On the uniqueness and regularity of the Primitive Equations imposing additional anisotropic regularity

On he uniqueness and egula i y o he P imi i e Equa ions imposing addi ional aniso opic egula i y. F. Guill´en-Gonz´alez♣∗† , M.A. Rod ´ıguez-Bellido♠ ♣Dp o. de Ecuaciones Di e enciales y An´alisis Num´e ico, Uni e sidad de Se illa, Ap do. 1160, 41080 Se illa, Spain. e-mail: guillen@nume .us.es ♠Dp o. de Ma em´a ica Aplicada I, E. T. S. de A qui ec u a, Uni e sidad de Se illa, A da. Reina Me cedes, 2, 41012 Se illa, Spain. e-mail: [email p o ec ed] Abs ac In his no e, we p o e ha gi en ua weak solu ion o he P imi i e Equa ions, impos- ing an addi ional condi ion on he e ical de i a i e o he eloci y u(conc e ely ∂zu∈ L∞(0,T;L2(Ω)) ∩L2(0,T;H1(Ω))), hen wo diffe en esul s hold; namely, uniqueness o weak solu ion (any weak solu ion associa ed o he same da a ha umus coincide wi h u) and global in ime s ong egula i y o u(wi hou “smallness assump ions” on he da a). Bo h esul s a e p o ed when ei he Di ichle o Robin ype condi ions on he bo om a e conside ed. In he las case, a domain wi h a s ic ly bounded om below dep h has o be imposed, e en o he uniqueness esul . Key wo ds: Weak-s ong uniqueness, P imi i e Equa ions, aniso opic es ima es, s ong solu- ion 1 In oduc ion The P imi i e Equa ions a e ela ed wi h a g ea a ie y o geophysical luids [9, 10, 12]. This sys em can be deduced asymp o ically om he Na ie -S okes equa ions wi h aniso opic (eddy) iscosi y, when he aspec a io (quo ien be ween e ical and ho izon al cha ac e is ic dimensions) ends o ze o [1, 2, 3]. The 3D sys em can be w i en as ollows: o ind u: (0,T)×Ω→R2, he ho izon al eloci y ield, and ps:(0,T)×S→R, a su ace po en ial unc ion (in ol ing he p essu e), e i ying: (PE)                  ∂ u−νH∆Hu−νz∂2 zzu+αu⊥+ +(u·∇H)u+u3∂zu+∇Hps=Fin (0,T)×Ω, ∇H·�u�=0 in(0,T)×S, u| =0 =u0in Ω, νz∂zu|Γs=Υ,u|Γl=0in (0,T), ei he u|Γb=0o ((νH∇Hu,ν z∂zu)·n+βu)|Γb=0in (0,T), whe e u3is he e ical eloci y, ha becomes now a diagnos ic a iable, depending on he ho izon al eloci y uas ollows: u3(x,z)=�0 z∇H·u(x,s)ds. (1) We conside he domain Ω={(x, y, z)=(x,z)∈R3/x∈S, −h(x)<z<0},wi hS⊂R2a bounded open se ( he su ace) and h:S→Ra non-nega i e con inuous unc ion ( he dep h). ∗Co esponding au ho †The i s au ho has been pa ially inanced by he p ojec BFM2000-1317. 1 2 THE MAIN RESULTS. 2 I s bounda y is decomposed as ∂Ω=Γ b∪Γl∪Γswhe e Γs={(x,0) : x∈S}is he su ace, Γl={(x,z)∈R3:x∈∂S, −h(x)<z<0}a e he side-walls and Γb={(x,z)∈R3:x∈ S, z =−h(x)}is he bo om. We ha e deno ed by �u�( ;x)=�0 −h(x)u( ;x,z)dz he e ical in eg a ion o u. The ho - izon al ope a o s ∆Hand ∇H ep esen ∂2 xx +∂2 yy and (∂x,∂ y) espec i ely. The cons an s νH,ν z>0 a e he iscosi y coefficien s and nis he ou wa d no mal ec o on he bo om. The ex e nal o ces a e da a deno ed by F:(0,T)×Ω→R2, and αu⊥=α(−u2,u 1) models he Co iolis o ces, wi h α∈Rdepending on he la i ude. We conside ei he homogeneous Di ichle o Robin ype bounda y condi ions on he bo om (wi h β:S→Ra non-nega i e da a unc ion depending on he ugosi y o he bo om) and Neumann bounda y condi ions on he su ace, whe e Υ:(0,T)×S→R2is a da a unc ion depending on he wind o ce. The Neumann condi ions on he bo om a e also conside ed aking β= 0. No ice ha P imi i e Equa ions a e a ian s o he Na ie -S okes equa ions. Now, he p essu e ield depends only on x(bu no on z). Howe e , he explici o m o u3gi en in (1) implies ha he sys em is no longe pa abolic espec o (u,u 3) and he egula i y o u3and ∇H·ua e compa able, hence he nonlinea e m co esponding o he e ical con ec ion u3∂zu is less egula ha in he Na ie -S okes case. The exis ence o weak solu ion o (PE) we e gi en in [10, 9]. The exis ence o local in ime s ong solu ion (o global o small enough da a) is p o ed in [7] o he 2D case (whe e Sis a eal in e al) and in [6] o he 3D case, using s ong egula i y esul s o he s a iona y linea case gi en in [14]. On he o he hand, some esul s o weak/s ong uniqueness we e gi en in [7, 6], always imposing addi ional egula i y hypo hesis o e he ho izon al and e ical de i a i es o u. In his wo k, we weaken hese addi ional hypo hesis ound in [7, 6] supposing only addi- ional egula i y o e he e ical de i a i e ∂zu(a oiding he addi ional egula i y o e ∇Hu). Mo eo e , we will also p o e ha his same addi ional egula i y implies global s ong egula i y when he da a a e mo e egula bu wi hou smallness assump ions. We hink ha he aniso opy be ween ho izon al and e ical scales could p oduce aniso opic egula i y o he solu ion. Indeed, his occu s in he 2D case; exis ence (and uniqueness) o weak solu ion u o he 2D model such ha ∂zuhas also weak egula i y, i.e. ∂zu∈L∞(0,T;L2(Ω)) ∩ L2(0,T;H1(Ω)), is p o ed in [4] o Robin bounda y condi ions on he bo om and in [5] o Di ichle condi ions. In his line, he exis ence o weak solu ion o (PE) wi h only weak egula i y o ∂zu(e en local in ime o global o small enough da a) is an in e es ing open p oblem, ha we a e going o analyse in a u u e wo k. 2 The main esul s. Basically, uis a weak solu ion o (PE)in(0,T), i u∈L2(0,T;H1(Ω))2∩L∞(0,T;L2(Ω)2) and e i ies he es ic ion ∇H·�u�= 0, he Di ichle condi ions in he ace sense and he momen um equa ions join ly wi h he Neumann and Robin condi ions in a a ia ional sense ([10, 6]). Mo eo e , i u∈L∞(0,T;H1(Ω)2)∩L2(0,T;H2(Ω)2) and ∂ u∈L2(0,T;L2(Ω)2), u is a s ong solu ion o (PE)in(0,T). Theo em 2.1 (Uniqueness o solu ion) Le ube a weak solu ion o (PE)in (0,T). I he e exis s a weak solu ion ¯ uo (PE)in (0,T) e i ying he addi ional egula i y: ∂z¯ u∈L∞(0,T;L2(Ω)2)∩L2(0,T;H1(Ω)2),(2) hen bo h solu ion coincided in [0,T). When Robin condi ions a e conside ed on he bo om, he assump ion h≥hmin >0in Shas o be imposed. Rema k 2.1 No ice ha we ha e educed he hypo heses on ¯ uimposed in [6] o ge ing unique- ness o weak/s ong solu ion. Conc e ely, in [6] we conside ed ∇H¯ u∈L2(0,T;L∞ zL2 x)and ∂z¯ u∈L∞(0,T;L2(Ω)2)∩L2(0,T;H1(Ω)2) 3 SOME AUXILIARY ANISOTROPIC ESTIMATES. 3 (see he nex sec ion o he de ini ion o he aniso opic space L∞ zL2 x). The e o e, we ha e emo ed he hypo hesis o ∇H¯ u. Theo em 2.2 (Global s ong egula i y) Le S⊆R2wi h ∂S∈C3and h∈C3(¯ S)wi h h≥hmin >0in ¯ S.Suppose ha u0∈H1(Ω)wi h ∇H·�u0�=0(and u0|Γb=0in he case o Di ichle condi ions on he bo om), F∈L2(0,T;L2(Ω)2)and Υ∈L2(0,T;H1/2+ε 0(Γs)2)∩ L∞(0,T;H−1/2(Γs)2) o some ε>0such ha ∂ Υ∈L2(0,T;H−3/2(Γs)2)wi h Υ(0) ∈ H−1/2(Γs)2.I ∂zu e i ies he addi ional egula i y o (2), hen uis a s ong solu ion o (PE) in (0,T). 3 Some auxilia y aniso opic es ima es. Le us o in oduce he aniso opic Lp,q spaces o any exponen s p, q ∈[1,+∞]. I will said say ha a unc ion belongs o Lq zLp x(Ω) i : (·,z)∈Lp(Sz) and � (·,z)�Lp(Sz)∈Lq(−hmax,0), whe e hmax = max S hand Sz={x∈S:(x,z)∈Ω} o each z∈(−hmax,0). We will use he ollowing h ee aniso opic esul s, he i s one has al eady been conside ed and p o ed in [6], and he o he ones a e new in his wo k (see Appendix o he p oo s). Lemma 3.1 a) Le ∈H1(Ω). Then ∈L2 zL4 x(Ω)and e i ies: � �L2 zL4 x≤C� �1/2 L2(Ω)� �1/2 H1(Ω)(3) b) Le ∈L2(Ω)2such ha ∇H· ∈L2(Ω),and 3de ined as in (1). Then, 3∈L∞ zL2 x(Ω) and � 3�L∞ zL2 x≤h1/2 max�∇H· �L2(Ω)(4) Lemma 3.2 Le ∈H1(Ω)such ha ∂z ∈H1(Ω)and |Γb=0. Then ∈L∞ zL4 x(Ω)and � �L∞ zL4 x≤C� �1/4 L2(Ω)� �1/4 H1(Ω)�∂z �1/4 L2(Ω)�∂z �1/4 H1(Ω)(5) Lemma 3.3 Assume h≥hmin >0in S. a) Le ∈L2(Ω)such ha ∂z ∈L2(Ω). Then ∈L∞ zL2 x(Ω)and hmin� �2 L∞ zL2 x≤� �2 L2(Ω)+2� �L2(Ω)�∂z �L2(Ω).(6) b) Le ∈H1(Ω)such ha ∂z ∈H1(Ω). Then ∈L∞ zL4 x(Ω)and h1/2 min� �L∞ zL4 x≤C� �1/4 L2(Ω)� �1/4 H1(Ω)�� �1/4 L2(Ω)� �1/4 H1(Ω)+�∂z �1/4 L2(Ω)�∂z �1/4 H1(Ω)�(7) 4 P oo o he main esul s in he Di ichle case. P oo o Theo em 2.1:We ollow he di ec me hod o p o e uniqueness, used o ins ance in [11] o he 3D Na ie -S okes equa ions. Deno ing =u−¯ uand 3=u3−¯u3, one has [6] (see [13] o mo e de ails): 1 2� ( )�2 L2(Ω)+ν� 0� (s)�2 H1(Ω)ds ≤− � 0�Ω [ ·∇H¯ u+ 3∂z¯ u]· dΩds := I1+I2, (8) 4 PROOF OF THE MAIN RESULTS IN THE DIRICHLET CASE. 4 whe e ν=min{νH,ν z}. Wi h espec o he p oo o uniqueness done in [6], we will change he ea men o he e m I1. Now, in eg a ing by pa s and applying (3) and (5) one has (he e, Di ichle condi ion on Γbis used) I1=� 0�Ω [( ·∇H) ·¯ u+(∇H· ) ·¯ u]dΩds ≤C� 0�Ω| ||∇H ||¯ u|dΩds ≤C� 0� �L2 zL4 x�∇H �L2(Ω)�¯ u�L∞ zL4 xds ≤C� 0� �1/2 L2(Ω)�∇H �3/2 L2(Ω)�¯ u�1/4 L2(Ω)�¯ u�1/4 H1(Ω)�∂z¯ u�1/4 L2(Ω)�∂z¯ u�1/4 H1(Ω)ds. Using he Young inequali y o he indexes (4/3,4), we ha e: I1≤ε� 0� �2 H1(Ω)ds +Cε� 0�¯ u�L2(Ω)�¯ u�H1(Ω)�∂z¯ u�L2(Ω)�∂z¯ u�H1(Ω)� �2 L2(Ω)ds. We bound I2as in [6] (using (3) and (4)), ob aining I2=� 0�Ω 3∂z¯ u· dΩds ≤� 0� 3�L∞ zL2 x�∂z¯ u�L2 zL4 x� �L2 zL4 xds ≤ε� 0� �2 H1(Ω)ds +Cε� 0�∂z¯ u�2 L2(Ω)�∂z¯ u�2 H1(Ω)� �2 L2(Ω)ds Using he p e ious bounds in (8), one has: � ( )�2 L2(Ω)+ν� 0� (s)�2 H1(Ω)ds ≤C� 0 a(s)� (s)�2 L2(Ω)ds (9) whe e a=�¯ u�L2(Ω)�¯ u�H1(Ω)�∂z¯ u�L2(Ω)�∂z¯ u�H1(Ω)+�∂z¯ u�2 L2(Ω)�∂z¯ u�2 H1(Ω).Sincea∈L1(0,T) ( hanks o he egula i y hypo hesis o ¯ uand ∂z¯ u), we a e in he hypo hesis o G onwall Lemma, hence he uniqueness is deduced. P oo o Theo em 2.2:Fi s , we li he bounda y da a Υusing an adequa e (s ong) solu ion (e,q s) o a s a iona y hyd os a ic S okes sys em. Obse e ha hypo hesis ∂ Υ∈ L2(0,T;H−3/2(Γs)2) implies ha ∂ e∈L2(0,T;L2(Ω)2) (see [8]). Then, we eason o e he homogeneous a iables ( , 3,π s)=(u−e,u 3−e3,p s−qs), e i ying:        ∂ −νH∆H −νz∂2 zz +u·∇H + 3∂zu+∇Hπs=Gin (0,T)×Ω, ∇H·� �=0in (0,T)×S, | =0 = 0in Ω, νz∂z |Γs=0, |Γb=0, |Γl=0in (0,T), (10) whe e 0=u0−e(0) and G=F−∂ e−u·∇He+e3∂zu.Thanks o he addi ional egula i y o ∂zuand he s ong egula i y o e, one has ha G∈L2(0,T;L2(Ω)2). Indeed, in he con ec i e e ms, we ha e p oduc s o u(and e3) belonging o L4 L∞ zL4 x,by∂zu(and ∇He) belonging o L4 L2 zL4 x(acco dingly Lemmas 3.1 and 3.2). Using he same a gumen han in [6], we apply a Gale kin me hod, using A mas es unc ions, whe e Ais he hyd os a ic S okes ope a o and mi s eigen unc ions. In o de o bound he con ec ion e ms, we use he inequali ies (3) and (5) as ollows ( o simplici y, we d op he m-indexes): �Ω u·∇H ·A dΩ≤�u�L∞ zL4 x�∇H �L2 zL4 x�A �L2(Ω) ≤C�u�1/4 L2(Ω)�u�1/4 H1(Ω)�∂zu�1/4 L2(Ω)�∂zu�1/4 H1(Ω)�∇H �1/2 L2(Ω)�A �3/2 L2(Ω) ≤ε�A �2 L2(Ω)+a( )�∇H �2 L2(Ω), (11) 5 PROOFS FOR ROBIN CONDITIONS ON THE BOTTOM 5 whe e a( )=C�u�L∞(0,T;L2(Ω))�u( )�H1(Ω)�∂zu�L∞(0,T;L2(Ω))�∂zu( )�H1(Ω), and �Ω 3∂zu·A dΩ≤� 3�L∞ zL4 x�∂zu�L2 zL4 x�A �L2(Ω) ≤C�∇H· �1/2 L2(Ω)�∂zu�1/2 L2(Ω)�∂zu�1/2 H1(Ω)�A �3/2 L2(Ω)≤ε�A �2 L2(Ω)+b( )�∇H �2 L2(Ω), (12) whe e b( )=C�∂zu�2 L∞(0,T;L2(Ω))�∂zu( )�2 H1(Ω). The addi ional egula i y o ∂zugua an ees ha a,b∈L1(0,T). Then, 1 2 d d �∇ �2 L2(Ω)+�A �2 L2(Ω)≤(a( )+b( ))�∇ �2 L2(Ω)+�G( )�2 L2(Ω). G onwall’s Lemma allows us o conclude ha ∈L∞(0,T;H1(Ω)2)∩L2(0,T;H2(Ω)2) and, hanks o he s ong egula i y o e, one has he same egula i y o u. Regula i y o ∂ uis ollowed by a s anda d way. 5 P oo s o Robin condi ions on he bo om P oo o Theo em 2.1:In his case, one a i es a (8) wi h he supplemen a y non-nega i e e m � 0�Γbβ| |2dσds in he le hand-side. Now, i is necessa y o change he bound o he e m I1in (8). Indeed, using di ec ly (8) (wi hou by pa s in eg a ion), we ge : I1≤� 0� �2 L2 zL4 x�∇H¯ u�L∞ zL2 x≤� 0� �L2(Ω)� �H1(Ω)�∇H¯ u�L∞ zL2 x(13) In o de o bound ∇H¯ u, we canno use he ollowing inequali y (p o ed in [6]) �∇H¯ u�L∞ zL2 x≤C�∇H¯ u�1/2 L2(Ω)�∇H¯ u�1/2 H1(Ω) because ∇H¯ u�∈ H1(Ω). Ins ead o his, we will use Lemma 3.3 a). Indeed, applying (6) o =∇H¯ uin (13), we a i e a I1≤ε� 0� �2 H1(Ω)ds +Cε hmin � 0��∇H¯ u�2 L2(Ω)+�∇H¯ u�L2(Ω)�∂z(∇H¯ u)�L2(Ω)�� �2 L2(Ω)ds Adding his exp ession o he es ima e o I2, one can also p o e uniqueness o solu ion. P oo o Theo em 2.2:The main diffe ence in he p oo is he use o inequali y (7) ins ead o (5) o Lemma 3.2 in o de o bound he con ec i e e ms. A Appendix P oo o Lemma 3.2.We will use he ollowing inequali y, p o ed in [6]; o any p, q ∈[1,+∞] wi h q>p, � �Lq xLp z≤� �Lp zLq x(14) Wi h he same a gumen s one can change he in eg a ion o de , i.e., � �Lq zLp x≤C� �Lp xLq z(15) Le in he hypo hesis o Lemma 3.2. Since |Γb= 0, we ha e (x,z)4=�2�z −h(x) (x,s)∂z (x,s)ds�2 ≤4� (x,·)�2 L2 z�∂z (x,·)�2 L2 z REFERENCES 6 Consequen ly, � (x,·)�L∞ z≤√2� (x,·)�1/2 L2 z�∂z (x,·)�1/2 L2 z.(16) Taking L4 x-no m, � �L4 xL∞ z≤√2� �1/2 L4 xL2 z�∂z �1/2 L4 xL2 z. Now using (14) and (3), we ob ain: � �L4 xL∞ z≤√2� �1/2 L2 zL4 x�∂z �1/2 L2 zL4 x≤C�u�1/4 L2(Ω)� �1/4 H1(Ω)�∂z �1/4 L2(Ω)�∂z �1/4 H1(Ω). To inish, i suffices o conside (15) in he le hand side, ge ing (5). P oo o Lemma 3.3.Fo any unc ion g=g(z)de inedinz∈(−h(x),0) wi h x∈S, we w i e g2(z)=g2(z�)+2�z z�g(s)∂zg(s)ds. In eg a ing in z�∈(−h(x),0), h(x)g2(z)≤ �g�2 L2 z+2�g�L2 z�∂zg�L2 z, hush(x)1/2�g�L∞ z≤�g�L2 z+√2�g�1/2 L2 z�∂zg�1/2 L2 z. Applying he p e ious inequali y o g= (x,·) and bounding om below h(x)≥hmin, we ge h1/2 min� (x,·)�L∞ z≤� (x,·)�L2 z+√2� (x,·)�1/2 L2 z�∂z (x,·)�1/2 L2 z.(17) In o de o p o e es ima e (7), we ollow he same a gumen ha in he p oo o Lemma 3.2, eplacing (16) by (17) and adap ing he calculus he ein. Finally, (6) ollows di ec ly aking L2 x-no m in (17). Re e ences [1] P. Az´e ad & F. Guill´en-Gonz´alez, Ma hema ical jus i ica ion o he hyd os a ic app oxima ion in he P imi i e Equa ions o Geophysical luid dynamics. Siam J. Ma h. Anal. ,Vol. 33, No. 4, 847-859 (2001). [2] 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, 855-865 (1992). [3] D. B esch, F. Guill´en-Gonz´alez, N. Masmoudi & M. A. Rod ´ıguez-Bellido, Asymp o ic de i a ion o a Na ie condi ion o he P imi i e Equa ions, o appea in Asymp o ic Analysis. [4] D. B esch, F. Guill´en-Gonz´alez, N. Masmoudi & M. A. 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Lions, Ma hema ical Topics in Fluids Mechanics, Vol.1, Incomp essible Models, Uni . Pa is- Dauphine & ´ Ecole Poly echnique, Ox o d Uni . P ess. Inc., NY, (1996). [12] J. Pedlosky, Geophysical luid dynamics, Sp inge -Ve lag, (1987). [13] M. A. Rod ´ıguez Bellido, Ph. Thesis, Uni e si y o Se ille (2001). [14] M. Ziane, Regula i y Resul s o S okes Type Sys ems. Applicable Analysis, Vol. 58, 263-292 (1995).