On the strong solutions of the primitive equations in 2D domains

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)                      ∂ �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)                ∂ 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)           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 )         −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)                      ∂ � −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                  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)                      ∂ 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:                      ∂ 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 [1] J. L. Lions, R. Teman, S. Wang, New o mula ion o he p imi i e equa ions o he a mosphe e and applica ions. Nonlinea i y, 5, 1992, 237-288. [2] J. L. Lions, R. Teman, S. Wang, On he equa ions o he la ge scale Ocean. Nonlinea i y, 5, 1992, 1007-1053. [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. [5] O. Besson & M. R. Laydi, Some Es ima es o he Aniso opic Na ie -S okes Equa ions and o he 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, 115-162. [12] R. A. Adams Sobole spaces, Academic P ess, New Yo k, 1975. [13] J. L. Lions, Quelques M´e hodes de R´esolu ion des P obl`emes aux Limi es Non Lin´eai es, Dunod, Pa is, 1969. 25