On he s ong solu ions o he P imi i e Equa ions in 2D
domains.
F. Guill´en-Gonz´alez1& M.A. Rod ´ıguez-Bellido2
Depa amen o de Ecuaciones Di e enciales y An´alisis Num´e ico, Uni e sidad de Se illa.
c/ Ta ia s/n, 41012 Se illa (SPAIN).
e-mails: [email p o ec ed], [email p o ec ed]
Co espondence and p oo s o co ec ions: F. Guill´en-Gonz´alez
Key wo ds: hyd os a ic p essu e, mixed bounda y condi ions, local and global exis-
ence, uniqueness, asymp o ic beha iou .
1 In oduc ion
Some geophysical luids can be modelled h ough he so-called “p imi i e equa ions” [1],
[2]. This model is ob ained o mally om he Na ie -S okes equa ions, wi h anyso opic
(eddy) iscosi y, assuming wo impo an simpli ica ions: hyd os a ic p essu e (depending
linea ly on he dep h) and he igid lid hypo hesis ( ix wa e su ace) [3] . Fo simplici y, we
ake cons an densi y and assume ha he effec s due o he empe a u e (and salini y)
can be decoupled om he dynamic o he low. Then, we ha e a h ee-dimensional
low induced by he wind ension on he su ace and by he cen ipe al and Co iolis
o ces. When he Ea h cu a u e is no conside ed, we can use ca esian coo dina es
ins ead o sphe ical coo dina es (see Lions-Teman-Wang [2] o he model wi h sphe ical
coo dina es), hence he domain is gi en by
Ω={(�x , z )∈IR 3;�x∈ω,−D(�x)<z<0},(1)
whe e ω⊆IR 2is an open domain and D:ω→IR +is he dep h unc ion. The diffe en
bounda ies o Ω(su ace, bo om and sidewalls) a e espec i ely: Γs={(�x , 0); �x∈ω},
Γb={(�x , −D(�x)); �x∈ω}and Γl={(�x , z ); �x∈∂ω,−D(�x)<z<0}. Including, as i is
usual ([4]), cen ipe al effec s in o he p essu e e m, he h ee-dimensional model is:
(EP)
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



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
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

∂ �u+(�u·∇)�u+u3∂z�u−νh∆�u−ν ∂2
zz�u+α�u⊥+∇ps=�
Fin (0,T)×Ω,
∇·��0
−D(�x )�u( ;�x , z )dz�=0 in(0,T)×ω,
�u| =0 =�u0in Ω,
ν ∂z�u|Γs=�τ ,�u|Γb∪Γl=�
0in(0,T).
1Pa ially suppo ed by C.I.C.Y.T p ojec MAR98-0486
2Suppo ed by C.I.C.Y.T p ojec MAR98-0486
1
He e, we deno e �x=(x, y), ∇=(∂x,∂
y)and∆=∂2
xx +∂2
yy. The unknowns a e he
ho izon al componen o low eloci y �u=(u1,u
2):(0,T)×Ω→IR 2and he su ace
p essu e ps:(0,T)×ω→IR, whe eas he e ical componen o he low eloci y is
u3( ;�x , z )=−�z
−D(�x )∇·�u( ;�x , s )ds, ∀ ∈(0,T),∀(�x , z )∈Ω.(2)
Mo eo e , νhand ν >0 a e posi i e cons an s, ep esen ing ho izon al and e ical (eddy)
iscosi y coefficien s espec i ely, �
F:(0,T)×Ω→IR 2is an ho izon al ex e nal o ce ield
(depending on empe a u e and salini y, o ins ance) and �τ :(0,T)×Γs→IR 2 ep esen s
he ho izon al s ess on he su ace p oduced by he wind. Finally, α�u⊥=α(−u2,u
1)
models Co iolis effec s, he no-slip condi ion is assumed on he bo om and e ical
slip ing is pe mi ed on he sidewalls.
To gi e a a ia ional o mula ion o p oblem (EP), le us de ine he ollowing unc ion
spaces:
C∞
b,l (Ω) = {�ϕ ∈C∞(Ω)2;supp(�ϕ )isacompac se ⊆Ω (Γb∪Γl)},
H1
b,l(Ω) = C∞
b,l
H1
(Ω) = {� ∈H1(Ω)2;� =0onΓ
b∪Γl},H
−1
b,l (Ω) = dual o H1
b,l(Ω),
V={�ϕ ∈C∞
b,l (Ω)2;∇·��ϕ �=0 inω},whe e ��ϕ �(�x)=�0
−D(�x )�ϕ (�x , z )dz,
H=VL2
={� ∈L2(Ω)2;∇·�� �=0 inω,�� �·�n|∂ω =0},
V=VH1
={� ∈H1(Ω)2;∇·�� �=0 inω,� |Γb∪Γl=�
0}.
De ini ion 1.1 Le �u0∈H,�
F∈L2(0,T;H−1
b,l (Ω)2)and �τ ∈L2(0,T;H−1/2(Γs)2).We
say ha �u:(0,T)×Ω→IR 2is a weak solu ion o (EP)in (0,T)i �u∈L∞(0,T;H)∩
L2(0,T;V), e i ies he a ia ional o mula ion:
�T
0�Ω�−�u·(∂ �ϕ +(�u·∇)�ϕ +u3∂z�ϕ )+νh∇�u:∇�ϕ +ν ∂z�u·∂z�ϕ +α�u⊥·�ϕ �dΩd
=�Ω�u0·�ϕ (0) dΩ+�T
0��
F,�ϕ�Ωd +�T
0��τ ,�ϕ �Γsd , ∀�ϕ ∈C1([0,T]; V)s. . �ϕ (T)=�
0,
and, mo eo e �usa is ies he ene gy inequali y:
1
2��u�2
L2(Ω)+�
0�νh�∇�u�2
L2(Ω)+ν �∂z�u�2
L2(Ω)�ds
≤1
2��u0�2
L2(Ω)+�
0��
F,�u�Ωds +�
0��τ ,�u�Γsds a.e. ∈(0,T).
(3)
He e, �·,·�Ωdeno es duali y be ween H−1
b,l (Ω)and H1
b,l(Ω), whe eas �·,·�Γsdeno es duali y
be ween H−1/2(Γs)and H1/2(Γs).
2
De ini ion 1.2 Le �u0∈V,�
F∈L2(0,T;L2(Ω)2),�τ ∈L2(0,T;H1/2(Γs)2)and ∂ �τ ∈
L2(0,T;H−3/2(Γs)2). Le �ube a weak solu ion o (EP)in (0,T), we say ha �uis a
s ong solu ion i e i ies he addi ional egula i y condi ions:
�u∈L∞(0,T;V)∩L2(0,T;H2(Ω)2∩V),∂
�u∈L2(0,T;H).
The exis ence o a weak solu ion is well known, see Lewando ski [3] and Lions-Teman-
Wang [2], always in domains wi h sidewalls (i.e. D≥Dmin >0inω). In hese wo ks,
compac ness me hod is used o ob ain he eloci y �uin a space wi h he es ic ion
∇·��u�= 0 and he p essu e is eco e ed, in he la e pa o he a gumen , by a speci ic
De Rham’s lemma on he su ace. In domains wi hou sidewalls, he exis ence o a weak
solu ion is ob ained as a consequence o a limi p ocess applied o he Na ie -S okes
equa ions wi h anyso opic iscosi y when he a io dep h o e ho izon al diame e (o
he domain) ends o ze o, see Besson-Laydi [5] o he s a iona y case and Aze ad-Guill´en
[6] o he e olu ion case. Finally, he exis ence o a weak solu ion in domains wi hou
sidewalls can be p o ed by in e nal app oxima ion a gumen s: a mixed ( eloci y-p essu e)
a ia ional o mula ion o he s a iona y p oblem is app oxima ed by a con o m Fini e
Elemen me hod in Chac´on-Guill´en [7] and a semi-disc e iza ion in ime o he e olu ion
p oblem is p o ed ha con e ges o con inuous p oblem in Guill´en-Redondo [8, 9].
Howe e , o as a as we know, he e a e no esul s abou he exis ence o s ong
solu ion o p oblem (EP), excep ing he s a iona y linea case [10]. One o he p incipal
p oblems in his s udy is he ea men o he bounda y condi ions; on he su ace we
ha e a non homogeneous Neumann condi ion, whe eas he sidewalls and he bo om ha e
homogeneous Di ichle condi ion. On he o he hand, he uniqueness o he solu ion o
p oblem (EP) is also an open p oblem, e en in he case o s ong solu ions.
1.1 The 2Dp oblem
In his wo k, we a e going o conside mainly he wo-dimensional p oblem (wi h only
one ho izon al di ec ion). In his case, Co iolis o ces ha e no sense. Now, he model is:
(EP2)
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
∂ u+u∂xu+u3∂zu−νh∂2
xxu−ν ∂2
zzu+∂xps=Fin (0,T)×Ω,
∂x�u�=0 in(0,T)×ω,
u| =0 =u0in Ω,
ν ∂zu|Γs=τ, u|Γb∪Γl=0 in(0,T).
Then, all he unknowns a e scala : he ho izon al componen o he lux eloci y u:
(0,T)×Ω→IR a n d h e s u p e i c i a l p e s s u e ps:(0,T)×ω→IR. The e ical componen
o he lux eloci y is u3( ;x, z)=−�z
−D(x)∂xu( ;x, s)ds. One impo an diffe ence espec
o 3D case is ha now ω⊆IR is an in e al, which changes he unc ion spaces o ee
di e gence. Now, we ha e he ollowing simple cha ac e iza ions:
V={ϕ∈C∞
b,l (Ω); �ϕ�=0 inω},
3
H={ ∈L2(Ω); � �=0 inω},
V={ ∈H1(Ω); � �=0 inω, |Γb∪Γl=0}.
Finally, de ini ions o weak and s ong solu ions a e simila o he 3D case (changing
ec o ial no a ion by scala no a ion in uand x, and anishing he Co iolis e m).
Rema k 1.1 Now, he 2nd. equa ion in (EP2)means ha �u�only depends on .
1.2 Main esul s
In his pape , we will ob ain he ollowing main esul s, all in he 2D case and in domains
wi h sidewalls.
Theo em 1.3 (S ong global solu ion o small da a.) Le ω⊆IR an in e al and
D∈C2(ω)such ha D≥Dmin >0in ω. We assume u0∈V,F∈L2(0,T;L2(Ω)) and
τ∈L2(0,T;H1/2+ε
0(Γs)), o some ε>0, wi h ∂ τ∈L2(0,T;H−1/2(Γs)). I he ollowing
“smallness es ic ion” is assumed: ∀ ∈[0,T],
(H)
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exp �−1
4C +�
0a(s)ds��2��u0�2
V+C2�τ(0)�2
H−1/2(Γs)�
+�
0exp �1
4Cs−�s
0a(σ)dσ�b(s)ds�<M
2,
whe e Mis a small enough posi i e cons an (see Lemma 5.2), Cis a cons an ha
appea s in (11) and a,ba e unc ions depending on he da a τand F(see (30) and (31)),
hen he e exis s a unique s ong solu ion (u, ps)o (EP2)in (0,T)(psis unique up o a
unc ion o ).
Co olla y 1.4 (Asymp o ic beha iou when ↑+∞.) Le ω⊆IR an in e al and
D∈C2(ω)such ha D≥Dmin >0in ω. We assume u0∈V,F∈L2(0,+∞;L2(Ω))
and τ∈L2(0,+∞;H1/2+ε
0(Γs)), o some ε>0, wi h ∂ τ∈L2(0,+∞;H−1/2(Γs)). I he
“smallness es ic ion” (H)is assumed ∀ ∈[0,+∞), hen he e exis a unique s ong
solu ion u∈L2(0,+∞;H2(Ω)∩V)) ∩L∞(0,+∞;V),∂ u∈L2(0,+∞;H). Mo eo e , i
�+∞
0exp �1
4C ���τ( )�2
H1/2+ε
0(Γs)+�∂ τ( )�2
H−1/2(Γs)+�F( )�2
L2(Ω)�d < +∞,(4)
he e exis s wo cons an s K1,K2>0such ha :
�∇u( )�2
L2(Ω)≤exp �−1
4C �K1�2�u0�2
V+C2�τ(0)�2
H−1/2(Γs)+K2�∀ ≥0,(5)
(i.e. he solu ion anishes exponen ially in he H1(Ω)-no m, as inc eases).
4
Theo em 1.5 (S ong local solu ion o any da a.) Unde hypo heses o Theo em
1.3, changing he es ic ion (H)by Dmax(= max
ωD)small enough, hen he e exis s
T∗∈(0,T]and a unique s ong solu ion (u, ps)o (EP2)in (0,T
∗).
Theo em 1.6 (Uniqueness o s ong/weak solu ion.) Le ube a weak solu ion o
(EP2)in (0,T). I he e exis s ano he weak solu ion ¯uo (EP2)in (0,T), such ha
e i ies he addi ional egula i y:
∂z¯u∈L4(0,T;L4(Ω)),(6)
hen bo h solu ions coincide in (0,T).
Rema k 1.2 The a gumen s o p o e all hese main esul s, will no be alid in he 3D
case. On he o he hand, he addi ional egula i y (6) ha implies uniqueness is e i ied by
he s ong solu ions o (EP2)(and no by only weak solu ions). Applying his uniqueness
a gumen o he 3Dcase, i is necessa y an addi ional egula i y ha is no e i ied by
he s ong solu ions.
To make he s udy abou exis ence o s ong solu ions, i will be con enien o decom-
pose he p oblem (EP) in wo: one linea p oblem (L) wi h nonhomogeneous bounda y
condi ions on he su ace, and a nonlinea p oblem (P) wi h homogeneous bounda y
condi ions.
This pape is o ganized as ollows. In Sec ion 2, we p o e some echnical inequali ies
ha we will use in he ollowing.
The linea p oblem (L) is s udied in Sec ions 3 and 4, whe eas he s udy o (P)(by
means o a Gale kin me hod) is made in Sec ion 5, whe e he p oo o Theo em 1.3is
inished. Indeed, in Sec ion 3, using he known esul s ([10]) abou s ong solu ion o he
linea s a iona y p oblem (Ls ), we deduce some p ope ies o he diffe en ial ope a o
associa ed, ha we apply in Sec ion 4, a i ing a he exis ence and uniqueness o s ong
solu ion o (L) (all hese esul s a e alid in any space dimension).
In Sec ion 6, we p esen he p oo o Theo em 1.5, based in a ixed poin a gumen (in
pa icula , i is no possible o make a Gale kin a gumen as in Theo em 1.3). Finally,
he uniqueness o weak solu ion assuming ha a s ong solu ion exis s (Theo em 1.6), is
p o ed in Sec ion 7.
2 Some echnical esul s
Fi s , we see h ee echnical lemmas ha we will used se e al imes in his pape :
Lemma 2.1 Le Ω⊆IR N(N=2o 3) be he domain conside ed in his wo k (de ined by
(1)). Then, o all � ∈W1,p(Ω)N−1(p>1), i we de ine 3(�x , z )=−�z
−D(�x )∇·� (�x , s )ds,
one has:
� 3�Lp(Ω)≤Dmax�∇ ·� �Lp(Ω)
5

P oo : I is a consequence o Fubini’s Theo em:
� 3�p
Lp(Ω)=�Ω������z
−D(�x )∇·� (�x , s )ds�����
p
dΩdz
≤�Ω��z
−D(�x )|∇·� (�x , s )|pds�(z+D(�x))p/p�dΩdz
=�ω�0
−D(�x )|∇·� (�x , s )|p��0
s(z+D(�x))p/p�dz�dΩds
≤Dp
max
p�ω�0
−D(�x )|∇·� (�x , z )|pdΩds =Dp
max
p�∇ ·� �p
Lp(Ω)
Lemma 2.2 (In e pola ion inequali ies.) Le Ω⊆IR Nbe a Lipschi z-con inuous
domain. The ollowing inequali y holds:
��u�Lp(Ω)≤C��u�1−q/p
W1,N (Ω)��u�q/p
Lq(Ω),∀�u∈W1,N (Ω)N−1,(7)
whe e N≤q≤p<+∞.
P oo : I is aken om he Ni enbe g’s pape [11], whe e is p o ed he esul when
Ω=IR
N. He e, we adap he p oo o a Lipschi z-con inuous domain Ω.
Fo his, we pass hese inequali ies o Ωusing a p olonga ion ope a o [12]
E:W1,1(Ω)N−1−→ W1,N (IRN)N−1,
e i ying E�u|Ω=�uand �E�u�W1,N (IRN)≤C��u�W1,N (Ω),∀�u∈W1,N (Ω)N−1, o some
C=C(Ω)>0. Ni enbe g’s esul says
�E�u�Lp(IRN)≤C�E�u�1−q/p
W1,N (IRN)�E�u�Lq(IRN).
The e o e, since ��u�Lp(Ω)≤�E�u�Lp(IRN)and �E�u�Lq(IRN)≤C��u�Lq(Ω), we a i e a (7).
An easy applica ion o he abo e Lemma and he Poinca ´e’s inequali y, gi e us he
ollowing:
Co olla y 2.3 Le Ω⊆IR Nbe he domain conside ed in his wo k. The ollowing in-
equali y holds:
��u�Lp(Ω)≤C�∇�u�1−q/p
LN(Ω)��u�q/p
Lq(Ω)∀�u∈W1,N
b,l (Ω)N−1,(8)
whe e N≤q≤p<+∞.
Rema k 2.1 The main ad an age o he 2D case is o conside (7) and (8) o N=2.
In he ollowing, we will call Gaglia do-Ni enbe g’s inequali y o (7) o (8) in he case
N=2,p=4and q=2, i.e.
�u�L4(Ω)≤C�u�1/2
L2(Ω)�u�1/2
H1(Ω)∀u∈H1(Ω),(9)
�u�L4(Ω)≤C�u�1/2
L2(Ω)�∇u�1/2
L2(Ω)∀u∈H1
b,l(Ω).(10)
6
3 The s a iona y linea case
In his Sec ion, we will see some p elimina y esul s abou he linea s a iona y sys em
(also called hyd os a ic S okes sys em):
(Ls )
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
−νh∆�u−ν ∂2
zz�u+∇ps=�gin Ω,
∇·��u�=0inω,
ν ∂z�u=�aon Γs,
�u=�
0onΓ
b∪Γl.
3.1 Known esul s abou exis ence and uniqueness
Lemma 3.1 (Weak solu ion o (Ls )) Le ω⊆IR d(d=1o 2) and le Ω⊆IR d+1,
de ined as in (1), be a Lipschi z-con inuous domain. I �g∈H−1
b,l (Ω)dand �a∈H−1/2(Γs)d,
hen he p oblem (Ls )has a unique solu ion �u∈H1(Ω)d. Mo eo e , one has he con in-
uous dependence, i.e. he e exis s a cons an C=C(Ω,ν
h,ν
)>0such ha
��u�V≤C���a�H−1/2(Γs)+��g�H−1
b,l (Ω)�.(11)
In [5], [7] and [3], he e a e diffe en p oo s o his esul (e en in he nonlinea case).
Lemma 3.2 (S ong solu ion o (Ls )) Le ω⊆IR d(d=1o 2)beaC2domain and
D∈C2(ω)wi h D≥Dmin >0in ω.I �g∈L2(Ω)dand �a∈H1/2+ε
0(Γs)d, o some ε>0,
hen he unique solu ion �uo he p oblem (Ls )belongs o H2(Ω)d∩V. Mo eo e , we ha e
he con inuous dependence, i.e. he e exis s a cons an C=C(Ω,ν
h,ν
)>0such ha :
��u�H2(Ω)≤C���a�H1/2+ε
0(Γs)+��g�L2(Ω)�.(12)
See [10] o he p oo o egula i y. The con inuous dependence can be deduced ol-
lowing he cons uc ion o he auxilia y p oblems made by Ziane in [10].
3.2 The hyd os a ic S okes ope a o
We de ine A, ha i will call “hyd os a ic S okes ope a o ”, as he esol en ope a o
ela ed o he homogeneous Neumann bounda y condi ions on he su ace and Di ichle
bounda y condi ions on he bo om and sidewalls, i.e. A:V→V�such ha
�A�u,� �V�,V =�Ω(νh∇�u:∇� +ν ∂z�u·∂z� )dΩ∀�u , � ∈V. (13)
Then, i we deno e �g=A�u∈V�, omLemma3.1, �uis he unique weak solu ion o he
hyd os a ic S okes p oblem (Ls ), wi h �a=�
0. Mo eo e , aking in o accoun Lemma 3.2,
Ais a sel -adjoin isomo phism om H2(Ω)2∩V o H. In pa icula , i A�u=�gwi h
�g∈H,�uis cha ac e ized as he unique s ong solu ion o he p oblem (Ls ), wi h �a=�
0.
Finally, he domain o A,de inedby
D(A)={�u;�u∈Vand A�u∈H}.
can be cha ac e ized as ollows:
7
Lemma 3.3 Le ω⊆IR d(d=1o 2)beaC2domain and D∈C2(ω)wi h D≥Dmin >0
in ω. Then
D(A)={�u;�u∈H2(Ω)d∩Vand ∂z�u=�
0on Γs}.(14)
Mo eo e , he e exis s C=C(Ω,ν
h,ν
)>0such ha
��u�H2(Ω)≤C�A�u�L2(Ω)∀�u∈D(A).(15)
P oo : Le Ybe he igh hand side o (14).
a) D(A)⊂Y:Le �u∈D(A). I we deno e �g=A�u, hen�uis he weak solu ion
o (Ls )wi h�a=�
0. As �g∈H, om he Ziane’s egula i y esul s [10], we deduce ha
�u∈Y,and hecon inuousdependence(12)says:
��u�H2(Ω)≤C��g�L2(Ω)=C�A�u�L2(Ω)
b) Y⊂D(A):Le �u∈Y.I wedeno e�
=−νh∆�u−ν ∂2
zz�u, hen�
∈L2(Ω)dand
A�u=P�
,whe ePis he o ogonal p ojec ion om L2(Ω)don o H. Hence A�u∈H,i.e.
�u∈D(A).
3.3 Cons uc ion o a special basis
In his subsec ion, we will p o e he ollowing esul :
Lemma 3.4 Unde he condi ions o Lemma 3.3, he e exis s a sequence {λj}j≥1⊆IR
wi h 0<λ
1≤λ2≤... ≤λj≤λj+1 ≤...,{λj}→+∞, and an o hono mal basis o H,
{�wj}j≥1, whe e each �wjis an eigen unc ion o Aassocia ed o eigen alue λj.
P oo : Le Λ:H−→ D(A)�→Hbe he ope a o ha associa es each �g∈H o
�u∈D(A), he unique s ong solu ion o he p oblem (Ls )wi h�a=�
0 (i.e. A�u=�g). This
is an compac (using Lemma 3.3 and he compac embedding o H2(Ω)2∩Vin o H)and
sel -adjoin ope a o
(Λ�g1,�g2)=(�u1,�g2)=(�u1,A�u
2)=(A�u1,�u
2)=(�g1,Λ�g2).
Then, as His sepa able, we can apply he Hilbe Schmid ’s Theo em (o spec al de-
composi ion), and he e exis s an o hogonal basis o H o med by eigen unc ions o Λ,
{� j}j≥1(Λ� j=µj� j,whe eµj�0asj�+∞). Le λj=1/µjand �zj=µj� j.Then
A�zj=λj�zj,and hesequence�wj=�zj/�λjis he o hono mal basis o he Lemma.
4 The e olu ion linea case
In his sec ion we will s udy he s ong solu ion o he nons a iona y linea p oblem:
(L)
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
∂ � −νh∆� −ν ∂2
zz� +∇qs=�
in (0,T)×Ω,
∇·�� �=0 in(0,T)×ω,
� | =0 =� 0in Ω,
ν ∂z� =�τ on (0,T)×Γs,
� =�
0on(0,T)×(Γb∪Γl).
8
Theo em 4.1 Le ω⊆IR d(d=1o 2) be a C2domain and D∈C2(ω)wi h D≥
Dmin >0in ω.I �
∈L2((0,T)×Ω)d,� 0∈V,�τ ∈L2(0,T;H1/2+ε
0(Γs)d), o some
ε>0, wi h ∂ �τ ∈L2(0,T;H−1/2(Γs)d), hen he e exis s a unique s ong solu ion � o (L)
in (0,T). Mo eo e , he e exis s C>0such ha
�� �2
L∞(V)+�� �2
L2(D(A)) +�∂ � �2
L2(H)≤C��� 0�2
V+��τ (0)�2
H−1/2(Γs)
+��
�2
L2(L2(Ω)) +��τ �2
L2(H1/2+ε
0(Γs)) +�∂ �τ �2
L2(H−1/2(Γs))�(16)
P oo : Uniqueness can be easily deduced om he linea i y o he p oblem (L). The
p oo o he exis ence will be sepa a e in se e al s eps.
S ep 1. Weak solu ion o (L).The weak solu ion � o (L)in(0,T)canbeob ained
as a limi o Gale kin app oxima ions � m∈C1([0,T]; Vm)(beingVmam-dimensional
subspace o V)such ha
(L)m
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d
d �Ω� m·�ϕ dΩ+νh�Ω∇� m:∇�ϕ dΩ+ν �Ω∂z� m·∂z�ϕ dΩ
=�Ω
�
m·�ϕ dΩ+�Γs
�τ m·�ϕ |Γsdσ∀�ϕ ∈Vm,
� m(0) being he p ojec ion o � 0on o Vm,
whe e �
m∈C0([0,T]; H−1
b,l (Ω)2)and�τ m∈C0([0,T]; H−1/2(Γs)2)a e espec i ely egula
app oxima ions o �
and �τ .
Taking � mas es unc ion in (L)m,onecandeduce ha hesequence� mis bounded
in L∞(0,T;H)∩L2(0,T;V). Passing o he limi in a s anda d way, we ob ain he weak
egula i y o � .
Rema k 4.1 (Weak solu ion o (EP)). Gale kin app oxima ions o nonline p oblem
(EP)a e simila o p oblem (L)m. The only diffe ences a e he nonlinea e ms:
�Ω�(�um·∇)�um+um3∂z�um�·�ϕ dΩ,
whe e um3is de ined om ∇·�umas in (2). Bu , hese e ms anish when �umis aken as
es unc ion, hence we can also deduce ha �umis bounded in L∞(0,T;H)∩L2(0,T;V).
Now, by using a compac ness esul (es ima ing ∂ �umin a con enien space), we could
pass o he limi and ob ain a weak solu ion �uo (EP2)in (0,T).
S ep 2. “Li ing” o he Neumann bounda y condi ions. We de ine he ope a o
B:�a∈H−1/2(Γs)d→�u=B�a∈V,whe e�uis he weak solu ion o he hyd os a ic
S okes p oblem (Ls )wi h�g=�
0, i.e.
�u∈Vsuch ha �A�u, �
ψ�V�,V =��a, �
ψ�Γs∀�
ψ∈V.
9
Lemma 5.2 Unde he hypo hesis (H)o he Theo em 1.3and supposing (29), le Mbe
a cons an such ha :
(a)1−C1DmaxM>1/2,
(b)C2M2<1/(4C),
(C1and C2a e he cons an s ha appea in (29) and C>0is he equi alence cons an
be ween �Au�L2and he H2-no m, see Lemma 3.3), hen
�wm( )�V<M, ∀ ∈[0,T].
P oo : A guing by con adic ion, we suppose he e exis s some ins an in (0,T)whe e
he bound Mis eached. Le ∗ he smalles o hese ins an s, i.e. �wm( )�V<M,
∀ ∈[0,
∗)and�wm( ∗)�V=M.Then,∀ ∈[0,
∗],
1−C1Dmax�wm( )�V≥1−C1DmaxM>1/2.
In he las es ima ion, we ha e used hypo hesis (a). I we deno e y( )=�wm( )�2
V,using
ha 1
C�wm�2
V≤�Awm�2
L2(Ω)(see (15) in Lemma 3.3), (29) yields:
y�( )+ 1
2Cy( )≤C2M2y( )+a( )y( )+b( ),∀ ∈[0,
∗].
Now, om hypo hesis (b),
y�( )+ 1
4Cy( )≤a( )y( )+b( ),∀ ∈[0,
∗].(32)
In eg a ing his diffe en ial inequali y be ween 0 and ∗, we ob ain:
y( ∗)≤exp �−1
4C ∗+� ∗
0a( )d ��y(0) + � ∗
0exp �1
4C −�
0a(s)ds�b( )d �
The e o e, since
y(0) = �wm0�2
V≤�w0�2
V≤2��u0�2
V+�e(0)�2
V�≤2��u0�2
V+C2�τ(0)�2
H−1/2(Γs)�,
hypo hesis (H) implies �wm( ∗)�V<M, hence we a i e a con adic ion.
S ep 3. P oo o Theo em 1.3: F om Lemma 5.2, wmis bounded in L∞(0,T;V).
Mo eo e , applying hypo hesis (a)o Lemma5.2 in (29), one has
d
d �wm�2
V+1
2�Awm�2
L2(Ω)≤C2M4+M2a( )+b( ),(33)
hence, in eg a ing in ime, we deduce ha wmis bounded in L2(0,T;H2(Ω)). On he o he
hand, aking ∂ wm( )∈Vmas a es unc ion in (24), in eg a ing in ime and using he
abo e egula i y, one deduces ha ∂ wmis bounded in L2(0,T;H). Then, by a s anda d
a gumen o passage o he limi , we ob ain ha w(and a su ace p essu e associa ed πs)
is a s ong global solu ion o (P). Finally, (u, ps)=(e+w,qs+πs) is a s ong solu ion o
(EP2)in(0,T). The uniqueness o s ong solu ion o (EP2)s ems omSec ion7.
16

Rema k 5.1 In he 3D case, we canno ob ain he abo e s ong es ima es. I is because
in he igh hand side o (27), i we es ima e he co esponding I2 e m, we ob ain a bound
o he o m
Dmax�Awm�5/2
L2(Ω)�wm�1/2
V
which canno be con ola ed wi h he le hand side o (27).
5.2 P oo o Co olla y 1.4.
Le us i s p o e exis ence o a s ong solu ion o (EP2)in(0,+∞). The a gumen is
based in S ep 1 and 2o he p oo o Theo em 1.3. In pa icula , i is no difficul o
ob ain he global weak es ima ions:
wmis bounded in L2(0,+∞;V)∩L∞(0,+∞;H).
Now, using hypo hesis (H)in[0,+∞), we can deduce ha �wm�V<M,∀ ∈[0 + ∞).
Le us change S ep 3. Ins ead o (33), we ew i e (29) as:
d
d �wm�2
V+1
2�Awm�2
L2(Ω)≤C2M2�wm�2
V+M2a( )+b( ).
Using ha wmis bounded in L2(0,+∞;V)anda,b∈L1(0,+∞) ( hanks o (31) and he
global egula i y o τ,∂ τand F), we ha e ha wmis bounded in L2(0,+∞;H2(Ω)∩
V). Then, we can conclude he exis ence o a s ong solu ion u∈L∞(0,+∞;V)∩
L2(0,+∞;H2(Ω)∩V)and∂ u∈L2(0,+∞;H).
Now, le us see he asymp o ic beha iou o u. Adding in bo h pa s o (32) d
d �e( )�2
V+
1
4C�e( )�2
V, aking in o accoun ha
d
d �e( )�2
V≤2�e( )�V�∂ e( )�V,
we ob ain o z( )=�wm( )�2
V+�e( )�2
V he inequali y:
z�( )+�1
4C−a( )�z( )≤b( )+ 1
2C�e( )�2
V+4C�∂ e( )�2
V.
Mul iplying by exp �1
4C −�
0a(s)ds�and in eg a ing on (0, ),
z( )≤exp �−1
4C +�
0a(s)ds��z(0)
+�
0exp �1
4Cs−�s
0a(σ)dσ��b(s)+ 1
2C�e(s)�2
V+4C�∂ e(s)�2
V�ds�.
(34)
Now, using ha �um( )�2
V≤2z( ),
�um( )�2
V≤exp �−1
4C �K1�z(0)
+2
K1�
0exp �1
4Cs��b(s)+ 1
2C�e(s)�2
V+4C�∂ e(s)�2
V�ds�,
17
whe e K1=2exp��a�L1(0,+∞)�.Sincez(0) ≤2�u0�2
V+C2�τ(0)�2
H−1/2(Γs), bounding in a
con enien way b,eand ∂ e(in unc ion o τ,∂ τand F), we can deduce he asymp o ic
beha iou (5) whene e he hypo hesis (4) holds.
6 Local s ong solu ion o any da a (p oo o Theo-
em 1.4)
We wan o apply now a ixed poin a gumen o ob ain s ong solu ion o (EP2), local
in ime, bu o any da a. Now, we s udy p oblem (Q), which is simila o (P)bu whose
solu ion is (w=u− , ˜πs=ps−qs), whe e ( ,qs) is he solu ion o (L)wi h 0=0
an = 0. Wi h his pu pose, we ew i e (Q)asa ixedpoin equa ionbymeanso a
linea isa ion. We de ine, o each T>0:
Y(T)=�¯w;¯w∈L2(0,T;D(A)) ∩L∞(0,T;V),∂
¯w∈L2(0,T;H),
¯w(0) = u0,�¯w�2
L∞(V)+�¯w�2
L2(D(A)) +�∂ ¯w�2
L2(H)≤R2�.
Gi en he s ong solu ion o (L)in(0,T)and ¯w∈Y(T), we conside he linea p oblem:
(Ql)
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∂ w−νh∂2
xxw−ν ∂2
zzw+∂xπs=G(¯w, )in(0,T)×Ω,
∂x�w�=0 in(0,T)×ω,
w| =0 =u0in Ω,
ν ∂zw=0 on(0,T)×Γs,
w=0 on(0,T)×(Γb∪Γl),
whe e G(¯w, )=F−(¯w+ )∂x(¯w+ )−(¯w3+ 3)∂z(¯w+ ). P oblem (Ql) is simila o
p oblem (R), which has al eady been s udied in Sec ion 4. The e o e, since u0∈Vand
G∈L2((0,T)×Ω), hen w∈L2(0,T;D(A)) ∩L∞(0,T;V)and∂ w∈L2(0,T;H).
Fi s , we a e going o p o e ha , he e exis s R2la ge enough such ha Y(T)�=∅,
∀T>0. Indeed, le w∗be he unique solu ion o he hyd os a ic S okes p oblem:
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∂ w∗−νh∂2
xxw∗−ν ∂2
zzw∗+∂xπs=0 in(0,T)×Ω,
∂x�w∗�=0 in(0,T)×ω,
w∗| =0 =u0in Ω,
ν ∂zw∗=0 on(0,T)×Γs,
w∗=0 on(0,T)×(Γb∪Γl).
Following he easoning o he p oblem (R), see (21) and (22), we know ha :
�w∗�2
L∞(V)+�w∗�2
L2(D(A)) +�∂ w∗�2
L2(H)≤�u0�2
V,(35)
he e o e, aking R2≥�u0�2
V, henw∗∈Y(T), ∀T>0.
18
Now, we in oduce he Banach space XT=L2(0,T;V) and he mapping
Φ:Y(T)−→ XT,gi en by Φ( ¯w)=w,
whe e wis he unique solu ion o (Ql). Ob iously, a ixed poin o Φsol es p oblem (Q).
A guing as in p oblem (R), we ha e:
�w�2
L∞(V)+�w�2
L2(D(A)) +�∂ w�2
L2(H)≤�u0�2
V+C�G(¯w, )�2
L2(L2(Ω)) (36)
On he o he hand, e i ies p oblem (L), wi h inicial da a ze o and homogeneous
second membe (i.e. 0=0and = 0) bu a nonhomogeneous Neumann bounda y
condi ion (τ)on hesu ace.Then, sa is ies he es ima e (see (16)):
� �2
L∞(V)+� �2
L2(H2(Ω)) +�∂ �2
L2(H)≤B(τ)2,(37)
whe e B(τ)2=C��τ(0)�2
H−1/2(Γs)+�τ�2
L2(H1/2+ε
0(Γs)) +�∂ τ�2
L2(H−1/2(Γs))�.
Now, we wan o ind condi ions o apply Schaude ’s Theo em.
1) ∃T∗∈(0,T]such ha Φ(Y(T∗)) ⊂Y(T∗):
Le ¯w∈Y(T)andw=Φ(¯w). Then:
�G(¯w, )�2
L2(L2(Ω)) ≤9��F�2
L2(L2(Ω)) +�(¯w+ )∂x(¯w+ )�2
L2(L2(Ω))
+�(¯w3+ 3)∂z(¯w+ )�2
L2(L2(Ω))�≡
3
�
i=1
Ii
(38)
We bound each e m Ii(cons an Gwill come om he Gaglia do-Ni enbe g’s inequal-
i ies, see Lemmas 2.2and2.3, whe eas Cwe will deno e diffe en cons an s independen
o R,B(τ), Dmax and T). Fi s , we es ima e
�(¯w+ )∂x(¯w+ )�2
L2(Ω)≤�¯w+ �2
L4(Ω)�∂x(¯w+ )�2
L4(Ω)
≤4��¯w�2
L4(Ω)+� �2
L4(Ω)���∂x¯w�2
L4(Ω)+�∂x �2
L4(Ω)�
≤4G2��¯w�V�¯w�L2(Ω)+� �V� �L2(Ω)���¯w�H2(Ω)�¯w�V+� �H2(Ω)� �V�.
In eg a ing in (0,T), aking in o accoun de ini ion o Y(T) and (37),
I2≤4G2T1/2��¯w�L∞(V)�¯w�L∞(H)+� �L∞(V)� �L∞(H)�
×��¯w�L∞(V)�¯w�L2(H2(Ω)) +� �L∞(V)� �L2(H2(Ω))�
≤CT1/2(B(τ)2+R2)2.
In a simila way, we bound he e ical eloci y e ms as ollows:
�(¯w3+ 3)∂z(¯w+ )�2
L2(Ω)≤�¯w3+ 3�2
L4(Ω)�∂z(¯w+ )�2
L4(Ω)
≤4��¯w3�2
L4(Ω)+� 3�2
L4(Ω)���∂z¯w�2
L4(Ω)+�∂z �2
L4(Ω)�
≤4D2
max ��∂x¯w�2
L4(Ω)+�∂x �2
L4(Ω)���∂z¯w�2
L4(Ω)+�∂z �2
L4(Ω)�
≤4G2D2
max ��¯w�H2(Ω)�¯w�V+� �H2(Ω)� �V�2.
19
The e o e, in eg a ing in (0,T),
I3≤4CD
2
max ��¯w�L2(H2(Ω))�¯w�L∞(V)+� �L2(H2(Ω))� �L∞(V)�2
≤CD
2
max (B(τ)2+R2)2.
In he las es ima es, we could no ob ain any powe o T, and his ac is he main
difficul y in ou a gumen . Indeed, inse ing all he abo e bound in (38),
�G(¯w, )�2
L2(L2(Ω)) ≤C��F�2
L2(L2(Ω)) +(B(τ)2+R2)2�D2
max +T1/2�� (39)
Then, om (36) and (39),
�w�2
L∞(V)+�w�2
L2(D(A)) +�∂ w�2
L2(H)≤�u0�2
V
+C��F�2
L2(L2(Ω)) +(B(τ)2+R2)2�D2
max +T1/2�� (40)
The abo e inequali y can be w i en as
�w�2
L∞(V)+�w�2
L2(H2(Ω)) +�∂ w�2
L2(H)≤aR4+bR2+c,
whe e, o some C=C(Ω,ν
h,ν
)>0,
a=C�D2
max +T1/2�,
b=2CB(τ)2�D2
max +T1/2�,
c=�u0�2
V+C��F�2
L2(0,T;L2(Ω)) +B(τ)4�D2
max +T1/2��.
Taking R2≥�u0�2
V(hence Y(T)�=∅,∀T>0), one has w∈Y(T)whene e
aR4+bR2+c≤R2.(41)
In he ollowing, we will see ha o any da a F,τ,u0,(41)is e i ied. Anecessa y
condi ion o (41) is b<1. Bu , i can also ind some sufficien condi ions. Indeed, one
possibili y is o impose he ollowing h ee condi ions:
Condi ion 1: Dmax and Ta e small enough such ha
b≤1
2.
Fo ins ance, 2 CB(τ)2D2
max ≤1/4and2CB(τ)2T1/2≤1/4.
Condi ion 2: R2big enough such ha
c≤1
4R2.
Condi ion 3: asmall enough (i.e. Dmax and Tsmall enough) such ha
aR
2≤1
4.
20
In conclusion, he e exis s T∗∈(0,T]andDmax >0 small enough, such ha o some
Rbig enough, one has Φ(Y(T∗)) ⊂Y(T∗).
2) Y(T∗)is ela i ely compac in XT∗.
Le WT∗={¯w;¯w∈L2(0,T
∗;D(A)) and ∂ ¯w∈L2(0,T
∗;H)}.Y(T∗)isaboundedse
o WT∗and WT∗is embedded in a compac way in XT∗.The e o e,Y(T∗) is ela i ely
compac in XT∗.
3) Y(T∗)is closed in XT∗.
Le {¯wn}n≥1⊆Y(T∗)such ha ¯wn−→ ¯ws ongly in XT∗(i.e. in he L2(0,T;V)-
no m). Le us see ha ¯w∈Y(T∗).As {¯wn}n≥1is bounded in WT∗, in pa icula , he e
exis s a subsequence {¯wk}o {¯wn}such ha :
¯wk�¯win L2(0,T
∗;D(A)∩V),
∂ ¯wk�∂
¯win L2(0,T
∗;H).(42)
Then, applying a compac ness esul o Aubin-Lions ype [13]:
¯wk−→ ¯win C([0,T
∗]; H).
The e o e, since ¯wk(0) = u0,∀k≥1, hen ¯w(0) = u0.Bylowe semi-con inui yo he
no m,
�¯w�2
L∞(V)+�¯w�2
L2(D(A)) +�∂ ¯w�2
L2(H)
≤lim in k→+∞��¯wk�2
L∞(V)+�¯wk�2
L2(D(A)) +�∂ ¯wk�2
L2(H)�≤R2,
hen ¯w∈Y(T∗), hence Y(T∗)isclosedinXT∗.Thisone,join lywi h2), imply ha Y(T∗)
is compac in XT∗.
4) Φ:Y(T∗)−→ Y(T∗)is con inuous espec o XT∗ opology.
Le {¯wn}n≥1⊆Y(T∗) such ha ¯wn→¯ws ongly in XT∗.Le usp o e ha :
Φ( ¯wn)=wn−→ Φ( ¯w)=ws ongly in XT∗.
As also {wn}n≥1⊆Y(T∗), he e a e subsequences {¯wk}o {¯wn}and {wk}o {wn}such
ha ¯wk�¯w, ¯w∈WT∗
wk�˜w, ˜w∈WT∗
(whe e he abo e con e gences a e as in (42)).
I we conside he sys em e i ied by wkand we pass o he limi as k→+∞,weob-
ain ha ˜wis a solu ion o he p oblem (Ql)wi hsecondmembe G(¯w, ). By uniqueness
˜w=Φ(¯w)=w.The e o e,wk−→ wweakly in WT∗and, by compac ness, wk−→ win
XT∗. Finally, all he sequence con e ges.
5) Exis ence o a ixed poin . As Y(T∗)isacon excompac se o XT∗and Φis
con inuous espec o XT∗ opology, applying he Schaude ’s Theo em we deduce he ex-
is ence o a ixed poin wo Φin Y(T∗). The e o e, wis a s ong solu ion o (Q)in(0,T
∗)
(i T∗ e i ies join ly wi h Dmax he condi ions 1 and 3).
21

Rema k 6.1 Again, in he 3D case we canno bound he nonlinea e ical con ec ion
�¯w3∂z¯w�2
L2(Ω)in unc ion o he s ong egula i y. Conc e ely, we ob ain a bound o he
o m
�¯w�3
D(A)�¯w�V
which canno be bounded using he de ini ion o Y(T). The e o e, we canno con inue wi h
he Fixed Poin easoning.
7 Uniqueness o weak/s ong solu ion (p oo o The-
o em 1.6)
We s a om a weak solu ion uo he sys em (EP) (see de ini ion 1.1), in pa icula u
e i ies he ene gy inequali y:
1
2�u( )�2
L2(Ω)+�
0�νh�∂xu�2
L2(Ω)+ν �∂zu�2
L2(Ω)�ds
≤1
2�u0�2
L2(Ω)+�
0�F,u�Ωds +�
0�τ,u�Γsds, a.e. ∈(0,T).
(43)
Suppose ha he e exis s ano he weak solu ion ¯umo e egula (associa ed o he same
da a u0and F). The idea is o ind unde wha egula i y condi ions, only on ¯u,weha e
ha u≡¯u. Obse e ha , s a ing om he weak a ia ional o mula ion o u(de ini ion
1.1), i is easy o e i y ha ∂ u∈L4/3(0,T;W�), whe e W={ψ∈V;∂zψ∈L4(Ω)}.
In ac , i we wan o ake ϕ=¯uas es unc ion in he weak a ia ional o mula ion o
u, he unique e m ha p esen s p oblems is �Ωu3∂z¯uudΩ. Then, wi h he addi ional
egula i y o he Theo em 1.6 o ¯u( ecall ∂z¯u∈L4(0,T;L4(Ω))) his e m has a sense,
hence one e i ies he ollowing equali y: a.e. ∈(0,T),
�u( ),¯u( )�Ω−�
0�∂ ¯u, u�Ωds +�
0�Ω(νh∂xu∂x¯u+ν ∂zu∂z¯u)dΩds
=�u0�2
L2(Ω)+�
0�F, ¯u�Ωds +�
0�Ω(u∂x¯u+u3∂z¯u)udΩds +�
0�τ,¯u�Γsds.
(44)
Now, we w i e he diffe en ial sys em o (¯u, ¯ps)as:
∂ ¯u+u∂x¯u+u3∂z¯u−νh∂2
xx ¯u−ν ∂2
zz ¯u+∂x¯ps
=F+(u−¯u)∂x¯u+(u3−¯u3)∂z¯u. (45)
Thanks o he addi ional egula i y o ¯u,wecanmul iply(45)byuand in eg a e on
Ω×(0, ):
�
0�∂ ¯u, u�Ωds +�
0�Ω�(u∂x¯u+u3∂z¯u)u+νh∂x¯u∂xu+ν ∂z¯u∂zu�dΩds
=�
0�F,u�Ωds +�
0�Ω�(u−¯u)∂x¯u+(u3−¯u3)∂z¯u�udΩds +�
0�τ,u�Γsds
(46)
22
Adding (44) and (46), he e ms �
0�∂ ¯u, u�Ωds and �
0�Ω(u∂x¯u+u3∂z¯u)udΩds a e
cancelled, ob aining:
�u( ),¯u( )�Ω+�
0�Ω2(νh∂xu∂x¯u+ν ∂zu∂z¯u)dΩds
=�u0�2
L2(Ω)+�
0�F,u +¯u�Ωds +�
0�τ,u +¯u�Γsds
+�
0�Ω�(u−¯u)∂x¯u+(u3−¯u3)∂z¯u�udΩds a.e. ∈(0,T).
(47)
Finally, we mul iply (45) by ¯uand in eg a e on Ω×(0, ), ob aining he ene gy equali y:
1
2�¯u( )�2
L2(Ω)+�
0�νh�∂x¯u�2
L2(Ω)+ν �∂z¯u�2
L2(Ω)�ds
=1
2�u0�2
L2(Ω)+�
0�F, ¯u�Ωds +�
0�τ,¯u�Γsds
+�
0�Ω�(u−¯u)∂x¯u+(u3−¯u3)∂z¯u�¯udΩds,
(48)
whe e he las e m on he igh hand o (48) anishes (by he ee di e gence condi ion).
Then, doing (43) + (48) −(47), we ob ain: a.e. ∈(0,T),
1
2�u( )−¯u( )�2
L2(Ω)+�
0�u(s)−¯u(s)�2
Vds
≤−�
0�Ω�(u−¯u)∂x¯u+(u3−¯u3)∂z¯u�udΩds
=−�
0�Ω�(u−¯u)∂x¯u+(u3−¯u3)∂z¯u�(u−¯u)dΩds
=−�
0�Ω|u−¯u|2∂x¯udΩds −�
0�Ω(u3−¯u3)∂z¯u(u−¯u)dΩds ≡I1+I2
(49)
We es ima e he Ii- e ms (using lemmas o Sec ion 2):
I1≤�
0�∂x¯u�L2(Ω)�(u−¯u)�2
L4(Ω)ds
≤�
0�∂x¯u�L2(Ω)�(u−¯u)�L2(Ω)�∇(u−¯u)�L2(Ω)ds
≤1
4�
0�(u−¯u)�2
Vds +C�
0�(u−¯u)�2
L2(Ω)�∂x¯u�2
L2(Ω)ds
23
I2≤�
0�u−¯u�L4(Ω)�∂z¯u�L4(Ω)�u3−¯u3�L2(Ω)ds
≤Dmax �
0�∇(u−¯u)�1/2
L2(Ω)�u(s)−¯u�1/2
L2(Ω)�∂z¯u�L4(Ω)�∂x(u3−¯u3)�L2(Ω)ds
≤CD
max �
0�u−¯u�3/2
V�u−¯u�1/2
L2(Ω)�∂z¯u�L4(Ω)ds
≤1
4�
0�(u−¯u)�2
L2(Ω)ds +CD
4
max �
0�∂z¯u�4
L4(Ω)�u−¯u�2
L2(Ω)ds
Hence, he inequali y (49) becomes: a.e. ∈(0,T),
�u( )−¯u( )�2
L2(Ω)+�
0�u(s)−¯u(s)�2
Vds
≤C�
0��∂x¯u(s)�2
L2(Ω)+D4
max�∂z¯u(s)�4
L4(Ω)��u(s)−¯u(s)�2
L2(Ω)ds
(50)
Then, om G onwall lemma, we can conclude he uniqueness.
Rema k 7.1 In he 3D case, he bound ob aining o I2is
1
4�
0��u(s)−�
¯u(s)�2
Vds +C�
0�∂z�
¯u(s)�8
L4(Ω)��u(s)−�
¯u(s)�2
L2(Ω)ds.
Now, o ob ain uniqueness, we ha e o impose in �
¯u he ollowing addi ional egula i y
∂z�
¯u∈L8(0,T;L4(Ω)2),
which i is no a consequence o he s ong egula i y.
Re e ences
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[3] R. Lewandowski, Analyse Ma h´ema ique e Oc´eanog aphie, Masson, 1997.
[4] J. Pedlosky, Geophysical luid dynamics, Sp inge -Ve lag, 1987.
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Hyd os a ic App oxima ion, M2AN-Mod. Ma h. Ana. Nume., Vol. 7, 1992, 855-865.
[6] P. Az´e ad & F. Guill´en, ´
Equa ions de Na ie -S okes en bassin peu p o ond: l’app oxima ion hyd o-
s a ique, C. R. Acad. Sci. Pa is, .329, S´e ie I, 1999, 961-966.
[7] T. Chac´on & F. Guill´en, An in insic analysis o exis ence o solu ions o he hyd os a ic app oxi-
ma ion o Na ie -S okes equa ions, submi ed.
[8] F. Guill´en & M. V. Redondo, Con e gencia de algunos esquemas num´e icos hacia el modelo e olu i o
de Ecuaciones P imi i as, Ac as XVI CEDYA, VI CMA, Uni e si y o Las Palmas de G an Cana ia
1999, 1165-1172.
24
[9] F. Guill´en & M. V. Redondo, wo k in p epa a ion.
[10] M. Ziane, Regula i y Resul s o S okes Type Sys ems. Applicable Analysis, Vol. 58, 1995, 263-292.
[11] L. Ni enbe g, On Ellip ic Pa ial Diffe en ial Equa ions, Ann. Scuo. No m. Sup. Pisa, 13 (3), 1959,
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[12] R. A. Adams Sobole spaces, Academic P ess, New Yo k, 1975.
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25