Asymp o ic analysis o he Na ie -S okes equa ions in a hin domain
wi h powe law slip bounda y condi ions
Ma ´ıa ANGUIANO∗and F ancisco Ja ie SU´
AREZ-GRAU†
Abs ac
This heo e ical s udy deals wi h he Na ie -S okes equa ions posed in a 3D hin domain wi h
hickness 0 < ε 1, assuming powe law slip bounda y condi ions, wi h an aniso opic enso ,
on he bo om. This condi ion, in oduced in (Djoko e al. Compu . Ma h. Appl. 128 (2022)
198–213), ep esen s a gene aliza ion o he Na ie slip bounda y condi ion. The goal is o s udy
he in luence o he powe law slip bounda y condi ions wi h an aniso opic enso o o de εγ
s, wi h
γ∈Rand low index 1 < s < 2, on he beha io o he luid wi h hickness εby using asymp o ic
analysis when ε→0, depending on he alues o γ. As a esul , we deduce he exis ence o a c i ical
alue o γgi en by γ∗
s= 3 −2sand so, h ee di e en limi bounda y condi ions a e de i ed. The
c i ical case γ=γ∗
sco esponds o a limi condi ion o ype powe law slip. The supe c i ical case
γ > γ∗
sco esponds o a limi bounda y condi ion o ype pe ec slip. The subc i ical case γ < γ∗
s
co esponds o a limi bounda y condi ion o ype no-slip.
AMS classi ica ion numbe s: 35Q35, 76A20, 76A05, 76M50.
Keywo ds: Thin domain; homogeniza ion; powe law slip bounda y condi ions; Na ie slip bounda y
condi ions; Na ie -S okes.
1 In oduc ion
The s a iona y Na ie -S okes equa ions in a domain Ω eads as ollows
−2νdi (D[u]) + (u· ∇)u+∇p= and di (u)=0,(1.1)
whe e udeno es he eloci y ield, D[u] = 1
2(Du + (Du)T) he de o ma ion enso associa ed wi h he
eloci y ield u,p he scala p essu e, he ex e nal o ces and ν > 0 he iscosi y. Conce ning he
bounda y condi ions, i is commonly accep ed ha iscous luids adhe e o su aces, and so he no-slip
condi ion a he su aces o a domain, gi en by
u= 0 on ∂Ω,
is widely used. Unde sui able egula i y condi ions on he domain and , his p oblem is well s udied
ma hema ically, see o ins ance Boye & Fab ie [11], Galdi [18] o Temam [26]. Howe e , his condi ion
∗Depa amen o de An´alisis Ma em´a ico. Facul ad de Ma em´a icas. Uni e sidad de Se illa. 41012-Se illa (Spain)
[email p o ec ed]
†Depa amen o de Ecuaciones Di e enciales y An´alisis Num´e ico. Facul ad de Ma em´a icas. Uni e sidad de Se illa.
41012-Se illa (Spain) [email p o ec ed]
1
Ma ´ıa Anguiano and F ancisco J. Su´a ez-G au
does no seem always alid physically, indeed some luids mel and solu ions slip agains he su ace.
Also, some imes he no-slip condi ion is no good enough because i is no possible o desc ibe he
beha io o he luid nea he bounda y. The e o e, i is necessa y o in oduce o he ype o bounda y
condi ions o desc ibe his beha io . In his sense, Na ie [22] p oposed he Na ie slip bounda y
condi ions in which i is assumed a hin laye o a luid nea o he bounda y and he angen ial
componen o he s ain enso should be p opo ional o he angen ial componen o he luid eloci y
on a pa o he bounda y Γ ⊂∂Ω, ha is
2ν[D[u]n]τ=−λ[u]τ, u ·n = 0,on Γ,(1.2)
whe e n deno es he ou side uni a y no mal ec o o Ω on Γ, λ > 0 is he ic ion coe icien and
he subsc ip τdeno es he o hogonal p ojec ion on he angen space o Γ, i.e. [u]τ=u−(u·n)n.
P oblem (1.1) wi h Na ie slip bounda y condi ions (1.2) has been s udied by many au ho s in di e en
con ex s, see o example Am ouche & Rejaiba [4], Clopeau e al. [14] and Solonniko & ˇ
Sˇcadilo [23].
No ice ha depending on he alue o λin (1.2), we shall conside he ollowing ype o bounda y
condi ions:
•Pe ec slip when λ= 0, i.e.
2ν[D[u]n]τ= 0, u ·n = 0,on Γ,(1.3)
•Pa ial slip when λ∈(0,+∞),
2ν[D[u]n]τ=−λ[u]τ, u ·n = 0,on Γ,(1.4)
•No-slip when λ= +∞, i.e.
[u]τ= 0, u ·n = 0,on Γ,
which implies u= 0 on Γ.
Rela ed o his, we e e o Ace edo e al. [1] o he s udy he limi ing beha io o he solu ion
(uλ, pλ) o p oblem (1.1) wi h Na ie slip bounda y condi ions (1.2), when he ic ion coe icien λ
goes o 0 o ∞. In ac , hey p o ed ha (uλ, pλ) weakly con e ges o (u0, p0) when λ→0 in sui able
Sobole spaces, whe e (u0, p0) is he solu ion o he Na ie -S okes sys em wi h Na ie slip bounda y
condi ions co esponding wi h λ= 0. Also, i holds ha (uλ, pλ) weakly con e ges o (u∞, p∞) when
λ→ ∞, whe e (u∞, p∞) is he solu ion o he Na ie -S okes sys em wi h no-slip bounda y condi ions,
i.e. he Na ie bounda y condi ions co esponding wi h λ= +∞.
In his wo k, we a e in e es ed in a gene aliza ion o he Na ie slip condi ion ecen ly in oduced
by Djoko e al. [16] (see also Aldbaissy e al. [2, 3] and Djoko e al. [17]), which a ises when he
con ac su ace is lub ica ed wi h a hin laye o a non-New onian luid. This condi ion is called powe
law slip bounda y condi ion and eads as ollows
2ν[D[u]n]τ=−|K[u]τ|s−2K2[u]τ, u ·n = 0,on Γ,(1.5)
whe e | |2= · is he Euclidean no m. We obse e ha in his condi ion, he angen ial shea is
a powe law unc ion o he angen ial eloci y, whe e K∈R2×2is an aniso opic enso , assumed
o be uni o mly posi i e de ini e, symme ic and bounded, and sis a eal, s ic ly posi i e numbe
ep esen ing he low beha io index.
2
Ma ´ıa Anguiano and F ancisco J. Su´a ez-G au
The bounda y condi ion (1.5) ep esen s a gene aliza ion o he Na ie slip bounda y condi ion
(1.2), since o s= 2 and K=λ1
2Iwi h λ > 0, hen he powe slip bounda y condi ion (1.5) educes o
Na ie slip bounda y condi ion (1.2). We also men ioned ha he powe slip bounda y condi ion (1.5)
is p esen in he con ex o lamina lows o New onian liquids (e.g. wa e ) o e complex su aces, also
when a ough o s uc u ed bounda y su ace is aniso opic, e.g. when i has ows o ible s, pilla s o
pe iodic pa e ns, he e ec i e slip condi ion is aniso opic, i.e., di ec ion dependen . When he su ace
is he e ogeneous, he e ec i e slip is also posi ion-dependen . This can occu , o example, when he
bounda y has a a ying deg ee o oughness o when he bounda y is a smoo h su ace wi h a a ying
hyd ophobic/hyd ophilic composi ion. Fo ins ance, we e e o he de i a ion o e ec i e slip bounda y
condi ions coming om ough bounda ies in Bonni a d e al. [8], Bonni a d & Su´a ez-G au [9, 10],
Bucu [12], Bucu e al. [13], Daliba d & G´e a d-Va e [15] and Su´a ez-G au [24, 25]. The exis ence o
solu ions o he S okes and Na ie -S okes equa ions wi h powe law bounda y condi ions (1.5) on a pa
o he bounda y was s udied in [16] o 1 <s<2, which co esponds o he angen ial shea hinning.
In he case s > 2 he exis ence o solu ions is no p o en, i was no able o p o e a in -sup condi on,
which is he key poin o ob ain he p essu e. In he case s= 2, epea ing he classical p oo o he
exis ence o solu ion o he S okes and Na ie -S okes p oblem wi h homogeneous Di ichle condi ions
(see o ins ance [1, Theo em 2.3], [19, Theo em 7.1] and [26, Theo em 10.1]) gi es he exis ence o
solu ion o he S okes and Na ie -S okes equa ions wi h Na ie slip bounda y condi ions.
Ou main in e es in his pape is o s udy a lub ica ion p oblem co esponding o he asymp o ic
in luence o he powe law bounda y condi ion (1.5), imposed on a pa o he bounda y wi h 1 <s<2,
on he beha io o he Na ie -S okes equa ions h ough a hin domain Ωε, whe e he small pa ame e
0< ε 1 ep esen s he hickness o he domain. Mo e p ecisely, we conside he ollowing 3D hin
domain (see Figu e 1)
Ωε={(x1, x2, x3)∈R2×R: (x1, x2)∈ω, 0< x3< εh(x1, x2)},
whe e ωis a smoo h, connec ed open se o R2and his a smoo h and posi i e unc ion (see Sec ion
2 o mo e de ails). To s udy he in luence o he slip bounda y condi ions on he beha io o he
Na ie -S okes equa ions in he hin domain Ωε( he subsc ip εis added o he unknowns o s ess he
dependence o he solu ion on he small pa ame e )
(−ν∆uε+ (uε· ∇)uε+∇pε= εin Ωε,
di (uε) = 0 in Ωε,
whe e we conside he case o he powe law slip bounda y condi ions (1.5) on Γ0wi h an aniso opic
enso Kε, depending on ε, o he o m
Kε=εγ
sK, wi h 1 <s<2 and γ∈R,
whe e Kis assumed o be uni o mly posi i e de ini e, symme ic and bounded. Thus, he powe slip
bounda y condi ion, whe e he angen ial shea is a powe law unc ion o he angen ial eloci y wi h
a coe icien depending on ε, is gi en by
2[D[uε]n]τ=−εγ|K[uε]τ|s−2K2[uε]τ, uε·n=0,on Γ0,(1.6)
and no-slip condi ion on he es o he bounda y, i.e.
uε= 0 on ∂Ωε Γ0.
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Ma ´ıa Anguiano and F ancisco J. Su´a ez-G au
A e he homogeniza ion p ocess (unde assump ions gi en in Sec ion 2) when ε→0 depending on
he alue o γ, we deduce (see Theo em 4.7) ha he limi eloci y eu= (eu0,0) and limi p essu e ep
sa is ies he educed 2D-S okes sys em
−ν∂2
z3eu0(z) = 0(z0)− ∇z0ep(z0) in Ω = {z∈R3:z0∈ω, 0< z3< h(z0)},
di z0 Zh(z0)
0eu0(z)dz3!= 0 in ω,
Zh(z0)
0eu0(z)dz3!·n = 0 on ∂ω,
eu0= 0 on Γ1=ω× {h(z0)},
(1.7)
whe e eu0= (u1, u2) and 0= ( 1, 2). Mo eo e , we p o e he exis ence o a c i ical alue o γgi en
by
γ∗
s= 3 −2s, wi h 1 <s<2,(1.8)
which le us de i e h ee di e en bounda y condi ions o eu0on he bo om Γ0:
•I γ=γ∗
s, hen he e ec i e bounda y condi ion on Γ0is a powe slip bounda y condi ion wi h
aniso opic enso K, i.e.
−ν∂z3eu0=−|Keu0|s−2K2eu0on Γ0.
Thus, o ake in o accoun he aniso opy, hen Kεhas o be o o de O(εγs∗
s).
•I γ > γ∗
s, hen he e ec i e bounda y condi ion on Γ0is he pe ec slip bounda y condi ion, i.e.
−ν∂z3eu0= 0 on Γ0.
This means ha o an aniso opy enso o o de smalle han O(εγ∗
s
s), hen he luid does no
ake in o accoun aniso opy and slides pe ec ly.
•I γ < γ∗
s, hen he e ec i e bounda y condi ion on Γ0is he no-slip condi ion, i.e.
eu0= 0 on Γ0.
This means ha o an aniso opy enso o o de g ea e han O(εγ∗
s
s), hen he aniso opy is so
s ong ha he luid is s opped on he bounda y.
Obse e ha o s= 2 and Kε=εγ
2λ1
2I, wi h λ > 0, whe e he powe slip condi ion (1.6) educes
o he Na ie slip condi ion wi h ic ion pa ame e λεγ, i holds ha he c i ical alue γ∗
2=−1, which
is he c i ical alue o he case o Na ie slip bounda y condi ion. Namely, o ake in o accoun he
ic ion coe icien λin he e ec i e bounda y condi ion, i.e.
−ν∂z3eu0=−λeu0on Γ0,
he o iginal ic ion coe icien has o be o o de O(ε−1). I he o iginal ic ion coe icien λεγis o
o de smalle han O(ε−1), hen he luid beha es on he bounda y as i he e we e no ic ion (pe ec
slippage), i.e.
−ν∂z3eu0= 0 on Γ0.
4
Ma ´ıa Anguiano and F ancisco J. Su´a ez-G au
Finally, i he ic ion coe icien λεγis o o de g ea e han O(ε−1), hen he ic ion coe icien is so
s ong ha he luid is s opped on he bounda y (no-slip condi ion), i.e.
eu0= 0 on Γ0.
To p o e hese esul s, we i s use he mul iscale expansion me hod, which is a o mal bu powe ul
ool o analyse homogeniza ion p oblems, see o ins ance he applica ion o his me hod in Bayada &
Chamba [7] and Mikeli´c [20]. Nex , once he esul s ha e been unde s ood, we igo ously jus i y hem
by means o he de i a ion o a p io i es ima es and some compac ness esul s.
As a as he au ho s know, he low o a New onian luids wi h powe law slip bounda y condi ions
has no been ye conside ed in he abo e desc ibed lub ica ion amewo k, which ep esen s he main
no el y o he pape . We obse e ha he ob ained indings a e amenable o he nume ical simula ions
wi h a conside able simpli ica ion wi h espec o he o iginal p oblem (which is compu a ionally mo e
expensi e), since he e ec i e sys em (1.7) is a wo dimensional o dina y di e en ial sys em wi h espec
o z3. The e o e, we belie e ha i could p o e use ul in he enginee ing p ac ice as well.
The pape is s uc u ed as ollows. In Sec ion 2, we in oduce he s a emen o he p oblem. In
Sec ion 3, we conside he o mal de i a ion, and in Sec ion 4 we will igo ously jus i y he esul s. We
inish he pape wi h a sec ion o e e ences.
2 Fo mula ion o he p oblem and p elimina ies
In his sec ion, we i s de ine he hin domain and some se s necessa y o s udy he asymp o ic beha io
o he solu ions. Nex , we in oduce he p oblem conside ed in he hin domain and also, he escaled
p oblem posed in he domain o ixed heigh , oge he wi h he espec i e weak a ia ional o mula ions.
The domain and some no a ion. Along his pape , 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.
We conside ωas an open, smoo h, bounded and connec ed se o R2, and a 3D hin domain gi en by
Ωε={(x0, x3)∈R2×R:x0∈ω, 0< x3< hε(x0)},
He e, he unc ion hε(x0) = εh(x0) ep esen s he eal gap be ween he wo su aces. The small pa a-
me e εis ela ed o he ilm hickness. Func ion his posi i e and smoo h C1bounded unc ion de ined
o x0. We de ine he bo om, op and la e al bounda ies o Ωεas ollows (see Figu e 1)
Γ0=ω× {0},Γε
1=(x0, x3)∈R3:x0∈ω, x3=hε(x0),Γε
`=∂Ωε (Γ0∪Γε
1).
Le us now in oduce some no a ion which will be use ul in he ollowing. Fo a ec o ial unc ion
ϕ= (ϕ0, ϕ3) and a scala unc ion φ, we in oduce he ope a o s ∆, di , Dand ∇by
∆ϕ= ∆x0ϕ+∂2
x3ϕ, di (eϕ) = di x0(ϕ0) + ∂x3ϕ3,
(Dϕ)ij =∂xjϕi o i= 1,2,3, j = 1,2,3,
∇φ= (∇x0φ, ∂x3φ) .
(2.9)
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
Ma ´ıa Anguiano and F ancisco J. Su´a ez-G au
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Figu e 1: Thin domain Ωε, op bounda y Γε
1and bo om bounda y Γ0
The p oblem and he escaling. As s a ed in he in oduc ion, we conside he 3D s a iona y
Na ie -S okes equa ions by se ing
uε= (u0
ε(x), u3,ε(x)), pε=pε(x),
a a poin x∈Ωε, which is gi en by
−ν∆uε+ (uε· ∇)uε+∇pε= εin Ωε,
di (uε) = 0 in Ωε,
uε= 0 on Γε
1∪Γε
`,
(2.10)
whe e ν > 0, and he powe law slip bounda y condi ions p esc ibed on Γ0gi en by
−ν∂x3u0
ε=−εγ|Ku0
ε|s−2K2u0
ε, uε,3= 0,on Γ0,(2.11)
whe e 1 <s<2 and γ∈R.
Rema k 2.1. He e, we ha e assumed he ollowing:
•Since di (uε) = 0, i holds 2 di (D[uε]) = ∆uε. Then, i also holds 2ν[D[uε] n]τ=ν[Duεn]τ.
•Since he bo om bounda y Γ0is la , he ou side no mal ec o n = −e3, whe e {ei}3
i=1 is he
canonical basis in R3, and so uε·n = 0 implies uε,3= 0. Also, he o hogonal p ojec ion
o a unc ion uεon he angen space o Γ0is [uε]τ=u0
ε, whe e u0
ε= (uε,1, uε,2), and hen,
ν[Duεn]τ=−ν∂x3u0
ε.
•Due o he hickness o he domain, i is usual o assume ha he e ical componen s o he
ex e nal o ces can be neglec ed and, mo eo e , he o ces can be conside ed independen o he
e ical a iable. Thus, o sake o simplici y, gi en 0= ( 1, 2)∈L2(ω)2, along he pape we
conside he ollowing ype o ex e nal o ces ε(see o ins ance [24, 25]):
ε= ( 0(x0),0) .
De ini ion 2.2. Fo ε > 0, we say ha (uε, pε)de ined on Ωεis a weak solu ion o p oblem (2.10)–
(2.11) i and only i he unc ions (uε, pε)∈V(Ωε)×L2
0(Ωε), whe e he co esponding unc ional space
o eloci y is
V(Ωε) = ϕ∈H1(Ωε)3:ϕ= 0 on ∂Ωε Γ0, ϕ3= 0 on Γ0,
6
Ma ´ıa Anguiano and F ancisco J. Su´a ez-G au
and he space o p essu e L2
0is he space o unc ions o L2wi h ze o mean alue, and sa is y
νZΩε
Duε:Dϕ dx +ZΩε
(uε· ∇)uε·ϕ dx +εγZΓ0
|Ku0
ε|s−2Ku0
ε·Kϕ0dσ −ZΩε
pεdi (ϕ)dx
=ZΩε
0·ϕ0dx, ∀ϕ∈V(Ωε),
(2.12)
and ZΩε
di (uε)ψ dx = 0,∀ψ∈L2(Ωε).(2.13)
Rema k 2.3. Unde p e ious assump ions, o e e y ε > 0, we ha e ha e e ence [16, P oposi ion
2.2] gi es he exis ence o a leas one weak solu ion (uε, pε)o p oblem (2.10)–(2.11).
The obje i e o his pape is o s udy he asymp o ic p oblems o he beha io o he sequence o
solu ions (uε, pε) o p e ious p oblems, when ε ends o ze o depending on he alue o γ. To do ha ,
we in oduce a classical change o a iables in hin domains, he dila a ion
z0=x0, z3=ε−1x3.(2.14)
This change ans o ms Ωεin o a ixed domain Ω, de ined by
Ω = (z0, z3)∈R2×R:z0∈ω, 0< z3< h(z0).(2.15)
The bounda y o Ω is deno ed by ∂Ω, whe e he op and la e al bounda ies o he escaled domain Ω
is de ined by
Γ1=(z0, z3)∈R2×R:z0∈ω, z3=h(z0),Γ`=∂Ω (Γ0∪Γ1).(2.16)
Acco dingly, we de ine he unc ions euεand epεby
euε(z) = uε(z0, εz3),epε(z) = pε(z0, εz3) a.e. z∈Ω.(2.17)
Obse e ha acco ding o he assump ion on ε, i holds e
ε(z)=( 0(z0),0) a.e. z∈Ω.
Le us now in oduce some no a ion which will be use ul in he ollowing. Fo a ec o ial unc ion
eϕ= (eϕ0,eϕ3) and a scala unc ion e
φob ained espec i ely om unc ions ϕand φby using he change
o a iables (2.14), we in oduce he ope a o s ∆ε, di ε,Dεand ∇εby
∆εeϕ= ∆z0eϕ+ε−2∂2
z3eϕ, di ε(eϕ) = di z0eϕ0+ε−1∂z3eϕ3,
(Dεeϕ)ij =∂zjeϕi o i= 1,2,3, j = 1,2,(Dεeϕ)i3=ε−1∂z3eϕi o i= 1,2,3,
∇εe
φ= (∇z0e
φ, ε−1∂z3e
φ).
(2.18)
Thus, using he change o a iables (2.14), he sys em (2.10)–(2.11) can be ew i en as
−ν∆εeuε+ (euε· ∇ε)euε+∇εepε=e
εin Ω,
di ε(euε) = 0 in Ω,
euε= 0 on Γ1∪Γ`,
(2.19)
7
Ma ´ıa Anguiano and F ancisco J. Su´a ez-G au
wi h powe law slip bounda y condi ions
−ν
ε∂z3eu0
ε=−εγ|Keu0
ε|s−2K2eu0
ε,euε,3= 0 on Γ0.(2.20)
Acco ding o he change o a iables (2.14) applied o he weak a ia ional o mula ions gi en in
De ini ion 2.2, hen, o ε > 0, a escaled weak solu ion (euε,epε)∈V(Ω)×L2
0(Ω), whe e he co esponding
unc ional space o eloci y is
V(Ω) = ϕ∈H1(Ω)3:ϕ= 0 on ∂Ω Γ0, ϕ3= 0 on Γ0,
sa is ies
νZΩ
Dεeuε:Dεeϕ dz +Ze
Ωε
(euε· ∇ε)euε·eϕ dz +εγ−1ZΓ0
|Keu0
ε|s−2Keu0
ε·Keϕ0dσ −ZΩepεdi ε(eϕ)dz
=ZΩ
0·eϕ0dz,
(2.21)
and ZΩ
di ε(euε)e
ψ dz = 0,(2.22)
o e e y eϕ∈V(Ω) and e
ψ∈L2(Ω) ob ained om (ϕ, ψ) by he change o a iables (2.14).
Now, he goal is o s udy he asymp o ic beha io o a sequence o solu ion (euε,epε) o p oblem
(2.19)–(2.20). In he nex sec ion, we will s udy he asymp o ic analysis in a o mal way, and in
Sec ion 4, we de elop he igo ous analysis.
3 The o mal asymp o ic expansion
In his sec ion, we apply he asymp o ic expansion me hod (see o ins ance [7, 20]) o he Na ie -S okes
equa ions (2.19) wi h powe slip bounda y condi ions (2.20). We will de i e a educed S okes sys em
wi h no-slip condi ion on he op bounda y Γ1and di e en bounda y condi ions on Γ0depending on
he alue o γ. The idea is o assume an expansion in εo he solu ion (euε,epε) gi en by
euε(z) = εβ 0(z) + ε 1(z) + ε2 2(z) + · · · ,epε(z) = p0(z) + εp1(z) + ε2p2(z) + ε3p3(z) + · · · (3.23)
To de e mine he e ec i e p oblem gi en by unc ions ( 0, p0), he expansion (3.23) is plugged in o he
PDE, we iden i y he a ious powe s o εand we ob ain a cascade o equa ions om which we e ain
only he leading ones ha cons i u e he e ec i e p oblem.
We ema k ha he alue βin he expansion (3.23) will be de e mined in he nex sec ion, by
de i ing a p io i es ima es o (euε,epε) (see Lemmas 4.4 and 4.5). Mo eo e , he e ec i e p oblem will
be jus i ied by co esponding compac ness esul s (see Lemma 4.6 and Theo em 4.7).
Theo em 3.1. Assume 1<s<2and de ine γ∗
s= 3 −2s. Conside (euε,epε)a sequence o solu ions
o p oblem (2.19)–(2.20). Assuming he asymp o ic expansion o he unknown (euε,epε)in he ollowing
o m
euε(z) = ε2 0(z) + ε3 1(z) + ε4 2(z) + · · · ,epε(z) = p0(z) + εp1(z) + ε2p2(z) + ε3p3(z) + · · · (3.24)
8
Ma ´ıa Anguiano and F ancisco J. Su´a ez-G au
o a.e. z∈Ω, whe e
i= (¯ i, i
3)wi h ¯ i= ( i
1, i
2), i = 0,1,...,
we deduce ha he main o de pai o unc ions ( 0, p0), wi h 0
3≡0and p0=p0(z0), sa is ies he
ollowing e ec i e educed S okes p oblem
−ν∂2
z3¯ 0(z) = 0(z0)− ∇x0p0(z0)in Ω,
di z0 Zh(z0)
0
¯ 0(z)dz3!= 0 in ω,
¯ 0= 0 on Γ1,
(3.25)
Mo eo e , ¯ 0sa is ies he ollowing bounda y condi ion on he bo om Γ0depending on he alue o γ:
•I γ=γ∗
s, hen i holds a powe law slip bounda y condi ion
−ν∂z3¯ 0=−|K¯ 0|s−2K2¯ 0on Γ0.(3.26)
•γ > γ∗
s, hen i holds a pe ec slip bounda y condi ion
−ν∂z3¯ 0= 0 on Γ0.(3.27)
•γ < γ∗
s, hen i holds a no-slip bounda y condi ion
¯ 0= 0 on Γ0.(3.28)
P oo . We i s p o e sys em (3.25). Fo his, we assume he asymp o ic expansion o he unknowns
(euε,epε) gi en by (3.24). Then, subs i u ing he expansion in o he p oblem (2.19)1,2, we ge
−νε2∆z0(¯ 0+O(ε)) −ν∂2
z3(¯ 0+O(ε)) + ε3( 0
3+O(ε))∂z3(¯ 0+O(ε)) + ∇z0(epε+O(ε)) = 0,
−νε2∆z0( 0
3+O(ε)) −ν∂2
z3( 0
3+O(ε)) + ε3( 0
3+O(ε))∂z3( 0
3+O(ε)) + 1
ε∂z3(epε+O(ε)) = 0,
ε2di z0(¯ 0+O(ε)) + ε∂z3( 0
3+ε 1
3+O(ε2)) = 0.
(3.29)
Collec ing he e ms o he same o de wi h espec o ε, we ha e
– The main o de e ms in (3.29)1,2a e
1 : −ν∂2
z3¯ 0+∇z0p0= 0in Ω,
1
ε:∂z3p0= 0 in Ω.(3.30)
– The main and nex o de e ms in (3.29)3a e
ε:∂z3 0
3= 0 in Ω,
ε2: di x0¯ 0+∂z3 1
3= 0 in Ω,(3.31)
9
Ma ´ıa Anguiano and F ancisco J. Su´a ez-G au
–P essu e. Es ima e (4.46)1implies, up o a subsequence, he exis ence o ep∈L2(Ω) such ha
con e gence (4.53) holds. Also, om (4.46)2, by no ing ha ε−1∂z3epεalso con e ges weakly in
H−1(Ω), we deduce ∂z3ep= 0. Then, epis independen o z3. To inish, i emains o p o e ha
p∈L2
0(Ω). Passing o he limi when ε ends o ze o in
ZΩepεdz = 0,
we espec i ely deduce ZΩep(z0)dz =Zω
h(z0)ep(z0)dz0= 0,
and so ha ephas null mean alue in Ω. This ends he p oo .
Nex , we p o e he main esul o his pape .
Theo em 4.7 (Main esul ).The limi pai o unc ions (eu, ep)∈Vz3×L2
0(Ω), wi h eu3≡0and
ep=ep(z0), gi en in Lemma 4.6 sa is ies he ollowing sys em in each case:
−ν∂2
z3eu0(z) = 0(z0)− ∇z0ep(z0)in Ω,
di z0 Zh(z0)
0eu0(z)dz3!= 0 in ω,
Zh(z0)
0eu0(z)dz3!·n=0 on ∂ω,
eu0= 0 on Γ1,
(4.54)
wi h ν > 0. Mo eo e , eu0sa is ies he ollowing bounda y condi ion on he bo om Γ0depending on he
alue o γ:
•γ=γ∗
s, hen i holds a powe law slip bounda y condi ion
−ν∂z3eu0=−|Keu0|s−2K2eu0on Γ0,(4.55)
whe e 1< s < 2and he aniso opic enso K∈R2×2is uni o mly posi i e de ini e, symme ic
and bounded.
•γ > γ∗
s, hen i holds a pe ec slip bounda y condi ion
−ν∂z3eu0= 0 on Γ0.(4.56)
•γ < γ∗
s, hen i holds a no-slip bounda y condi ion
eu0= 0 on Γ0.(4.57)
Rema k 4.8. By uniqueness o solu ions o p oblems gi en in Theo em 4.7, we obse e ha he pai o
unc ions (eu, ep)a e he same as hose unc ions ( 0, p0)ob ained in Theo em 3.1 by o mal a gumen s.
16
Ma ´ıa Anguiano and F ancisco J. Su´a ez-G au
P oo o Theo em 4.7. We will di ide he p oo in h ee s eps.
S ep 1. Le us i s conside he case γ > γ∗
s. F om Lemma 4.6, o p o e (4.54), i jus emains
o p o e he equa ions (4.54)1. To do his, we conside eϕ∈C1
c(ω×(0, h(z0)))3such ha eϕ3≡0 and
eϕ= 0 on Γ1. Thanks o eϕequaling ze o o z0ou side a compac subse o ω, hen
eϕ= 0 on ∂Ω Γ0and eϕ3= 0 on Γ0,
and so, we can ake i as es unc ion in (2.21), which is gi en by
νZΩ
Dεeuε:Dεeϕ dz +ZΩ
(euε· ∇ε)euε·eϕ dz +εγ−1ZΓ0
|Keu0
ε|s−2Keu0
ε·Keϕ0dσ
−ZΩepεdi ε(eϕ)dz =ZΩ
0·eϕ0dz.
(4.58)
Le us now pass o he limi when ε ends o ze o in e e y e ms o (4.58):
– Fi s e m on he le -hand side. Taking in o accoun con e gence (4.49), we ge
νZΩ
Dεeuε:Dεeϕ dz =νZΩ
ε−2∂z3eu0
ε·∂z3eϕ0dz +Oε=νZΩ
∂z3eu0·∂z2eϕ0dz +Oε.
– Second e m on he le -hand side. Taking in o accoun he egula i y o ϕ0, applying Cauchy-
Schwa z’s inequali y and es ima es (4.43), we ge
ZΩ
(euε· ∇ε)euεeϕ dz≤ keuεkL2(Ω)3kDεeuεkL2(Ω)3×3keϕkL∞(Ω)3≤Cε3,(4.59)
which implies ZΩ
(euε· ∇ε)euε·eϕ dz →0.
– Thi d e m on he le -hand side. We obse e ha since Γ0is la , hen he su ace measu e
associa ed o Γ0gi en by dσ =dz0. F om H¨olde ’s inequali y, Kis bounded, he Sobole
embedding L2,→Ls, he ace es ima e (4.40)1applied o eϕ0, he ace es ima e (4.40)2applied
o eu0
ε, and he es ima e (4.43) o Dεeu0
ε, we ge
εγ−1ZΓ0
|Keu0
ε|s−2Keu0
ε·Keϕ0dσ≤εγ−1kKeu0
εks−1
Ls(Γ0)2kKeϕ0kLs(Γ0)2
≤εγ−1kKks
L∞(Γ0)2×2keu0
εks−1
Ls(Γ0)2keϕ0kLs(Γ0)2
≤Cεγ−1keu0
εks−1
L2(Γ0)2keϕ0kL2(Γ0)2
≤Cεγ−1εs−1kDεeu0
εks−1
L2(Γ0)3×2kDzeϕ0kL2(Ω)3×2
≤Cεγ−γ∗
skDzeϕ0kL2(Γ0)3×2
≤Cεγ−γ∗
s,
which ends o ze o because γ > γ∗
s. Then, we ge
εγ−1ZΓ0
|Keu0
ε|s−2Keu0
ε·Keϕ0dσ →0.
17
Ma ´ıa Anguiano and F ancisco J. Su´a ez-G au
– Fou h e m on he le -hand side o (4.58). Taking in o accoun ha eϕ3≡0 and con e gence
(4.53), we ge
ZΩepεdi ε(eϕ)dz =ZΩepεdi x0(eϕ0)dz =ZΩepdi x0(eϕ0)dz +Oε.
Finally, om he abo e con e gences when ε→0, we de i e he ollowing limi sys em
νZΩ
∂z3eu0·∂z3eϕ0dz −ZΩepdi z0(eϕ0)dz =ZΩ
0·eϕ0dz, (4.60)
o e e y eϕ0∈C1
c(ω×(0, h(z0)))2wi h ϕ0= 0 on Γ1. By densi y, his equali y holds ue o e e y
eϕ0∈H1(0, h(z0); L2(ω)2) such ha eϕ0= 0 on Γ1. We obse e ha p oblem (4.60) has a unique solu ion
(eu0,ep) and he p oblem is equi alen o (4.54)1wi h bounda y condi ion (4.56). Uniqueness o solu ion
o (4.63) implies ha limi does no depend on he subsequence.
S ep 2. Nex , we conside he case γ < γ∗
s. Acco ding o Lemma 4.6, we p oceed simila ly o he
S ep 1, bu he e we also conside eϕ= 0 on Γ0, i.e. we conside a es unc ion eϕ∈C1
c(Ω)3wi h eϕ3≡0.
This means ha he e is no bounda y e m in he a ia ional o mula ion (4.58). Thus, p oceeding as
S ep 1, we deduce he limi a ia ional o mula ion (4.60), which holds o e e y eϕ0∈H1
0(Ω)2. This
p oblem has a unique solu ion and is equi alen o p oblem (4.54)1wi h bounda y condi ion (4.57).
Uniqueness o solu ion o (4.63) implies ha limi does no depend on he subsequence.
S ep 3. Finally, we conside he case γ=γ∗
s. Due o he nonlinea bounda y e m in (4.58), we
need o use mono onici y a gumen s o pass o he limi . Fo his, o simpli y he no a ion, we de ine
he applica ion e 7→ Aε(e ) as ollows
(Aε(e ),ew) = νZΩ
Dεe :Dεew dz +εγ−1ZΓ0
|Ke 0|s−2Ke 0·Kew0dσ,
o all e , ew∈H1(Ω)3such ha eϕ=ew= 0 on ∂Ω Γ0and eϕ3≡ew3≡0 on Γ0. F om [16, Lemma 2.3],
o e e y ε > 0, he mapping Aεis s ic ly mono one, i.e.
(Aε( )− Aε(w), −w)≥0.(4.61)
Acco ding o Lemma 4.6 and simila o S ep 1, we conside eϕ∈C1
c(ω×(0, h(z0)))3wi h eϕ3≡0 and
ϕ= 0 on Γ1, and we choose e εde ined by
e ε=eϕ−euε,
as es unc ion in (4.58). So we ge
(Aε(euε),e ε)−ZΩepεdi ε(e ε)dz =ZΩ
0·e 0
εdz −ZΩ
(euε· ∇ε)euε·e εdz,
which is equi alen o
(Aε(eϕ)− Aε(euε),e ε)−(Aε(eϕ),e ε) + ZΩepεdi ε(e ε)dz =−ZΩ
0·e 0
εdz +ZΩ
(euε· ∇ε)euε·e εdz.
18
Ma ´ıa Anguiano and F ancisco J. Su´a ez-G au
Due o (4.61), we can deduce
(Aε(eϕ),e ε)−ZΩepεdi ε(e ε)dz ≥ZΩ
0·e 0
εdz −ZΩ
(euε· ∇ε)euε·e εdz,
i.e. using he exp ession o Aε, we ha e
νZΩ
Dεeϕ:Dεe εdz +εγ−1ZΓ0
|Keϕ0|s−2Keϕ0·Ke 0
εdσ −ZΩepεdi ε(e ε)dz
≥ZΩ
0·e 0
εdz −ZΩ
(euε· ∇ε)euε·e εdz.
(4.62)
Since eϕ3≡0 and di ε(euε) = 0 in Ω, i holds
ZΩepεdi ε(e ε)dz =ZΩepεdi z0(eϕ0)dz,
and om RΩ(euε· ∇εeuε)euε·euεdz = 0, we deduce ha (4.62) eads
νZΩ
Dεeϕ:Dε(eϕ−euε)dz +εγ−1ZΓ0
|Keϕ0|s−2Keϕ0·K(eϕ0−eu0
ε)dσ −ZΩepεdi z0(eϕ0)dz
≥ZΩ
0·(eϕ0−eu0
ε)dz −ZΩ
(euε· ∇ε)euε·eϕ dz.
Replacing ϕby ε2ϕand di iding by ε2gi es
νZΩ
ε2Dεeϕ:Dε(eϕ−ε−2euε)dz +εγ−γ∗
sZΓ0
|Keϕ0|s−2Keϕ0·K(eϕ0−ε−2eu0
ε)dσ −ZΩepεdi z0(eϕ0)dz
≥ZΩ
0·(eϕ0−ε−2eu0
ε)dz −ZΩ
(euε· ∇ε)euε·eϕ dz.
Nex , we pass o he limi when ε ends o ze o:
– Fi s e m in he le -hand side. F om con e gence (4.49), we ge
νZΩ
ε2Dεeϕ:Dε(eϕ−ε−2euε)dz =νZΩ
∂z3eϕ0·∂z3(eϕ−ε−2eu0
ε)dz +Oε
=νZΩ
∂z3eϕ0·∂z3(eϕ−eu0)dz +Oε.
– Second e m in he le -hand side. Since γ=γ∗
s, om he con inuous embedding o H1(0, h(z0);
L2(ω)) in o L2(Γ0) and con e gence (4.49), we ge
εγ−γ∗
sZΓ0
|Keϕ0|s−2Keϕ0·K(eϕ0−ε−2eu0
ε)dσ =ZΓ0
|Keϕ0|s−2Keϕ0·K(eϕ0−ε−2eu0
ε)dσ +Oε
=ZΓ0
|Keϕ0|s−2Keϕ0·K(eϕ0−eu0)dσ +Oε.
19
Ma ´ıa Anguiano and F ancisco J. Su´a ez-G au
– Thi d e m in he le -hand side. F om con e gence (4.53), we ge
ZΩepεdi z0(eϕ0)dz =ZΩepdi z0(eϕ0)dz +Oε,
Mo eo e , since epis independen o z3and om condi ion (4.50), we ha e
ZΩep(z0) di z0(eu0)dz =Zωep(z0) di z0 Zh(z0)
0eu0dz3!dz0= 0,
so he hi d e m is w i en as ollows
ZΩepεdi z0(eϕ0)dz =ZΩepdi z0(eϕ0−eu0)dz +Oε.
– Fi s e m in he igh -hand side. F om con e gence (4.49), we ha e
ZΩ
0·(eϕ0−ε−2eu0
ε)dz =ZΩ
0·(eϕ0−eu0)dz +Oε.
– Second e m in he igh -hand side. F om H¨olde ’s inequali y and es ima es (4.43), we ge
ZΩ
(euε· ∇ε)euεeϕ dz≤ keuεkL2(Ω)3kDεeuεkL2(Ω)3×3keϕkL∞(Ω)3≤Cε3,
which implies ZΩ
(euε· ∇ε)euεeϕ dz →0.
Finally, om p e ious con e gences, we deduce he ollowing limi a ia ional inequali y
νZΩ
∂z3eϕ0·∂z3(eϕ−eu0)dz +ZΓ0
|Keϕ0|s−2Keϕ0·K(eϕ0−eu0)dσ −ZΩepdi z0(eϕ0−eu0)dz
≥ZΩ
0·(eϕ0−eu0)dz.
Since eϕ0is a bi a y, by Min y’s lemma, see [19, Chap e 3, Lemma 1.2], we deduce
νZΩ
∂z3eu0·∂z3eϕ0dz +ZΓ0
|Keu0|s−2Keu0·Keϕ0dσ −ZΩepdi x0(eϕ0)dz =ZΩ
0·eϕ0dz, (4.63)
o e e y eϕ0∈C1
c(ω×(0, h(z0)))2such ha eϕ0= 0 on ∂Ω Γ0. By densi y, his equali y holds ue
o e e y eϕ0∈Vz3such ha eϕ0= 0 on ∂Ω Γ0. F om [16, Theo em 2.1], p oblem (4.63) has a unique
solu ion (eu0,ep) and is equi alen o (4.54)1wi h bounda y condi ion (4.55). Uniqueness o solu ion o
(4.63) implies ha limi does no depend on he subsequence.
20
Ma ´ıa Anguiano and F ancisco J. Su´a ez-G au
Rema k 4.9. In he case s= 2 and Kε=εγ
2λ1
2I, wi h λ > 0, whe e he powe slip condi ion (1.6)
educes o he ollowing Na ie slip condi ion, wi h ic ion pa ame e depending on ε,
−ν∂x3u0
ε=−εγu0
ε, uε,3= 0,on Γ0.(4.64)
Repea ing he classical p oo o he exis ence o solu ion o he Na ie -S okes p oblem wi h homogeneous
Di ichle condi ions (see o ins ance [1, Theo em 2.3], [19, Theo em 7.1] and [26, Theo em 10.1]) gi es
he exis ence o a leas a weak solu ion (uε, pε)∈V(Ωε)×L2
0(Ωε)o p oblem (2.10) and (4.64).
P oceeding simila ly o he p oo o Theo em 4.7, bu wi hou mono onici y a gumen s, we can p o e
ha he limi sys em sa is ied by (eu0,ep)is (4.54), and ha he e exis s a c i ical alue o γis −1, which
ag ees wi h γ∗
2, such ha he e ec i e limi condi ions on Γ0a e he ollowing ones
•I γ=−1, hen
−ν∂z3eu0=−λeu0on Γ0.
•I γ > −1, hen
−ν∂z3eu0= 0 on Γ0.
•I γ < −1, hen
eu0= 0 on Γ0.
By classical a gumen s o he exis ence and uniqueness o solu ion o he S okes p oblem wi h homo-
geneous Di ichle condi ions, he e exis s a unique weak solu ion (eu0,ep)o p oblem (4.54) wi h co es-
ponding bounda y condi ions on Γ0depending on he alue o γgi en abo e.
Acknowledgmen s
We would like o hank all hose people (publishe s, edi o s, e e ees and esea che s) o all he suppo
ecei ed o he de elopmen o ou lines o esea ch. In pa icula , Ma ´ıa would like o dedica e his
a icle o he a he , Julio, o his uncondi ional suppo .
Con lic o in e es
The au ho s con i m ha he e is no con lic o in e es o epo .
Da a a ailabili y s a emen
Da a sha ing no applicable o his a icle as no da ase s we e gene a ed o analysed du ing he cu en
s udy.
21
Ma ´ıa Anguiano and F ancisco J. Su´a ez-G au
Re e ences
[1] P. Ace edo, C. Am ouche, C. Conca, and A. Ghosh, S okes and Na ie –S okes equa ions wi h
Na ie bounda y condi ions, J. Di e . Equ. 285 (2021), 258–320
[2] R. Aldbaissy, N. Chalhoub, J.K. Djoko, and T. Sayah, Full disc e isa ion o he ime dependen
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