′and is g ea e han 1 unless ηis a cons an . Acco dingly, we ha e lim ε→0Tε=T0=νl1l2khmi|g|2 ν(1 + hmi) + l3khmi> T =νl1l2k|g|2 2ν+l3k. ⊔⊓ Rema k 2.2 No ice ha 1 1 + |∇ (x′)|2≤ hmi(x′)≤1 + ( (x′))2 1 + |∇ (x′)|2 1 |S|ZS|∇y′η(x′, y′)|dy′1/2 . The las quan i y is bounded om abo e by he local asymp o ic a io o he Rε-a ea and he R-a ea. In o he wo ds, o any x′∈S, we ha e hm′i(x′)≤lima→0limε→0|RεSB(x′;a)| lima→0|RSB(x′;a)| whe e B(x′;a) is he ball cen e ed a (x′, (x′)) o adius a.⊔⊓ Rema k 2.3 The d ag Tεcan also be w i en in he o m Tε=g·ZPσε·n ds =−kg ·ZP(uε−g)ds. (16) Indeed, he ollowing equali ies hold : −g·ZRε σε·n ds =−g·ZRε∪Pσε·n ds +g·ZPσε·n ds =−g·ZΩε∇·σεds +g·ZPσε·n ds. A simila equali y holds o T0.⊔⊓ 5 3 Exis ence, uniqueness and es ima es We will p o e ha he e exis s exac ly one solu ion (uε, pε) o (??) and exac ly one solu ion (u0, p0) o (??). No ice ha (??) is simila o (??) wi h a a ying ic ion coe icien , since bo h walls Rεand Ra e he g aphs o pe iodic Lipschi z unc ions. The e o e, in o de o pu hese wo p oblems in he same amewo k, we will assume in his Sec ion ha k∈L∞ pe (Ωε), k ≥κ > 0,(17) whe e κis a eal numbe . We will use he ollowing a ia ional o mula ion :            uε∈(H1 pe (Ωε))3, pε∈L2 pe (Ωε), 2νZΩε e(uε) : e(ϕ)−ZΩε pε∇·ϕ+ZRε∪Pkuε·ϕ=ZPkg ·ϕ∀ϕ∈(H1 pe (Ωε))3, ∇·uε= 0, (18) whe e e(ϕ) = 1 2(∇ϕ+ ∇ϕ) and e(u) : e(ϕ) = Pi,j eij(u)eij(ϕ). P oposi ion 3.1 P oblem (??)is equi alen o (??)and possesses exac ly one so- lu ion. Fu he mo e, one has kuεkH1(Ωε)+kpεkL2(Ωε)≤C, (19) whe e Cis independen o ε.⊔⊓ Ob iously, his esul also p o ides he exis ence and uniqueness o a solu ion (u0, p0) o (??). In o de o p o e his p oposi ion, we need some p e ious esul s. In pa icula , we need a Ko n inequali y o a special class o s a -shaped domains. By de ini ion, Dis s a -shaped, wi h espec o a ball Bi he segmen connec ing any wo poin s x∈Band y∈Dlies in D. Lemma 3.1 The e exis s C > 0such ha , o any bounded domain D⊂IR3o diame e Rwhich is s a -shaped wi h espec o a ball Bo adius ρand o any ∈(H1(D))3, he ollowing inequali y holds k∇ k2 L2(D)≤CR ρ3ke( )k2 L2(D)+k∇ k2 L2(B).(20) ⊔⊓ 6 Fo he p oo see O.A. Oleinik, A.S. Shamae and G.A. Yosi ian, [?, Theo- em 2.10, p. 23]. Le us pu D⋆={x∈IR3:x′∈S, 0< x3<l3 2}. Lemma 3.2 The e exis s C > 0, only depending on Sand l3, such ha , o all ∈(H1(D⋆))3, k∇ k2 L2(D⋆)≤Cke( )k2 L2(D⋆)+ZP| |2ds.(21) P oo : Suppose he asse ion in his lemma is alse. Then, o each m≥1, he e exis s wm∈(H1(D⋆))3such ha k∇wmk2 L2(D⋆)> mke(wm)k2 L2(D⋆)+ZP|wm|2ds. Le us pu m=wm/k∇wmkL2(D⋆). Then k∇ mkL2(D⋆)= 1 and ke( m)k2 L2(D⋆)+ZP| m|2<1 m, whence we ob ain he ollowing as m→ ∞ : e( m)→0 in (L2(D⋆))3×3,ZP| m|2ds →0.(22) On he o he hand, he ollowing es ima e holds o all ∈(H1(D⋆))3: ZD⋆| |2ds ≤l3ZP| |2+l3 2ZD⋆|∇ |2.(23) Indeed, o any egula and any x= (x′, x3) in D⋆, one has (x) = (x′,0) + Zx3 0∂x3 (x′, y3)dy3. Consequen ly, | (x)|2≤2| (x′,0)|2+ 2Zl3/2 0|∂x3 (x′, y3)|dy32, whence Zl3/2 0| (x′, x3)|2dx3≤l3| (x′,0)|2+Zl3/2 0|∂x3 (x′, y3)|dy32 7 and ZSZl3/2 0| (x′, x3)|2dx3dx′ ≤l3ZP| (x′,0)|2dx′+1 2ZSZl3/2 0|∂x3 (x′, y3)|2l3dy3dx′. This p o es (??), a leas when is egula enough. By densi y, (??) holds o all in (H1(D⋆))3. I ollows om (??) ha mis uni o mly bounded in (H1(D⋆))3. F om he compac ness o he embedding H1(D⋆)֒→L2(D⋆), he e exis s a subsequence, s ill deno ed m, ha con e ges s ongly in (L2(D⋆))3×3 o some ∈(H1(D⋆))3. In iew o Ko n inequali y in Lipschi z domains, one has k m′− mk2 H1(D⋆)≤C(ke( m′)−e( m)k2 L2(D⋆)+k m′− mk2 L2(D⋆)), whe e he cons an Cdepends only on Sand l3, see [?], [?]. These inequali ies and (??) show ha mcon e ges s ongly in H1(D⋆) o and, also, ha k∇ kL2(D⋆)= 1,ke( )kL2(D⋆)= 0,ZP| |2ds = 0.(24) Bu he equali y e( ) = 0 implies ha is a igid displacemen , i.e. =Ax +b whe e Ais a skew-symme ic cons an ma ix and bis a cons an ec o . This ac , oge he wi h he hi d equali y in (??), implies = 0. This leads o a con adic ion and p o es he lemma. ⊔⊓ Lemma 3.3 Le εsa is y (??). Then, o any ∈(H1(Ωε))3, k k2 H1(Ωε)≤Cke( )k2 L2(Ωε)+ZP| |2,(25) whe e Cdepends only o l1,l2, and η. P oo : The unc ion εde ined by (??) is Lipschi z-con inous, wi h a Lipschi z cons an independen o ε. I is also bounded om below by a posi i e numbe independen o ε. The e o e, i l1and l2a e small enough, he e exis s a ball B independen o εsuch ha Ωεis s a -shaped wi h espec o B. Mo eo e , Bcan be chosen in D⋆. Fo a bi a ily gi en l1and l2, by di iding Sin su icien ly small squa es, i ollows ha Ωεis he union o mdomains Ωi εwhich a e espec i ely s a - shaped wi h espec o he balls Bi, wi h mand Biindependen o ε. Lemma 3.1 yields he ollowing o each iand o all ∈(H1(Ωε))3: k∇ k2 L2(Ωi ε)≤Cke( )k2 L2(Ωi ε)+k∇ k2 L2(Bi). Adding hese inequali ies o i= 1, ..., m, we ind ha k∇ k2 L2(Ωε)≤Cke( )k2 L2(Ωε)+k∇ k2 L2(D⋆) 8 5 Some echnical lemmas Lemma 5.1 Le γ=γ(x′, y′)be a Lipschi z unc ion on S2, pe iodic wi h espec o y′and sa is ying ZSγ(x′, y′)dy′= 0 o any x′∈S. The e exis s a eal numbe Csuch ha : i)Fo all unc ions ϕand ψin H1(Ω) and 0< ε ≤1, we ha e ZΩγ(x′,x′ ε)ψ(x)ϕ(x)dx≤CεkψkH1(Ω)kϕkH1(Ω).(53) ii)Fo all unc ions ψ∈H2(Ω) and ϕ∈H1(Ω) and 0< ε ≤1, we ha e ZSγ(x′,x′ ε)ϕ(x′, (x′))ψ(x′, (x′)) dx′≤C√εkψkH2(Ω)kϕkH1(Ω).(54) ⊔⊓ Fo he p oo , see O.A. Oleinik, A.S. Shamae and G.A. Yosi ian [?, Lemma 1.6, p. 8]. Lemma 5.2 Assume ∈W3,∞(S). The e exis s a posi i e numbe C, independen o u0and ε, wi h he ollowing p ope ies : i)Fo 0< ε ≤1, we ha e k∇·( Nεu)kL2(Ω) ≤Cεku0kH2(Ω).(55) ii)Fo any unc ion ϕ∈(H1(Ω))3and 0< ε ≤1, we ha e ZΩ(b eε(u) : b eε(ϕ))jε−ZΩe(u0) : e(ϕ)≤C√εku0kH3(Ω)kϕkH1(Ω).(56) P oo : No ice ha u0∈(H3(Ω))3, since ∈W3,∞(S). Le us i s p o e (??). Since by de ini ion Nε= (1 + εηε)Mε, we see ha ∇·( Nεw) = ∂x1((1 + εηε)w1) + ∂x2((1 + εηε)w2) +∂x3(−x3ε(∂x1ηε)w1−x3ε(∂x2ηε)w2+w3) =∇·w+εηε(∂x1w1+∂x2w2)−x3(ε(∂x1ηε)∂x3w1+ε(∂x2ηε)∂x3w2), 15 o any unc ion w. Wi h he pa icula choice w=u=u0+x3εηε∂x3u0, since ∇·u0= 0, we ob ain ∇·u=x3(ε(∂x1ηε)∂x3u01 +ε(∂x2ηε)∂x3u02) + εηε∂x3u03 and εηε∂x1u1−x3ε(∂x1ηε)∂x3u1=εηε∂x1u01 −x3ε(∂x1ηε)∂x3u01 +x3(εηε)2∂2 x1x3u01 −x2 3ε2ηε(∂x1ηε)∂2 x3x3u01. Using he co esponding simila equali y o u2, we deduce ha ∇·( Nεu) = x3(εηε)2(∂2 x1x3u01 +∂2 x2x3u02) −x2 3εηε(ε(∂x1ηε)∂2 x3x3u01 +ε(∂x2ηε)∂2 x3x3u02). The inequali y (??) ollows, since |ηε(x′)| ≤ kηkL∞(S2)and |ε∂xiηε(x′)|=ε∂xiη(x′,x′ ε) + ∂yiη(x′,x′ ε)≤2kηkW1,∞(S2). Le us now p o e (??). The de ini ion (??) o b eεyields he ollowing o all u and ϕ: (b eε(u) : b eε(ϕ)) = 1 2Mε∇u:Mε∇ϕ+1 2Mε∇u: (Mε∇ϕ). Mo eo e , he de ini ion o Mεleads o he iden i ies, o 1 ≤k≤3, (Mε∇ϕ)ik =            ∂xiϕk−1 jε x3ε(∂xiηε)∂x3ϕki= 1,2, ∂x3ϕk−1 jε εηε∂x3ϕki= 3. Fo i= 3, we ha e used jε= 1 + εηεand hus 1/jε= 1 −εηε/jε. In pa icula , we ob ain (Mε∇u)ik =            ∂xiu0k+x3εηε∂2 xix3u0k−1 jε ε2ηε∂xiηε∂2 x3x3u0ki= 1,2, ∂x3u0k+1 jε x3εηε∂2 x3x3u0ki= 3. The e o e, b eε(u)jε:b eε(ϕ)−e(u0) : e(ϕ) = 2 X i=1 3 X k=1 aik +X α εηεbα, whe e aik =−1 2x3ε(∂xiηε)∂x3ϕk(∂xiu0k+∂xku0i) 16 and each bαis he p oduc o a i s o second de i a i e o u0, a de i a i e o ϕand some o he ollowing unc ions : x3,jε, (jε)−1,εηεand ε∂xiηε. Since |εηε| ≤ Cε, we see ha ZΩ|εηεbα| ≤ Cεku0kH2(Ω)kϕkH1(Ω). Since ε∂xiηε(x′) = ε∂xiη(x′,x′ ε) + ∂yiη(x′,x′ ε), we can also pu aik =−1 2x3∂yiηε∂x3ϕk(∂xiu0k+∂xku0i) + ε∂xiηεb′ ik,(57) whe e he b′ ik ha e he same s uc u e as bα. Hence, ZΩ|ε∂xiηεb′ ik| ≤ Cεku0kH2(Ω)kϕkH1(Ω). The in eg als o he o he e ms in he igh -hand side o (??) a e as ollows : ZΩ−x3∂yjηε(∂x3ϕk)(∂xlu0m) =ZΩ∂yjηε∂x3(x3∂xlu0m)ϕk−Z∂Ωx3∂yjηεϕk(∂xlu0m)n3ds. Since 1/ε is an in ege and ηis pe iodic wi h espec o all i s a iables, we ha e ZS∂yjηε=ZS∂yiη(x′,x′ ε)dx′= 0.(58) The e o e, hanks o lemma 5.1, pa i), ZΩ∂yjηε∂x3(x3∂xlu0m)ϕkdx≤Cεku0mkH2(Ω)kϕkkH1(Ω). On he o he hand, om lemma 5.1, pa ii), we ha e Z∂Ω∂yjηεx3(∂xlu0m)ϕkn3ds=ZS∂yjη(x′,x′ ε)(x3(∂xlu0m)ϕk)(x′, (x′)) dx′ ≤√εku0mkH3(Ωε)kϕkkH1(Ωε). He e, we ha e used ha n3ds =dx′. This p o es ha ZΩai,k≤Cεku0kH3(Ω)kϕkkH1(Ω). This comple es he p oo o (??). 17 Lemma 5.3 Le us assume ha ∈W3,∞(S). Then, o any εsa is ying (??)and any ϕ∈(H1 pe (Ω))3, we ha e ZΩp0∇·( Nεϕ−ϕ)≤C√εkp0kH2(Ω)kϕkH1(Ω),(59) whe e Cis independen o ε,p0and ϕ. P oo : Le us i s no ice ha p0∈H2(Ω), since ∈W3,∞(S). We ha e ∇·( Nεϕ−ϕ) = εηε(∂x1ϕ1+∂x2ϕ2)−x3 2 X j=1 (ε∂xjηε+∂yjηε)∂x3ϕj. Then, mul iplying by p0and in eg a ing by pa s he las e m, we see ha ZΩp0∇·( Nεϕ−ϕ) = ZΩεp0 2 X j=1 (ηε∂xjϕj−x3∂xjηε∂x3ϕj) +ZΩ∂x3(x3p0) 2 X j=1 ∂yjηεϕj−ZS (x′)p0(x′, (x′)) 2 X j=1 ∂yjηε(x′)ϕj(x′, (x′)) dx′. In he igh -hand side, he i s in eg al is bounded by ZΩεp0 2 X j=1 (ηε∂jϕj−εx3∂xjηε∂x3ϕj)≤Cεkp0kL2kϕjkH1(Ω). Using lemma 5.1, pa i) and (??), he second in eg al is bounded by ZΩ∂x3(x3p0) 2 X j=1 ∂yjηεϕj≤Cεkp0kH2(Ω)kϕjkH1(Ω). On he o he hand, om lemma 5.1, pa ii), he bounda y in eg al sa is ies ZS (x′)p0(x′, (x′)) 2 X j=1 ∂yjηε(x′)ϕj(x′, (x′)) dx′≤C√εkp0kH2(Ω)kϕkH1(Ω). Hence, lemma 5.3 is p o ed. ⊔⊓ Lemma 5.4 Assume ha ψ∈L2 pe (Ω) is such ha ZΩψ= 0. Then he e exis s ϕ∈(H1 0(Ω))3such ha ∇· (Nεϕ) = ψin Ω,kϕkH1(Ω) ≤CkψkL2(Ω),(60) whe e Cis independen o ε,ψand ϕ. 18 P oo : Le us pu ψε=1 1 + εηε(ψ◦L−1 ε). Clea ly, ψε∈L2 pe (Ωε). F om lemma 3.4 wi h w0= 0, he e exis s wε∈(H1 0(Ωε))3 such ha ∇·wε=ψεin Ωεand kwεkH1(Ωε)≤CkψεkL2(Ωε), whe e Cdoes no depend o ε,ψεand wε. In acco dance wi h (??), he unc ion ϕ=b wεsa is ies ∇·( Nεϕ) = ψin Ω. Mo eo e , ZΩε|ψε|2=ZΩ 1 1 + εηε|ψ|2≤ZΩ|ψ|2. ⊔⊓ 6 The p oo s o p oposi ion 4.1 and heo em 2.1 P oo o p oposi ion 4.1: We will i s p oo he es ima es (??). Le us w i e he weak o mula ion o he p oblem sa is ied by (u0, p0) : (u0, p0)∈(H1 pe (Ω))3×L2 pe (Ω),(61) 2νZΩe(u0) : e(ϕ)−ZΩp0∇·ϕ+ZRKuε·ϕ+kZPuε·ϕ=kZPg·ϕ ∀ϕ∈(H1 pe (Ω))3, (62) ∇·u0= 0 in Ω.(63) Subs ac ing (??) om (??), we ob ain, o all ϕ∈(H1 pe (Ω))3, 2νZΩb eε(b uε) : b eε(ϕ)jε−e(u0) : e(ϕ)−ZΩb pε∇·( Nεϕ)−p0∇·ϕ +ZR(kµεb uε−Ku0)·ϕ ds +kZP(b uε−u0)·ϕ ds = 0. (64) Le us se b uε=u+zε, whe e u=u0+x3εηε∂x3u0. Then (??) eads kzεkH1(Ω) ≤C√ε. (65) Since u=u0on P, (??) gi es : 2νZΩ(b eε(zε) : b eε(ϕ))jε+kZRzε·ϕµεds +kZPzε·ϕ ds =−2νZΩb eε(u) : b eε(ϕ)jε−e(u0) : e(ϕ)+ZΩ(b pε−p0)∇·( Nεϕ) +ZΩp0∇·( Nεϕ−ϕ)−ZR(kµεu−Ku0)·ϕ ds. (66) 19 Le us choose ϕ=zε. Then he le -hand side is la ge han Ckzεk2 (H1(Ω))3 hanks o Ko n inequali y (??). Using (??) o es ima e he i s in eg al in he igh -hand side o (??), we see ha kzεk2 H1(Ω) ≤C√εku0kH3(Ω)kzεkH1(Ω) +A+B+ZR(kµεu−Ku0)·zεds,(67) whe e Cis independen o εand Aand Ba e espec i ely gi en by A=ZΩ(b pε−p0)∇·( Nεzε), B =ZΩp0∇·( Nεzε−zε). Thanks o (??), ∇·( Nεzε) = ∇·( Nεb uε)−∇·( Nεu) = −∇·( Nεu). Thus, (??) and (??) gi e A≤Cε(kb pεkL2(Ω) +kp0kL2(Ω))ku0kH2(Ω) ≤Cε. (68) On he o he hand, (??) implies B≤C√εkp0kH2(Ω)kzεkH1(Ω) ≤C√εkzεkH1(Ω).(69) In o de o es ima e he in eg al o e Rin (??), le us pu kµεu−Ku0= (kmε−K)u0+k(µε−mε)u0+kµε(u−u0),(70) whe e mε(x′) = m(x′, x′/ε). F om he de ini ion o K, we see ha (kmε−K)(x′) = kγ(x′, x′/ε), whe e γ(x′, y′) = m(x′, y′)−hmi(x′). Thus, lemma 5.1, pa ii), gi es : ZR(kmε−K)u0·zεds≤C√εku0kH2(Ω)kzεkH1(Ω).(71) F om (??), (??) and (??), we deduce ha µ2 ε−(mε)2=|ε ∇x′ηε+εηε∇ |2+ 2(ε ∇x′ηε+εηε∇ )( ∇y′ηε+∇ ) 1 + |∇ |2. This implies ha |µε−mε| ≤ Cε and ZRk(µε−mε)u0·zεds≤Cεku0kH1(Ω)kzεkH1(Ω).(72) 20 Finally, |u−u0|=|x3ηε∂x3u0| ≤ C|∂x3u0|and ZRkµε(u−u0)·zεds≤Cεku0kH2(Ω)kzεkH1(Ω).(73) F om (??), (??), (??) and (??), we see ha ZR(kµεu−Ku0)·zεds≤C√εkzεkH1(Ω).(74) Using (??), (??) and (??) in (??), we ob ain kzεk2 H1(Ω) ≤C√εkzεkH1(Ω) +Cε. This implies (??) and, he e o e, (??) holds. Le us now p o e (??). Choosing ϕ∈(H1 0(Ω))3in (??), we ind 2νZΩ(b eε(zε) : b eε(ϕ))jε=ZΩ(b pε−p0)∇·( Nεϕ) + ZΩp0∇·( Nεϕ−ϕ) −2νZΩ(b eε(u) : b eε(ϕ))jε−e(u0) : e(ϕ). (75) Le ψbe a unc ion in L2 pe (Ω) and le ϕ∈(H1 0(Ω))3be he unc ion u nished by lemma 5.4. Then (??) eads ZΩ(b pε−p0)ψ= 2νZΩ(b eε(zε) : b eε(ϕ))jε +2νZΩ(b eε(u) : b eε(ϕ))jε−e(u0) : e(ϕ)−ZΩp0∇·( Nεϕ−ϕ). Using (??), (??), (??) and (??), i is no ha d o see ha ZΩ(b pε−p0)ψ≤C√εkψkL2. Since ψis a bi a y, his implies (??). This comple es he p oo . ⊔⊓ P oo o heo em 2.1: Fi s , le us p o e ha (??) ollows om (??). Le δ∗= 1/(2kηkL∞(S2)). Thanks o (??), we ha e |Tε−T0|=kg ·ZP(u0−uε)≤k|g|Cδ∗ku0−uεkH1(ωδ∗). The e o e, (??) implies (??). Le us now p o e (??). I su ices o assume δ≤δ∗, since (??) o δ=δ∗implies (??) o any δ≥δ∗. Using (??) and pu ing b ωδ=L−1 ε(ωδ), we ha e kϕk2 H1(ωδ)=Zbωδ jε(|b ϕ|2+|Mε∇b ϕ|2)≤Ckb ϕkH1(bωδ). 21 Then, i su ices o p o e ha kb uε−b u0kH1(bωδ)≤C√ε (whe e Ccan depend o δ). F om (??), we see ha i is su icien o p o e ha ku−b u0kH1(bωδ)≤C√ε. (76) Since u=u0=b u0on P, we only ha e o es ima e he L2no ms o he i s de i a i es ∂xj(u−b u0). In ac , we a e going o p o e ha , o all x∈b ωδ, |∂xj(u−u0)(x)| ≤ C√ε. (77) Recall ha u=u0+εx3ηε∂x3u0. Then, ∂xju=   ∂xju0+x3(∂yjηε+ε∂xjηε)∂x3u0+εx3ηε∂2 x3xju0j= 1,2, ∂x3u0+εηε∂x3u0+εx3ηε∂2 x3x3u0j= 3. Thanks o (??), ∂xjb u0=  d ∂xju0+x3(∂yjηε+ε∂xjηε)d ∂x3u0j= 1,2, (1 + εηε)d ∂x3u0j= 3. Since ηε,∂xjηεand ∂yjηεa e bounded, i ollows ha |∂xj(u−b u0)| ≤ C 3 X k=1 |∂xku0−d ∂xku0|+Cε(|d ∂x3u0|+|∂x3u0|+|∂x3xju0|)). Since ∈W3,∞(S), we ha e u0∈(H3(Ω))3and hus Mo ey-Sobole heo em gi es u0∈(C1,1/2(Ω))3, wi h a no m in his space bounded by Cku0kH3(Ω). The e o e, o all x∈b ωδ, |(∂xku0−d ∂xku0)(x)| ≤ Cδku0kH3(Ω)|x−Lε(x)|1/2≤Cδ√ε and (??) is es ablished. Thanks o Poinca ´e inequali y, (??) holds oo. This com- ple es he p oo o (??). Finally, le us p o e (??). A guing as be o e, i su ices o show ha kb pε−b p0kL2(bωδ)≤C√ε. F om (??), i su ices o p o e ha kp0−b p0kL2(bωδ)≤C√ε. (78) 22 Bu we al eady know ha p0∈H2(Ω), whence (again om Mo ey-Sobole embed- ding) p0∈C0,1/2(Ω) and i s no m in his space is bounded by Ckp0kH2(Ω). This leads o he ollowing inequali ies, o all x∈b ωδ, |(p0−b p0)(x)| ≤ Cδkp0kH2(Ω)|x−Lε(x)|1/2≤Cδ√ε. This implies (??) and, hus, (??) holds. The p oo o heo em 2.1 is now comple ed.⊔⊓ Acknowledgemen s: The wo k o he second and hi d au ho s has been pa ially suppo ed by D.G.E.S.–Spain, P oyec o PB98-1242. The wo k o he ou h au ho has been pa ially suppo ed by IBERDROLA isi ing esea che s p og amme. Re e ences [1] G. Allai e – Homogeniza ion o he Na ie -Sokes equa ions wi h a slip bound- a y condi ion, Comm. Pu e Appl. Ma h., XLIV, 6, p. 605–642, 1991. [2] Y. Ami a , J. Simon – In luence de la ugosi ´e en hyd odynamique lin´eai e, C. R. Acad. Sci. Pa is, . 323, S´e ie I, p. 313–318, 1996. [3] Y. Ami a , J. Simon – Rible s and d ag minimiza ion. In Op imiza ion me h- ods in pde’s, S. Cox and I. Lasiecka eds., Con empo a y Ma hema ics, AMS, p. 9–17, 1997. [4] Y. Ami a , D. B esch, J. Lemoine, J. Simon – E ec o ugosi y on a low go e ned by Na ie -S okes equa ions, Qua e ly Appl. Ma h., ( o appea ). [5] J.A. Bello –L egula i y o he S okes and Na ie -S okes p oblems, Ann. Ma . Pu a. Appl. Vol. 170, p. 187–206, 1996. [6] D. Cio anescu, P. Dona o and H. I. Ene – Homogeniza ion o he S okes p oblem wi h non-homogeneous slip bounda y condi ions, Ma h. Me hods Appl. Sci., Vol. 19, p. 857–882, 1996. [7] C. Conca – On he applica ion o he homogeniza ion heo y o a class o p oblems a ising in luid mechanics, J. Ma h. Pu es e Appl., 64, p. 31–75, 1985. [8] C. Conca – Nume ical esul s on he homogeneiza ion o S okes and Na ie - S okes equa ions modeling a class o p oblems om luid mechanics, Compu . Me hods Appl. Mech. Eng g., 53(3), p. 223–258, 1985. 23 [9] G.A. Chechkin, A. F iedman, A.L. Piani ski – The bounda y- alue p oblem in domains wi h e y apidly oscilla ing bounda y, J. Ma h. Anal. Appl., 231, p. 213–234, 1999. [10] G. Du au , J.L. Lions –Les in´equa ions en M´ecanique e Physique, Dunod, Pa is 1972. [11] G.P. Galdi –An in oduc ion o he ma hema ical heo y o he Na ie -S okes equa ions I : Linea ized s eady p oblems, Sp inge -Ve lag, New Yo k 1994. [12] V. Gi aul , P.A. Ra ia –Fini e elemen me hods o Na ie -S okes Equa- ions, Sp inge -Ve lag, Be lin 1986. [13] W. J¨ age , A. Mikeli´ c– On he oughness-induced bounda y condi ions o an incomp essible iscous low, J. Di e en ial Equa ions ( o appea ). [14] R. L. Pan on –Incomp essible low, Wiley-In e science, New-Yo k, 1984. [15] O.A. Oleinik, A.S. Shamae , G.A. Yosi ian –Ma hema ical p oblems in elas ici y and homogeniza ion, No h-Holland, Ams e dam 1992. [16] E. Sanchez-Palencia –Non-homogeneous media and ib a ion heo y, Lec- u es No es in Physics no. 127, Sp ige -Ve lag, Be lin 1980. 24