Derivation of a quasi-stationary coupled Darcy-Reynolds equation for incompressible viscous fluid flow through a thin porous medium with a fissure

De i a ion o a quasi-s a iona y coupled Da cy-Reynolds equa ion o incomp essible iscous luid low h ough a hin po ous medium wi h a issu e Ma ´ıa ANGUIANO Depa amen o de An´alisis Ma em´a ico Uni e sidad de Se illa, P. O. Box 1160, 41080-Se illa (Spain) [email p o ec ed] Abs ac We conside a non-s a iona y S okes sys em in a hin po ous medium o hickness εwhich is pe o a ed by pe iodically dis ibu ed solid cylinde s o size ε, and con aining a issu e o wid h ηε. Passing o he limi when εgoes o ze o, we ind a c i ical size ηε≈ε2 3in which he low is desc ibed by a 2D quasi-s a iona y Da cy law coupled wi h a 1D quasi-s a iona y Reynolds p oblem. AMS classi ica ion numbe s: 75A05, 76A20, 76M50, 35B27. Keywo ds: S okes equa ion; Da cy’s law; Reynolds equa ion; hin po ous medium; issu e. 1 1 In oduc ion The aim o his wo k is o p o e he con e gence o he homogeniza ion p ocess o he non-s a iona y S okes sys em in a hin po ous medium Dεηεo hickness εwhich is pe o a ed by pe iodically dis- ibu ed solid cylinde s o size εand con ains a issu e {0≤x2≤ηε}o wid h ηε. We conside he luid low h ough a pe iodic dis ibu ion o e ical cylinde s and a issu e. The pe iodic dis ibu ion o e ical cylinde s and he issu e a e con ined be ween wo pa allel pla es (see Figu e 1). A ep esen a i e elemen a y olume o he hin po ous medium is a cube o la e al leng h εand e ical len g h ε. The cube is epea ed pe iodically in he space be ween he pla es. Each cube can be di ided in o luid pa and a solid pa , whe e he solid pa has he shape o a e ical cylinde o heigh ε. " " " ⌘" x3 x2 x1 Figu e 1: View o he domain Dεηε The ques ion o a medium con aining a issu e wi h p ope ies di e en om hose o he es o he ma e ial has been he subjec o many s udies p e iously, see Cia le e al [1], Panasenko [2] and Chap e 13 o Sanchez-Palencia [3] among o he s. A simila p oblem o he one conside ed in his pape wi h a ixed heigh domain, bu o he Laplace’s equa ion, was s udied in Bou gea and Tapie o [4]. The peculia beha io obse ed o he Laplace’s equa ion when ηε≈ε2 3has mo i a ed he analogous s udy o he S okes sys em in Bou gea e al [5] (see [6] o he Na ie -S okes sys em and [7] o a non-s a iona y S okes sys em). In Anguiano [8], we conside a non-s a iona y S okes sys em in a hin po ous medium o hickness εwhich is pe o a ed by pe iodically dis ibu ed solid cylinde s o size aε. We apply an adap a ion o he un olding me hod in o de o ob ain igo ously quasi-s a iona y Da cy’s laws. The beha io obse ed when aε≈εhas mo i a ed he ac o conside ing a hin po ous medium con aining a issu e. In his sense, ou aim in he p esen pape is o ex end he s udy o Bou gea e al [5] o he case o a non-s a iona y S okes sys em in a domain o small heigh ε, pe o a ed by pe iodically dis ibu ed solid cylinde s o size ε, con aining a issu e o wid h ηε, which makes necessa y o escale in he heigh a iable in o de o wo k wi h a domain o heigh one. We ind he same c i ical size as in Bou gea e al [5], wha means ha he e olu i e model and he hin hickness o he domain do no modi y he c i ical size. Howe e , he hin hickness o he domain leads us o use echniques o educ ion o he dimension oge he wi h homogeniza ion in o de o ob ain mo e simpli ied e ec i e models han hose ob ained in Bou gea e al [5]. Mo e p ecisely, we ob ain he ollowing esul s co esponding o h ee cha ac e is ic si ua ions depending on he pa ame e ηεwi h espec o ε: •I ηεε2 3 he issu e is no gi ing any con ibu ion. In his case, in o de o ind he limi , we 2 use he esul s de eloped in Anguiano [8] and we ob ain a 2D quasi-s a iona y Da cy’s law. •I ηεε2 3 he issu e is dominan . We in oduce a escaling o he issu e in o de o wo k wi h a domain wi h size one, and hen we p o e ha he limi o he eloci y is a Di ac measu e concen a ed on he line {x2= 0} ∩ {x3= 0} ep esen ing he co esponding angen ial line low. Meanwhile in he po ous medium he e ec i e eloci y is equal o ze o. •I ηε≈ε2 3wi h ηε/ε2 3→λ, 0 < λ < +∞, i appea s a coupling e ec and he e ec i e low beha es as 2D quasi-s a iona y Da cy low in he po ous medium coupled wi h he angen ial low o he line {x2= 0}∩{x3= 0}. Compa ed o he i s case ηεε2 3, he e ec i e eloci y has now an addi ional angen ial componen concen a ed on {x2= 0}∩{x3= 0}. Mo eo e , he limi p oblem is now gi en by a new a ia ional equa ion, in which appea s he pa ame e λ, and consis s o a 2D quasi-s a iona y Da cy law in he po ous medium coupled wi h a 1D quasi-s a iona y Reynolds p oblem on he line {x2= 0}∩{x3= 0}. 2 The domain and some no a ions 2.1 The domain Le ω⊂R2be smoo h bounded connec ed open se and Ω = ω×(0,1) ⊂R3. We de ine Ω+= Ω ∩ {x2>0},Ω−= Ω ∩ {x2<0},Σ=Ω∩ {x2= 0},Σ1= Σ ∩ {x3= 0}. Fo some η0>0 we de ine he domains D= Ω−∪(η0e2+ Ω+)∪(Σ ×[0, η0]e2), D0=D∩ {x3= 0}, wi h e2= (0,1,0). Le ε > 0 be a small pa ame e de o ed o end o ze o and 0 < ηε< η0be a small pa ame e de o ed o end o ze o wi h ε. A pe iodic po ous medium is de ined by a domain ωand an associa ed mic os uc u e, o pe iodic cell Y0= [0,1]2, which is made o wo complemen a y pa s: he luid pa Y0 , and he solid pa Y0 s (Y0 SY0 s=Y0and Y0 TY0 s=∅). Mo e p ecisely, we assume ha Y0 sis a smoo h and connec ed se s ic ly included in Y0. Fo k0= (k1, k2)∈Z2, each cell Y0 k0=k0+Y0is di ided in a luid pa Y0 k0and a solid pa Y0 sk0. We de ine Y=Y0×(0,1) ⊂R3, and is di ided in a luid pa Y and a solid pa Ys. We also deno e Y− s=[ k0∈Z2 − Ysk0, Y + s=[ k0∈Z2 + Ysk0, all he solid pa s in R2×(0,1), whe e Z2 −={k0∈Z2, k2<0}and Z2 +={k0∈Z2, k2>0}. I is ob ious ha E =(R2×(0,1)) (Y− s∪Y+ s)∩Ω is he luid pa in Ω. Following [9], we make he ollowing assump ions on Y ,E ,Ysand Y∗ s=Y+ s∪Y− s: i) Y is an open connec ed se o s ic ly posi i e measu e, wi h a locally Lipschi z bounda y. ii) Yshas s ic ly posi i e measu e in Y. 3 iii) E and he in e io o Y∗ sa e open se s wi h bounda ies o class C0,1and a e locally loca ed on one side o hei bounda ies. Mo eo e E is connec ed. We also de ine Y− s,ε =εY 0− s×(0,1), Y + s,εηε= (ηεe2+εY 0+ s)×(0,1),e Sεηε=∂(Y− s,ε ∪Y+ s,εηε). We deno e by e Aεηε= (Y− s,ε ∪Y+ s,εηε)∩D- he solid pa o he domain D, e Dεηε=D e Aεηε- he luid pa o he domain D(including he issu e), e Iηε= Σ ×(0, ηε)e2- he issu e in D, e Ωεηε=e Dεηε e Iηε- he luid pa o he po ous medium in D. Le us de ine a domain wi h hickness ε, gi en by Ωε= Ω ∩ {0< x3< ε} ⊂ R3. We also de ine Ωε += Ω+∩ {0< x3< ε},Ωε −= Ω−∩ {0< x3< ε},Σε= Ωε∩ {x2= 0}, and Dε= Ωε −∪η0e2+ Ωε +∪(Σε×[0, η0]e2). The mic oscale o a po ous medium is he small posi i e numbe ε. The domain ωis co e ed by a egula mesh o size ε: o k0= (k1, k2)∈Z2, each cell Y0 k0,ε =εk0+εY 0is di ided in a luid pa Y0 k0,ε and a solid pa Y0 sk0,ε, i.e. is simila o he uni cell Y0 escaled o size ε. We de ine Yk0,ε =Y0 k0,ε ×(0,1) ⊂R3, which is also di ided in a luid pa Y k0,ε and a solid pa Ysk0,ε. Now, we deno e by Aεηε,Dεηε,Iηεand Ωεηε he se s e Aεηε,e Dεηε,e Iηεand e Ωεηε, espec i ely, wi h hickness ε, i.e., Aεηε=e Aεηε∩ {0< x3< ε}- he solid pa o he domain Dε, Dεηε=e Dεηε∩ {0< x3< ε}- he luid pa o he domain Dε(including he issu e), Iηε=e Iηε∩ {0< x3< ε}- he issu e in Dε, Ωεηε=e Ωεηε∩ {0< x3< ε}- he luid pa o he po ous medium in Dε. Finally we de ine Ω+ εηε=Dεηε∩ {x2> ηε},Ω− εηε=Dεηε∩ {x2<0},Γηε=∂Σε×(0, ηε)e2, and D+=D∩ {x2>0}, D−= Ω−. 4 " " ⌘" x3 x2 ⌦+ "⌘" ⌦ "⌘" I⌘" " " x2=0 x2=⌘" x1 x2 Figu e 2: View o he domain Dεηε om abo e (le ) and la e al ( igh ) 2.2 Some no a ions Le us in oduce some no a ions which will be use ul in he ollowing. Fo a ec o ial unc ion = ( 1, 2, 3) and a scala unc ion w, we in oduce he ope a o s: Dε,∇εand di εby (Dε )i,j =∂xj i o i= 1,2,3, j = 1,2, (Dε )i,3=1 ε∂y3 i o i= 1,2,3, ∇εw= (∇x0w, 1 ε∂y3w) , di ε = di x0 0+1 ε∂y3 3, and mo eo e he ope a o s Dηε,∇ηεand di ηεby (Dηε )i,1=∂x1 i o i= 1,2,3, (Dηε )i,2=1 ηε ∂y2 i o i= 1,2,3, (Dηε )i,3=1 ε∂y3 i o i= 1,2,3, ∇ηεw= (∂x1w, 1 ηε ∂y2w, 1 ε∂y3w) , di ηε =∂x1 1+1 ηε ∂y2 2+1 ε∂y3 3. We deno e by Oεa gene ic eal sequence which ends o ze o wi h εand can change om line o line. We deno e by Ca gene ic posi i e cons an which can change om line o line. 5 3 Se ing and main esul s He eina e , he poin s x∈R3will be decomposed as x= (x0, x3) wi h x0∈R2,x3∈R. We also use he no a ion x0 o deno e a gene ic ec o o R2. In his sec ion, we desc ibe he asymp o ic beha io o an incomp essible iscous luid in a hin po ous medium wi h a issu e. The p oo o he co esponding esul s will be gi en in he nex sec ions. Ou esul s a e e e ed o he non-s a iona y S okes sys em. Namely, o ∈C([0, T]×D)3le us conside a sequence (uε, pε)∈L2(0, T;H1 0(Dεηε))3×L2(0, T;L2(Dεηε)), which sa is ies      ∂uε ∂ −µ∆uε+∇pε= in (0, T)×Dεηε, di uε= 0 in (0, T)×Dεηε, uε(0, x)=0, x ∈Dεηε, (3.1) whe e T > 0, µ > 0 is he iscosi y and Dεηεis de ined in Sec ion 2. The igh -hand side is o he o m ( , x) = ( 0( , x0),0),a.e. x∈D, (3.2) whe e 0∈C([0, T]×D)2.(3.3) This choice o is usual when we deal wi h hin domains. Since he hickness o he domain εis small hen he e ical componen o he o ce can be neglec ed and, mo eo e he o ce can be conside ed independen o he e ical a iable. Finally, we may conside Di ichle bounda y condi ions wi hou al e ing he gene ali y o he p ob- lem unde conside a ion, uε= 0 on (0, T)×∂Dεηε.(3.4) Fo any ixed ε, unde he assump ions o and u0 ε, a classical esul (see Temam [10]) shows ha (3.1)-(3.4) has a leas one weak solu ion (uε, pε)∈L2(0, T;H1 0(Dεηε))3×L2(0, T;L2(Dεηε)), whe e pεis uniquely de ined up o an addi i e cons an , ha is, i is uniquely de ined i we conside he co esponding equi alence class: pε∈L2(0, T;L2(Dεηε)/R). Ou aim is o s udy he asymp o ic beha io o uεand pεwhen ε ends o ze o. Fo his pu pose, we use he dila a ion in he a iable x3 y3=x3 ε,(3.5) in o de o ha e he unc ions de ined in an open se wi h ixed heigh e Dεηεgi en in Sec ion 2. Namely, we de ine ˜uε∈L2(0, T;H1 0(e Dεηε))3, ˜pε∈L2(0, T;L2(e Dεηε)/R) by ˜uε( , x0, y3) = uε( , x0, εy3),˜pε( , x0, y3) = pε( , x0, εy3), a.e. ( , x0, y3)∈(0, T )×e Dεηε. Using he ans o ma ion (3.5), he sys em (3.1) can be ew i en as        ∂˜uε ∂ −µ∆ε˜uε+∇ε˜pε= in (0, T)×e Dεηε, di ε˜uε= 0 in (0, T)×e Dεηε, ˜uε(0, x0, y3)=0,(x0, y3)∈e Dεηε, (3.6) 6 wi h Di ichle bounda y condi ions ˜uε= 0 on (0, T)×∂e Dεηε,(3.7) whe e we se ∆εw= ∆x0w+ε−2∂2 y3wand e Dεηεis de ined in Sec ion 2. Ou goal hen is o desc ibe he asymp o ic beha io o his new sequence (˜uε, ˜pε). Mo eo e , in o de o s udy he beha io o ˜uε, ˜pεin he issu e we ew i e ou equa ions in he uni cylinde e I1= Σ ×(0,1)e2by in oducing he change o a iable y2=x2 ηε ,(3.8) which ans o m e Iηεin a ixed domain e I1. We de ine he new unc ions ˜ Uε( , x1, y2, y3) = ˜uε( , x1, ηεy2, y3),˜ Pε( , x1, y2, y3) = ˜pε( , x1, ηεy2, y3)−cεηε,(3.9) wi h cεηε=1 |e Iηε|Ze Iηε ˜pε( , x0, y3)dx0dy3.(3.10) Using he ans o ma ion (3.8), he sys em (3.6) can be ew i en as        ∂˜ Uε ∂ −µ∆ηε˜ Uε+∇ηε˜ Pε= ( , x1, ηεy2) in (0, T)×e I1, di ηε˜ Uε= 0 in (0, T)×e I1, ˜ Uε(0, x1, ηεy2, y3)=0,(x1, ηεy2, y3)∈e I1, (3.11) wi h Di ichle bounda y condi ions ˜ Uε= 0 on (0, T)×∂e I1,(3.12) whe e we se ∆ηεw=∂2 x1w+η−2 ε∂2 y2w+ε−2∂2 y3w. Ou main esul e e ed o he asymp o ic beha io o he solu ion o (3.6) is gi en by he ollowing heo em. Theo em 3.1. We dis ingue h ee cases depending on he ela ion be ween he pa ame e ηεwi h espec o ε: i) i ηεε2 3, hen he e exis s (˜ , ˜p)∈L2((0, T)×D)3×L2(0, T ;L2(D)/R), wi h ˜ 3= 0 and ˜p independen o y3, such ha he solu ion (ε−2˜uε,˜pε)o p oblem (3.6)-(3.7) sa is ies ε−2˜uε*˜ in L2((0, T )×D)3,˜pε→˜pin L2(0, T;L2(D)/R).(3.13) Mo eo e , ˜p∈L2(0, T;H1(D)/R)and (˜ V , ˜p)is he unique solu ion o he 2D quasi-s a iona y Da cy law (whe e is only a pa ame e )        ˜ V0( , x0) = 1 µK 0( , x0)− ∇x0˜p( , x0)in (0, T )×D0, di x0˜ V( , x0)=0in (0, T)×D0, ˜ V( , x0)·n= 0 in (0, T)×∂D0, (3.14) 7 whe e ˜ V( , x0) = R1 0˜ ( , x0, y3)dy3and K∈R2×2is a symme ic, posi i e, enso de ined by i s en ies Kij =ZY Dywi(y) : Dywj(y)dy, i, j = 1,2,(3.15) whe e wi(y),i= 1,2, wi h RY wi 3dy = 0, deno es he unique solu ion in H1 #(Y )3o he local s a iona y S okes p oblems in 3D        −∆ywi+∇yqi=eiin Y , di ywi= 0 in Y , wi= 0 in ∂(Y Y ), wi, qiY0−pe iodic. (3.16) ii) i ηεε2 3and le (˜ Uε,˜ Pε)be a solu ion o (3.11)-(3.12). Then he e exis ˜ U ∈ L2((0, T)×e I1)3, independen o y3, wi h ˜ U2=˜ U3= 0, and ˜ P∈L2(0, T;L2(e I1)/R)only depending on and x1, such ha o a subsequence, ηε−2˜ Uε*˜ Uin L2((0, T)×e I1)3,˜ Pε*˜ Pin L2(0, T;L2(e I1)/R), whe e ˜ U1( , x1, y2) = y2(1 −y2) 2 1( , x1,0) −∂x1˜ P( , x1).(3.17) Mo eo e , i holds ha ηε−3˜uε? *˜ VδΣ1in L2(0, T;M(D))3,(3.18) whe e ˜ V ∈ L2((0, T)×Σ1)3, wi h ˜ V2=˜ V3= 0, such ha ˜ V1( , x1) = Z1 0 ˜ U1( , x1, y2)dy2=1 12  1( , x1,0) −∂x1˜ P( , x1),(3.19) and, in ac ˜ P∈L2(0, T;H1(Σ1)/R)is he unique solu ion o he 1D quasi-s a iona y Reynolds p oblem on Σ1(whe e is only a pa ame e )    ∂x1 1( , x1,0) −∂x1˜ P( , x1))= 0 in (0, T)×Σ1,  1( , x1,0) −∂x1˜ P( , x1)·n= 0 on (0, T)×∂Σ1.(3.20) iii) i ηε≈ε2 3, wi h ηε/ε2 3→λ,0<λ<+∞, hen he e exis a Da cy eloci y ˜ , a Reynolds eloci y ˜ Vand a p essu e ield ˜psuch ha ε−2˜uε? *˜ +λ3˜ VδΣ1in L2(0, T;M(D))3, ˜pε→˜pin L2(0, T ;L2(D)/R),(3.21) whe e δΣ1is he Di ac measu e concen a ed on Σ1, and M(D)3is he space o Radon meau es on D. The eloci ies ˜ and ˜ Va e linked wi h he p essu e ˜p h ough he 2D Da cy law (3.14) in (0, T)×D0and he 1D Reynolds p oblem (3.20) on (0, T)×Σ1. The p essu e ield ˜p∈ L2(0, T;H1(D0)/R)wi h ˜p(·,0) ∈L2(0, T;H1(Σ1)/R), is he unique solu ion o he a ia ional p oblem ZT 0ZD0 1 µK 0( , x0)− ∇x0˜p( , x0)· ∇x0ϕ( , x0)dx0d +λ3 12 ZT 0ZΣ1 ( 1( , x1,0) −∂x1˜p( , x1)) ∂x1ϕ( , x1,0) dx1d = 0, (3.22) o e e y ϕ∈L2(0, T;H1(D0)) wi h ϕ(·,0) ∈L2(0, T;H1(Σ1)). 8 Rema k 3.2. The coupled p oblem (3.22) co esponding o he c i ical case ηε≈ε2 3, wi h ηε/ε2 3→λ, 0< λ < +∞, can be conside ed as he gene al one. In ac , i λ ends o in ini y in (3.22) we eco e he 1D quasi-s a iona y Reynolds p oblem (3.20), meanwhile i λ ends o ze o we eco e he 2D quasi-s a iona y Da cy law (3.14). 4 A P io i Es ima es Le us begin wi h a lemma on Poinca ´e inequali y in he po ous medium e Ωεηε, which will be e y use ul (see o example Lemma 4.1 in [8]). Lemma 4.1. The e exis s a cons an Cindependen o ε, such ha , o any unc ion ∈H1(e Dεηε)3 and = 0 on e Sεηε, one has k kL2(e Ωεηε)3≤Cε kDε kL2(e Ωεηε)3×3.(4.23) Nex , we gi e an use ul es ima e in he issu e e Iηε. Lemma 4.2. The e exis s a cons an Cindependen o ε, such ha , o any unc ion ∈H1(e Dεηε)3 and = 0 on e Sεηε, one has k kL2(e Iηε)3≤Cηε 1 2(ηε+ε)1 2kDε kL2(e Dεηε)3×3.(4.24) P oo . Fo any unc ion w(y)∈H1(e I1)3wi h w= 0 in ∂e I1, he Poinca ´e inequali y in e I1s a es ha Ze I1 |w|2dz ≤CZe I1 |∂z2w|2dz, (4.25) whe e he cons an Cdepends only on e I1. Fo e e y k0∈Z2, by he change o a iable z1=x1, z2=x2 ηε , z3=x3 ε, dz =dx εηε , ∂z2=ηε∂x2,(4.26) we escale (4.25) om e I1 o Iηε. This yields ha , o any unc ion w(x)∈H1(Iηε)3wi h w= 0 in ∂Iηε, one has ZIηε |w|2dx ≤Cη2 εZIηε |∂x2w|2dx ≤Cη2 εZIηε |Dxw|2dx, (4.27) wi h he same cons an Cas in (4.25). Finally, applying he dila a ion (3.5) in (4.27), we ob ain Z˜ Iηε |w|2dx0dy3≤Cη2 εZ˜ Iηε |Dεw|2dx0dy3, which gi es k kL2(e Iηε)3≤CηεkDε kL2(e Iηε)3×3.(4.28) Nex , i we choose a poin y∈Aεηε, which is close o he poin x∈Iηε, hen we ha e (x)− (y) = D (ξ)(x−y)≤(ε+ηε)|D |. 9 On he o he hand, we ha e ZT 0 |<∇ε˜pε( ), ϕ( )(wε−w)>D|d =ZT 0<∇x0˜pε( ), ϕ( )Rε(w0 ε−w0)>e Dεηεd =ZT 0hµ∆x0˜ 0 ε( ), ϕ( )Rε(w0 ε−w0)ie Dεηε+h 0( ), ϕ( )Rε(w0 ε−w0)ie Dεηε−h∂˜ 0 ε( ) ∂ , ϕ( )Rε(w0 ε−w0)ie Dεηεd , and using Cauchy-Schwa z’s inequali y, es ima e (4.32), he i s es ima e in (4.34), he es ima es o he es ic ed ope a o Rεapplied o Dx0ins ead o Dε, and aking in o accoun ha ηεε2 3and ε1, we ge ZT 0 |<∇ε˜pε( ), ϕ( )(wε−w)>D|d ≤C ZT 0 ϕ( )2kw0 ε−w0k2 L2(D)2d 1/2 +εZT 0 ϕ( )2kDx0w0 ε−Dx0w0k2 L2(D)2×2d 1/2!→0 as ε→0, by i ue o (5.45) and he Rellich Theo em. This implies ha ∇ε˜pε→ ∇x0˜ps ongly in L2(0, T;H−1(D))3, which implies he s ong con e gence o he p essu e gi en in (5.43). Lemma 5.2. Le ηεε2 3and le (˜ ε,˜pε)be he ex ended solu ion o (3.6)-(3.7). Le (˜ , ˜p)∈L2((0, T )× D)3×L2(0, T;L2(D)/R)be gi en by Lemma 5.1. Then, ˜p∈L2(0, T ;H1(D)/R)and (˜ , ˜p)is he unique solu ion o Da cy’s law (3.14). P oo . We apply Theo em 3.1-(i) in [8], because in he p esen pape aε≈εin he po ous pa , in o de o ob ain ha (˜ , ˜p) is he unique solu ion o Da cy’s law (3.14). Finally, he classical heo y o he ellip ic equa ion implies exis ence o he unique solu ion ˜pbelongs o L2(0, T;H1(D)/R). P oo o Theo em 3.1-i).I emains o p o e con e gence (3.13) o he whole eloci y ˜uε, i.e. o p o e ε−2k˜uεkL2((0,T )×e Iηε)3→0.(5.46) Fo his, i is su icien o p o e ha ε−2k˜uεkL2((0,T )×e Iηε)3→0 o ηεε, (5.47) and ε−2k˜uεkLq((0,T )×e Iηε)3→0 o εηεε1 α,1< α < 3 2,(5.48) o a qwhich will be de ined below. Using (4.31) and using ηεε, we ha e ε−2k˜uεkL2((0,T )×e Iηε)3≤C ηε 5 2 ε2+ηε ε+ηε ε1 2!, 16 so ha (5.47) easily holds. Using H¨olde ’s inequali y wi h he conjuga e exponen s 2 qand 2 2−qwe ob ain ε−2k˜uεkLq((0,T )×e Iηε)3≤C ηε 1 q+2 ε2+ηε 1 q+1 2 ε+ηε 1 q ε1 2!. Now we ake ηε=ε1 α. Then we ind ha ε−2k˜uεkLq((0,T )×e Iηε)3≤Cε1 α1 q+2−2+ε1 α1 q+1 2−1+ε1 qα −1 2.(5.49) We seek an op imal qsuch ha he igh hand side in (5.49) ends o ze o. I is easy o p o e ha we ha e a con e gence o ze o o any q∈1,2 2(α−1)+1. The e o e, (5.48) holds and so we ha e (5.46). 5.2 P oblem in he issu e pa ηεε 2 3 The p oo o Theo em 3.1-ii) will be de eloped in di e en lemmas. Lemma 5.3. Le ηεε2 3and le (˜ Uε,˜ Pε)be he solu ion o (3.11)-(3.12). Then he e exis subse- quences o ˜ Uεand ˜ Pεs ill deno ed by he same, and unc ions ˜ U ∈ L2((0, T)×e I1)3, independen o y3, wi h ˜ U2=˜ U3= 0,˜ P∈L2(0, T;L2(e I1)/R)such ha ηε−2˜ Uε*˜ Uin L2((0, T)×e I1)3,˜ Pε*˜ Pin L2(0, T;L2(e I1)/R).(5.50) Mo eo e , ˜ P=˜ P(x1)and ˜ U1is gi en by exp ession (3.17). P oo . Taking in o accoun ηεε2 3and es ima es (4.31), (4.32), (4.33), (4.41) wi h he change o a iable (3.8), we ha e k˜ UεkL2((0,T)×e I1)3≤Cηε2,(5.51) k∂x1˜ UεkL2((0,T)×e I1)3≤Cηε,k∂y2˜ UεkL2((0,T)×e I1)3≤Cηε2,(5.52) k∂y3˜ UεkL2((0,T)×e I1)3≤Cε ηε,(5.53) k˜ UεkL∞(0,T;L2(e I1))3≤Cηε,(5.54) k˜ PεkL2(0,T;L2(e I1)/R)≤C. (5.55) F om he es ima es (5.51) and (5.55), he e exis ˜ U ∈ L2((0, T)×e I1)3,˜ P∈L2(0, T;L2(e I1)/R) such ha con e gence (5.50) holds. Mo eo e ηε−2∂y2˜ Uε* ∂y2˜ Uin L2((0, T)×e I1)3,(5.56) and om (5.54), he e exis s ˜ W ∈ L∞(0, T;L2(e I1))3such ha ηε−1˜ Uε∗ *˜ Win L∞(0, T;L2(e I1))3.(5.57) The es ima e (5.53) implies ha ε−1η−1 ε∂y3˜ Uεis bounded in L2((0, T)×e I1)3. This oge he wi h ηεε2 3implies ha η−2 ε∂y3˜ Uε ends o ∂y3˜ U= 0. This implies ha ˜ Udoes no depend on y3. 17 As ˜ Udoes no depend on y3, le ϕ∈C∞ 0((0, T)×e I1)3independen o y3. Taking in o accoun ha di ηε˜ Uε= 0 in (0, T )×e I1, we ha e ηε−1ZT 0Ze I1∂x1˜ Uε 1+ηε−1∂y2˜ Uε 2+ε−1∂y3˜ Uε 3ϕ dx1dy2dy3d =−ηε−1ZT 0Ze I1 ˜ Uε 1∂x1ϕ dx1dy2dy3d −ηε−2ZT 0Ze I1 ˜ Uε 2·∂y2ϕ dx1dy2dy3d = 0. Taking he limi ε→0 we ob ain ZT 0Ze I1 ˜ U2∂y2ϕ dx1dy2dy3d = 0, so ha ˜ U2=˜ U2( , x1). Since ˜ U,∂y2˜ U ∈ L2((0, T)×e I1)3 he aces ˜ U( , x1,0), ˜ U( , x1,1) a e well de ined in L2((0, T)×Σ)3. Analogously o he p oo o Lemma 4.2 we choose a poin β(x1,y3)∈e Aεηε, which is close o he poin α(x1,y3)∈Σ, hen we ha e ZT 0ZΣ |˜ Uε( , x0,0, y3)|2dx1dy3d =ZT 0ZΣ |˜uε( , x1,0, y3)|2dx1dy3d ≤CZT 0ZΣ Z(β(x1,y3),α(x1,y3)) Dε˜uε·(α(x1,y3)−β(x1,y3))d`!2 dx1dy3d , so ha , by Cauchy-Schwa z’s inequali y, k˜ Uε( , x1,0, y3)k2 L2((0,T)×Σ)3≤CεkDε˜uεk2 L2((0,T)×e Dεηε)3×3. Taking in o accoun es ima e (4.32) and ηεε2 3, we ha e ηε−2k˜ Uε( , x1,0, y3)k2 L2((0,T)×Σ)3≤Cεηε→0 as ε→0, which implies ha ˜ U( , x1,0) = 0 , and analogously ˜ U( , x1,1) = 0 . Consequen ly ˜ U2= 0 . I emains o p o e ha ˜ U3= 0. In o de o do ha , as ˜ Udoes no depend on y3, we ake a es unc ion = (0,0, 3(x1, y2)) in (3.11), d d Ze I1 ˜ Uε 3( ) 3dx1dy2dy3+Ze I1 ∂2 x1˜ Uε 3( ) 3dx1dy2dy3+1 η2 εZe I1 ∂2 y2˜ Uε 3( ) 3dx1dy2dy3= 0, in D0(0, T). We conside ϕ∈C1 c([0, T]) such ha ϕ(T) = 0 and ϕ(0) 6= 0. Mul iplying by ϕand in eg a ing be ween 0 and T, we ha e −ZT 0 d d ϕ( )Ze I1 ˜ Uε 3( ) 3dx1dy2dy3d +ZT 0 ϕ( )Ze I1 ∂2 x1˜ Uε 3( ) 3dx1dy2dy3d +1 η2 εZT 0 ϕ( )Ze I1 ∂2 y2˜ Uε 3( ) 3dx1dy2dy3d = 0. 18 We pass o he limi when ε ends o ze o, and using he con e gences (5.56) and (5.57) wi h 3ϕ( )∈L2((0, T)×e I1), 3 d d ϕ( )∈L1(0, T;L2(e I1)), we can deduce ha ˜ U3= 0. Finally, we compu e he exp ession o ˜ Ugi en in (3.17). Fi s , we ake a es unc ion = (0,0, ε 3) in (3.11), and we ob ain εd d Ze I1 ˜ Uε 3( ) 3dx1dy2dy3+εZe I1 ∂2 x1˜ Uε 3( ) 3dx1dy2dy3+ε η2 εZe I1 ∂2 y2˜ Uε 3( ) 3dx1dy2dy3 +1 εZe I1 ∂2 y3˜ Uε 3( ) 3dx1dy2dy3−Ze I1 ˜ Pε∂y3 3dx1dy2dy3= 0, in D0(0, T). Mul iplying by ϕand in eg a ing be ween 0 and T, we ha e −εZT 0 d d ϕ( )Ze I1 ˜ Uε 3( ) 3dx1dy2dy3d +εZT 0 ϕ( )Ze I1 ∂2 x1˜ Uε 3( ) 3dx1dy2dy3d +ε η2 εZT 0 ϕ( )Ze I1 ∂2 y2˜ Uε 3( ) 3dx1dy2dy3d +1 εZT 0 ϕ( )Ze I1 ∂2 y3˜ Uε 3( ) 3dx1dy2dy3d −ZT 0 ϕ( )Ze I1 ˜ Pε∂y3 3dx1dy2dy3d = 0. We pass o he limi when ε ends o ze o, and using he es ima e (5.53), he con e gences (5.50) and (5.57) wi h 3ϕ( )∈L2((0, T)×e I1), 3 d d ϕ( )∈L1(0, T;L2(e I1)), we can deduce ha ˜ Pdoes no depend on y3. We ake a es unc ion = (0, ηε 2,0), independen o y3, in (3.11), and we ob ain ηε d d Ze I1 ˜ Uε 2( ) 2dx1dy2dy3+ηεZe I1 ∂2 x1˜ Uε 2( ) 2dx1dy2dy3+1 ηεZe I1 ∂2 y2˜ Uε 2( ) 2dx1dy2dy3 −Ze I1 ˜ Pε∂y2 2dx1dy2dy3=ηεZe I1 2 2dx1dy2dy3, in D0(0, T). Mul iplying by ϕand in eg a ing be ween 0 and T, we ha e −ηεZT 0 d d ϕ( )Ze I1 ˜ Uε 2( ) 2dx1dy2dy3d +ηεZT 0 ϕ( )Ze I1 ∂2 x1˜ Uε 2( ) 2dx1dy2dy3d +1 ηεZT 0 ϕ( )Ze I1 ∂2 y2˜ Uε 2( ) 2dx1dy2dy3d −ZT 0 ϕ( )Ze I1 ˜ Pε∂y2 2dx1dy2dy3d =ηεZT 0 ϕ( )Ze I1 2 2dx1dy2dy3d . We pass o he limi when ε ends o ze o, and using he con e gences (5.50) and (5.57) wi h 2ϕ( )∈L2((0, T)×e I1), 2 d d ϕ( )∈L1(0, T;L2(e I1)), 19 we can deduce ha ˜ P=˜ P( , x1). Now, aking in o accoun ha ˜ Udoes no depend on y3and ˜ U2=˜ U3= 0, we ake a es unc ion = ( 1(x1, y2),0,0) in (3.11), d d Ze I1 ˜ Uε 1( ) 1dx1dy2dy3+Ze I1 ∂2 x1˜ Uε 1( ) 1dx1dy2dy3+1 η2 εZe I1 ∂2 y2˜ Uε 1( ) 1dx1dy2dy3 −Ze I1 ˜ Pε∂x1 1dx1dy2dy3=Ze I1 1( , x1, ηεy2) 1dx1dy2dy3, in D0(0, T). Mul iplying by ϕand in eg a ing be ween 0 and T, we ha e −ZT 0 d d ϕ( )Ze I1 ˜ Uε 1( ) 1dx1dy2dy3d +ZT 0 ϕ( )Ze I1 ∂2 x1˜ Uε 1( ) 1dx1dy2dy3d +1 η2 εZT 0 ϕ( )Ze I1 ∂2 y2˜ Uε 1( ) 1dx1dy2dy3d −ZT 0 ϕ( )Ze I1 ˜ Pε∂x1 1dx1dy2dy3d =ZT 0 ϕ( )Ze I1 1( , x1, ηεy2) 1dx1dy2dy3d . We pass o he limi when ε ends o ze o, and using he con e gences (5.50) and (5.57) wi h 1ϕ( )∈L2((0, T)×e I1), 1 d d ϕ( )∈L1(0, T;L2(e I1)), we ob ain he ODE    −∂2 y2˜ U1( , x1, y2) = 1( , x1,0) −∂x1˜ P( , x1), ˜ U1( , x1,0) = ˜ U1( , x1,1) = 0, which gi es he exp ession (3.17) o ˜ U1. P oo o Theo em 3.1-ii).I emains o p o e he con e gence (3.18) o he whole eloci y o he unc- ion Vgi en by (3.19), and also p o e ha ˜ P∈L2(0, T;H1(Σ)/R) is he unique solu ion o he Reynolds p oblem (3.20). Taking as es unc ion ϕ∈C∞((0, T)×D), independen o y3, in he equa ion di ε˜uε= 0 in (0, T)×D, we ob ain ZT 0ZD di ε˜uεϕ dx0dy3d =−ZT 0ZD ˜ 0 ε·∇x0ϕ dx0dy3d −ηεZT 0Ze I1 (˜ Uε)0·∇x0ϕ( , x1, ηεy2)dx1dy2dy3d = 0, so ha mul iplying by ηε−3, ZT 0Ze I1 ηε−2˜ Uε 1∂x1ϕ( , x1, ηεy2)dx1dy2dy3d (5.58) =−ZT 0ZD ηε−3˜ ε· ∇x0ϕ dx0dy3d −ZT 0Ze I1 ηε−2˜ Uε 2∂x2ϕ( , x1, ηεy2)dx1dy2dy3d . 20 Using (4.30) and aking in o accoun ηεε2 3, we ob ain ηε−3k˜ εkL2((0,T )×D)3≤C ε ηε 3 2 +ε2 ηε3!→0 as ε→0.(5.59) Taking he limi in (5.58) as ε→0, using con e gence (5.50), ˜ U2= 0 and ˜ U1independen o y3, we ha e ZT 0ZΣ ˜ U1∂x1ϕ(x1,0) dx1dy2d = 0, and by de ini ion (3.19), we ge ZT 0ZΣ1 1( , x1,0) −∂x1˜ P( , x1)∂x1ϕ( , x1,0) dx1d = 0. Consequen ly, ˜ P∈L2(0, T;H1(Σ1)/R) and is he unique solu ion o (3.20). Finally, we conside ϕ∈C0((0, T)×D)3, independen o y3, and so we ha e ZT 0ZD ηε−3˜uε·ϕ dx0dy3d =ZT 0ZD ηε−3˜ ε·ϕ dx0dy3d +ZT 0Ze I1 ηε−2˜ Uε·ϕ( , x1, ηεy2)dx1dy2dy3d . Using (5.59), con e gence (5.50) and ˜ U2=˜ U3= 0, we ob ain ZT 0ZD ηε−3˜uε·ϕ dx0dy3d →ZT 0ZΣ ˜ U1( , x1, y2)ϕ1( , x1,0) dx1dy2d =ZT 0ZΣ1 ˜ V1( , x1)ϕ1( , x1,0) dx1=ZT 0 h˜ V1( , x1)δΣ1, ϕiM(D)3,C0(D)3d , which implies (3.18). 5.3 E ec s o coupling ηε≈ε 2 3 The conclusion o he p e ious wo subsec ions is ha o any sequence o solu ions (˜ ε,˜pε) wi h ηεε2 3and ( ˜ Uε,˜ Pε) wi h ηεε2 3, and le ing ε→0, we can ex ac subsequences s ill deno ed by ˜ ε,˜pε,˜ Uε,˜ Pεand ind unc ions ˜ ∈L2(0, T ;H1(0,1; L2(ω)3)) wi h ˜ 3= 0, ˜p∈L2(0, T;H1(D)/R), ˜ U ∈ L2((0, T)×e I1)3, independen o y3, wi h ˜ U2=˜ U3= 0, ˜ P∈L2(0, T;H1(Σ)/R) such ha ε−2˜ ε*(˜ 0,0) in L2(0, T;H1(0,1; L2(ω)3)),˜pε→˜pin L2(0, T;L2(D)/R), ηε−2˜ Uε*(˜ U1,0,0) in L2((0, T)×e I1)3,˜ Pε*˜ Pin L2(0, T;L2(e I1)/R). (5.60) Mo eo e such limi unc ions ˜ , ˜p, ˜ U,˜ Pnecessa ily sa is y he equa ions ˜ V0( , x0) = 1 µK 0( , x0)− ∇x0˜p( , x0)in (0, T )×D0, ˜ U1( , x1, y2) = y2(1 −y2) 2 1( , x1,0) −∂x1˜ P( , x1)in (0, T)×e I1, (5.61) whe e ˜ V0( , x0) = R1 0˜ 0( , x0, y3)dy3. We a e going o ind he connec ion be ween he unc ions ˜pand ˜ P, i.e. o ind he coupling e ec s be ween he solu ion in he po ous pa and in he issu e. 21 Lemma 5.4. Le ηε≈ε2 3, wi h ηε/ε2 3→λ,0< λ < +∞, and le ˜pε∈L2(0, T;L2(D)/R),˜p∈ L2(0, T;H1(D)/R),˜ P∈L2(0, T;H1(Σ)/R)be such ha (5.60) and (5.61) hold. Then, ZT 0ZD0 1 µK 0( , x0)− ∇x0˜p( , x0)· ∇x0ϕ( , x0)dx0d +λ3 12 ZT 0ZΣ1 1( , x1,0) −∂x1˜ P( , x1)∂x1ϕ( , x1,0) dx1d = 0, (5.62) o e e y ϕ∈L2(0, T;H1(D0)) wi h ϕ( , ·,0) ∈L2(0, T;H1(Σ1)). P oo . Le ϕε( , x0, y3) = ϕ( , x0, εy3)∈L2(0, T;H1(D)) wi h ϕ∈L2(0, T;H1(D)) and ϕ( , ·,0) ∈ L2(0, T;H1(Σ)). Taking in o accoun he de ini ions (5.42) o ˜ εand (3.9) o ˜ Uε, and om di ε˜uε= 0 in (0, T)×Dwe ha e ZT 0ZD ε−2˜uε· ∇εϕεdx0dy3d =ZT 0ZD ε−2˜ ε· ∇εϕεdx0dy3d +ηε ε2 33ZT 0Ze I1 ηε−2˜ Uε· ∇εϕε( , x1, ηεy2, y3)dx1dy2dy3d = 0, and by he de ini ion o ϕε, we can deduce ZT 0ZD ε−2˜ ε· ∇ϕ( , x0, εy3)dx0dy3d +ηε ε2 33ZT 0Ze I1 ηε−2˜ Uε· ∇ϕ( , x1, ηεy2, εy3)dx1dy2dy3d = 0. Taking he limi as ε→0, using (5.60), ˜ 3=˜ U2=˜ U3= 0, ηε/ε2 3→λ, and aking in o accoun ha ˜ U1does no depend on y3, we ob ain ZT 0ZD ˜ 0( , x0, y3)· ∇x0ϕ( , x0,0) dx0dy3d +λ3ZT 0ZΣ ˜ U1( , x1, y2)∂x1ϕ( , x1,0,0) dx1dy2d = 0, and aking in o accoun exp essions (5.61) and (3.19), we ge (5.62). We a e going o p o e he ela ion ˜p( , x1,0) = ˜ P( , x1) + C, wi h C∈R. Then (3.22) ollows om (5.62). Lemma 5.5. Le ηε≈ε2 3,ηε/ε2 3→λ,0< λ < +∞, and le ˜p,˜ Pbe he limi p essu es om (5.60). Then, he e exis s C∈Rsuch ha ˜p( , x1,0) = ˜ P( , x1) + C, (5.63) and ˜p∈L2(0, T ;H1(D0)/R)wi h ˜p( , ·,0) ∈L2(0, T;H1(Σ1)/R)is he unique solu ion o he a ia ional p oblem (3.22). P oo . We need o ex end he es unc ions conside ed in he p oo o Lemma 5.2 o he issu e e Iηε. To do his, we de ine I0 ηε=e Iηε∩ {x3= 0},Bηε=D0 −∪Σ1∪I0 ηεand Y1=Y ∩ {x2= 0}, and we conside φ(y0)∈C∞ #(Bηε)3be such ha φ(y0) = 0 in Y0 Y0 . We de ine φε(x0) =      φx0 εin D0 −, K2e2in I0 ηε,whe e K2=ZY1 φ2(y1,0)dy1. 22 Le ϕ∈C∞ 0(B1), wi h B1=D−∪Σ∪e I1be such ha ZΣ ϕ(x1,0, y3)dx1dy3= 0.(5.64) Taking in (3.6) as es unc ion wε(x0, y3) =    ϕ(x0, y3)φx0 εin D−, ϕx1,x2 ηε, y3K2e2in e Iηε, we ob ain d d ZBηε ˜uε( )·wεdx0dy3!+µZBηε Dε˜uε( ) : Dεwεdx0dy3=ZBηε 0( )·w0 εdx0dy3+ZBηε ˜pε( ) di εwεdx0dy3. We conside ψ∈C1 c([0, T]) such ha ψ(T) = 0 and ψ(0) 6= 0. Mul iplying by ψand in eg a ing be ween 0 and T, we ha e −ZT 0 d d ψ( )ZBηε ˜uε( )·wεdx0dy3d +µZT 0 ψ( )ZBηε Dε˜uε( ) : Dεwεdx0dy3d (5.65) =ZT 0 ψ( )ZBηε 0( )·w0 εdx0dy3d +ZT 0 ψ( )ZBηε ˜pε( ) di εwεdx0dy3d . Using (5.51), we ha e K2ZT 0 d d ψ( )Ze Iηε ˜ Uε 2( )·ϕx1,x2 ηε , y3dx0dy3d  =K2ηεZT 0 d d ψ( )Ze Iηε ˜ Uε 2( )·ϕ(x1, y2, y3)dx1dy2dy3d ≤Cη3 ε→0 as ε→0. We obse e ha K2ZT 0 ψ( )Ze Iηε 0( )·ϕ0x1,x2 ηε , y3e2dx0dy3d =ηεK2ZT 0 ψ( )Ze I1 0( )·ϕ0(x1, y2, y3)e2dx1dy2dy3d →0 as ε→0, and by he de ini ion o wεin e Iηεand using es ima es (5.52), (5.53), we deduce K2ZT 0 ψ( )Ze Iηε Dε˜ Uε( )∂x2ϕ(x1,x2 ηε , y3)dx0dy3d  =K2ZT 0 ψ( )Ze I1 Dηε˜ Uε( )∂y2ϕ(x1, y2, y3)dx1dy2dy3d ≤Cηε→0 as ε→0, 23 Then, om (5.65), we can deduce ha −ZT 0 d d ψ( )ZD− ˜uε( )·wεdx0dy3d +ZT 0 ψ( )ZD− Dε˜ ε( ) : Dεwεdx0dy3d (5.66) =ZT 0 ψ( )ZD− 0( )·w0 εdx0dy3d +ZT 0 ψ( )ZD− ˜pε( )di εwεdx0dy3d +K2ZT 0 ψ( )Ze Iηε ˜pε( )∂x2ϕ(x1,x2 ηε , y3)dx0dy3d +Oε. Fo he las e m on he igh hand side, we ha e K2ZT 0 ψ( )Ze Iηε ˜pε( )∂x2ϕ(x1,x2 ηε , y3)dx0dy3d =K2ZT 0 ψ( )Ze Iηε cεηε( )∂x2ϕ(x1,x2 ηε , y3)dx0dy3d +K2ZT 0 ψ( )Ze Iηε (˜pε( )−cεηε( ))∂x2ϕ(x1,x2 ηε , y3)dx0dy3d , whe e cεηεis de ined in (3.10). Using (5.60), we ob ain K2ZT 0 ψ( )Ze Iηε (˜pε( )−cεηε( ))∂x2ϕ(x1,x2 ηε , y3)dx0dy3d =K2ZT 0 ψ( )Ze I1 ˜ Pε( )∂y2ϕ(x1, y2, y3)dx1dy2dy3d →K2ZT 0 ψ( )Ze I1 ˜ P( , x1)∂y2ϕ(x1, y2, y3)dx1dy2dy3d =−K2ZT 0 ψ( )ZΣ ˜ P( , x1)ϕ(x1,0, y3)dx1dy3d , (5.67) as ε→0, whe e ˜ Pεis gi en by (3.9), and using (5.64), we ha e K2ZT 0 ψ( )cεηε( )Ze Iηε ∂x2ϕ(x1,x2 ηε , y3)dx0dy3d =K2ZT 0 ψ( )cεηε( )Ze I1 ∂y2ϕ(x1, y2, y3)dx1dy2dy3d = 0. Passing o he limi in (5.66) simila ly as in he p oo o Theo em 6.1-(i) in [8] by using an adap a ion o he un olding me hod, and aking in o accoun (5.67) and ZT 0 ψ( )ZD0 −×Y ˜p( , x0) di x0(ϕ(x0, y3)φ(y0)) dx0dyd =−ZT 0 ψ( )ZD0 −×Y ∇x0˜p( , x0)ϕ(x0, y3)φ(y0)dx0dyd +ZT 0 ψ( )ZΣ×Y1 ˜p( , x1,0)ϕ(x1,0, y3)φ2(y1,0) dx1dy1dy3d =−ZT 0 ψ( )ZD0 −×Y ∇x0˜p( , x0)ϕ(x0, y3)φ(y0)dx0dyd +K2ZT 0 ψ( )ZΣ ˜p( , x1,0)ϕ(x1,0, y3)dx1dy3d , hen we ha e ZT 0 ψ( )ZΣ˜p( , x1,0) −˜ P( , x1)ϕ(x1,0, y3)dx1dy3d = 0, 24 so ha Z(0,T)×Σ1˜p( , x1,0) −˜ P( , x1)ϑ( , x1)dx1d = 0, o e e y ϑ∈C∞ 0((0, T)×Σ1) such ha RΣϑ dx1= 0, a.e. ∈(0, T). Finally we conclude ha he e exis s a cons an C∈Rsuch ha (5.63) holds and ˜p( , x1,0) ∈L2(0, T ;H1(Σ1)/R). Using (5.63) in o (5.62), we ob ain he a ia ional o mula ion (3.22) o he limi p essu e ˜pin he Banach space o unc ions ∈L2(0, T;H1(D0)) such ha ( , x1,0) ∈L2(0, T ;H1(Σ1)). Since K∈R2×2is a symme ic, posi i e, enso gi en by (3.15), i can be p o ed ha (3.22) has a unique solu ion in ha Banach space wi h he no m | |L2(0,T ;H1(D0)) +| (x1,0)|L2(0,T;H1(Σ1)). P oo o Theo em 3.1-iii).I emains o p o e he con e gence (3.21) o he whole eloci y. Le ϕ∈C0((0, T)×D)3. Then ZT 0ZD ε−2˜uε·ϕ dx0dy3d =ZT 0ZD ε−2˜ ε·ϕ dx0dy3d +ηε ε2 33ZT 0Ze I1 ηε−2˜ Uε·ϕ( , x1, ηεy2, y3)dx1dy2dy3d = 0. Taking he limi as ε→0, using (5.60), ˜ 3=˜ U2=˜ U3= 0 and ηε/ε2 3→λ, we ob ain ZT 0ZD ε−2˜uε·ϕ dx0dy3d →ZT 0ZD ˜ 0·ϕ0dx0dy3d +λ3ZT 0Ze I1 ˜ U1( , x1, y2)ϕ( , x1,0, y3)dx1dy2dy3d . Taking in o accoun ha ZT 0Ze I1 ˜ U( , x1, y2)ϕ( , x1,0, y3)dx1dy2dy3d =ZT 0ZΣ1 V( , x1)Z1 0 ϕ( , x1,0, y3)dy3dx1d =ZT 0 hVδΣ1, ϕiM(D)3,C0(D)3d , whe e V( , x1) is gi en by (3.19), we ge (3.21). Acknowledgmen s The au ho would like o hank he e e ees o he de ailed ema ks which allowed o imp o e his pape . The au ho has been suppo ed by Jun a de Andaluc´ıa (Spain), P oyec o de Excelencia P12- FQM-2466, and in pa by Eu opean Commission, Excellen Science-Eu opean Resea ch Council (ERC) H2020-EU.1.1.-639227. Re e ences [1] Cia le P-G, Led e H, Nzwenga R. Mod´elisa ion de la jonc ion en e un co ps ´elas ique idimen- sionnel e une plaque. C. R. Acad. Sci., Pa is, S´e ie I. 1987; 305: 55-58. 25