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.
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