Uniqueness of solution for the 2D Primitive Equations with friction condition on the bottom

Uniqueness o solu ion o he 2D P imi i e Equa ions wi h ic ion condi ion on he bo om∗ D. B esch† , F. Guill´en-Gonz´alez‡ , N. Masmoudi§ , M. A. Rod ´ıguez-Bellido¶ Monog a ´ıas del Semin. Ma em. Ga c´ıa de Galdeano. 27: 135–143, (2003). Abs ac Uniqueness o solu ion o he P imi i e Equa ions wi h Di ichle condi ions on he bo om is an open p oblem e en in 2D domains. In his wo k we p o e a esul o addi ional egula i y o a weak solu ion o he P imi i e Equa ions when we eplace Di ichle bounda y condi ions by ic ion condi ions. This allows o ob ain uniqueness o weak solu ion global in ime, o such a sys em [3]. Indeed, we show weak egula i y o he e ical de i a i e o he solu ion, ∂z o all ime. This is because his de i a i e e i ies a linea pde o con ec ion-di usion ype wi h con ec ion eloci y , and he p essu e belongs o a L2-space in ime wi h alues in a weigh ed space. Keywo ds: Bounda y condi ions o ype Na ie , 2D P imi i e Equa ions, uniqueness AMS Classi ica ion: 35Q30, 35B40, 76D05 1 In oduc ion and mo i a ion. P imi i e Equa ions a e one o he models used o o ecas he luid eloci y and p essu e in he ocean. Such equa ions a e ob ained om he dimensionless Na ie -S okes equa ions, le ing he aspec a io (quo ien be ween e ical dimension and ho izon al dimensions) go o ze o. The i s esul s abou exis ence o solu ion (weak, in he sense o he Na ie - S okes equa ions) a e p o ed o bounda y condi ions o Di ichle ype on he bo om o he domain and wi h wind ac ion on he su ace, in he wo ks by Lions-Temam-Wang, ∗The second and ou h au ho s ha e been inanced by he C.I.C.Y.T p ojec MAR98-0486. †Labo a oi e de Ma h´ema iques Appliqu´ees CNRS 6620 Uni . Blaise Pascal, 63177 Aubi`e e (FRANCE), b esc[email p o ec ed]cle mon . ‡Dp o. Ecuaciones Di e enciales y An´alisis Num´e ico, Fac. Ma em´a icas, Uni . Se illa, C/ Ta ia, s/n - 41012 Se illa (SPAIN), guillen@nume .us.es §Cou an Ins i u e o Ma hema ical Science, New Yo k Uni e si y, [email p o ec ed]yu.edu ¶Dp o. Ma em´a ica Aplicada I, E. T. S. de A qui ec u a, Uni . Se illa, A da. Reina Me cedes, s/n - 41012 Se illa (SPAIN), [email p o ec ed] 135 [5, 6], o domains wi h e ical walls and in he wo k o Az´e ad-Guill´en, [1], o domains wi hou e ical walls. Howe e , uniqueness o solu ion emained as an open p oblem due o he necessi y o a mo e egula solu ion. In he case o e ical sidewalls, he au ho s p o ed in [4] he exis ence o a mo e egula solu ion, global in ime o small da a o local in ime o any da a. In hese cases, uniqueness o solu ion is gua an eed. Bu , om a physical poin o iew, homogeneous Di ichle bounda y condi ions (on he bo om) a e only jus i ied when conside ing a molecula iscosi y luid. In many geophysical luids, he ole o his iscosi y is negligible, being mo e ele an he iscosi y due o u bulen e ec s. I seems hen logical o use Na ie bounda y condi ions o he P imi i e Equa ions. Mo eo e , hey p e en he appea ance o a bounda y laye phenomena on he bo om. The au ho s ob ained he P imi i e Equa ions model wi h Na ie ype bounda y con- di ions om he Na ie -S okes equa ions in [2]. He e, we will ocus on he uniqueness p oblem in he 2D case, see also [3]. We will p esen wha we conside is he i s esul o uniqueness o weak solu ion o he 2D P imi i e Equa ions. 2 The model. The domain conside ed is de ined by: Ω = {(x, z)∈R2/ x ∈S, −h(x)<z<0}, whe e S(ocean su ace) is an open in e al and h:¯ S→R+is a nonnega i e con inuous unc ion de ined on ¯ S ha anishes on ∂S. The bounda y o he domain is ∂Ω = ¯ Γb∪Γs, whe e he bo om is Γb={(x, z)∈R2:x∈S, z =−h(x)}and he su ace Γs={(x, 0) : x∈S}. The e o e, he luid eloci y ( , w) and he p essu e psa is y he ollowing equa ions: (PE)                                    ∂ + ∂x +w ∂z −νh∂2 xx −ν ∂2 zz +∂xp= in (0, T)×Ω, ∂zp= 0, w( , x, z) = Z0 z ∂x ( , x, s)ds, h i= 0 in (0, T)×S, ν ∂z =α| ai |( ai − ) on (0, T)×Γs, ν ∂z =β(x) on (0, T)×Γb, | =0 = 0in Ω, whe e h i( ;x) = Z0 −h(x) ( ;x, z)dz, ai is he ho izon al eloci y o he wind a he su ace, 0 he ho izon al ini ial eloci y, (νh, ν ) he aniso opic u bulen iscosi y, α∈R 136 a posi i e cons an and β=β(x) a posi i e unc ion de ined on S. Rema k 2.1 The model o P imi i e Equa ions wi h Na ie condi ions deduced in [2] was o med by (P E)1,∂zp= 0 and ∂x +∂zw= 0 in (0, T)×Ω,ν ∂z =α( ai − )and w= 0 on (0, T)×Γs,ν ∂z =β and ( , w)·n= 0 on (0, T)×Γband | =0 = 0in Ω. The equa ion ∂x +∂zw= 0 and bounda y condi ions o wimply ha w( ;x, z) = R0 z∂x ( ;x, s)ds and ∂xh i= 0. Finally, as h iis a 1-dimensional unc ion, he hypo hesis h i= 0 on (0, T )×∂S implies ha h i= 0 on (0, T )×S. 3 De ini ions and p e ious esul s. Fo he eloci y , we in oduce he ollowing spaces: V={ϕ∈C∞ s(Ω) : hϕi= 0 in S,} whe e C∞ s(Ω) is he space o C∞- unc ions ha anish in a neighbou ghood o ∂Γs. We will deno e by Hand Vi s closu es in he L2(Ω) and H1(Ω)−no ms espec i ely. De ini ion 3.1 (Weak solu ion) We say ha is a weak solu ion o (PE)in (0, T ) i : ∈L∞(0, T;H)∩L2(0, T;V), sa is ies he a ia ional o mula ion: ∀ϕ∈C1([0, T ]; V)wi h ϕ(T) = 0,                        −ZT 0ZΩ (∂ ϕ+ ∂xϕ+w∂zϕ) +ZT 0ZΩ (νh∂x ∂xϕ+ν ∂z ∂zϕ) +ZT 0ZS δ(x) |Γbϕ|Γb+ZT 0ZS α| ai |( |Γs− ai )ϕ|Γs =ZΩ 0ϕ(0) + ZT 0ZΩ ϕ +νhZT 0ZS |Γb∂x[ϕ|Γbh0(x)], wi h w=R0 z∂x and sa is ies he ollowing ene gy inequali y            1 2k ( )k2 L2(Ω) +νhZ 0k∂x (s)k2 L2(Ω) +ν Z 0k∂z (s)k2 L2(Ω) +Z 0ZS γ(x)| |Γb|2+1 2Z 0ZS α| ai || |Γs|2≤1 2k 0k2 L2(Ω) +1 2Z 0ZS α| ai |3 wi h δ(x) = β(x)1 + νh ν |h0(x)|2and γ(x) = δ(x)−νh 2h00(x). Rema k 3.1 In o de o ensu e ha he sys em is dissipa i e (necessa y p ope y om a physical poin o iew), we assume ha γ(x)≥0. 137 Rema k 3.2 No ice ha he bounda y condi ion on he bo om is no s anda d because ∂z is no he Neumann condi ion espec o he laplacian ope a o . This ac p oduces he e m νhRT 0RS |Γb∂x[ϕ|Γbh0(x)] in he a ia ional o mula ion. In o he wo ds, gi ing a weak solu ion , we can ge an associa e p essu e p h ough he De Rham Lemma (as a Lag ange mul iplie ) in such a way ha ( , w, p) e i y he di e en ial p oblem (PE) in he dis ibu ion sense (see [3] o mo e de ails). In pa icula , he ollowing mixed a ia ional o mula ion can be ob ained: ∀ϕ∈C1([0, T]; C∞ s(Ω)), wi h ϕ(T) = 0, he e exis s a unc ion ψsmoo h enough, sa is ying (ϕ, ψ)·n|∂Ω= 0 such ha : −ZT 0ZΩ (∂ ϕ+ ∂xϕ+w∂zϕ) +ZT 0ZΩ (νh∂x ∂xϕ+ν ∂z ∂zϕ) +ZT 0ZS δ(x) |Γbϕ|Γb+ZT 0ZS α| ai |( |Γs− ai )ϕ|Γs =ZΩ 0ϕ(0) + ZT 0ZΩ ϕ +νhZT 0ZS |Γs∂x(ϕ|Γbh0) + ZT 0ZΩ p∇·(ϕ, ψ). (1) Theo em 3.2 (See [2] o a p oo o his esul .) Suppose ha h∈H2(S)wi h |h0|>0 on ∂S,β∈L∞(S), ∈L2(0, T ;L2(Ω)), ai ∈L3(0, T ;L3(S)), 0∈Hand γ(x)≥0on S. Then, he e exis s a weak solu ion o (P E)in (0, T). De ini ion 3.3 (Weak- o ici y solu ion) We will say ha is a weak- o ici y solu- ion o (P E)in (0, T)i i is a weak solu ion ha also sa is ies he addi ional egula i y: ∂z ∈L∞(0, T;L2(Ω)) ∩L2(0, T;H1(Ω)). Rema k 3.3 ∂z can be seen as he o ici y associa ed o he P imi i e Equa ions. In- deed, i we conside he o ici y o he 2D Na ie -S okes equa ions, ωNS =∂z NS − ∂xwNS, le ing he aspec a io go o ze o we a i e a ∂z . 4 Main esul . Theo em 4.1 (Uniqueness o weak solu ion) Unde he hypo hesis o Theo em 3.2, i we also conside ha β∈H1 0(S), ai ∈L∞(0, T;H1 0(S)),∂ ai ∈L2(0, T;L1(S)),∂z ∈ L2(0, T;H−1(Ω)),∂z 0∈L2(Ω) and he dep h unc ion h e i ies |h0|/h ≤c/dis (x, ∂S), hen he e exis s a unique weak solu ion o (P E). Mo eo e , his solu ion is a weak- o ici y solu ion. Ou line o he p oo : He e, we will explain he main ideas ha we ha e ollowed o p o e Theo em 4.1. Fo a comple e p oo o his esul see [3]. Following he me hod o P. L. Lions, [7], o p o e uniqueness o weak solu ion o he Na ie -S okes equa ions we obse ed ha addi ional egula i y is necessa y o one 138 o he wo solu ions compa ed. Applying he a gumen o (PE), we obse ed ha his egula i y should be ∂z ∈L4(0, T;L4(Ω)). In o de o ob ain mo e egula i y o ∂z , we sea ch o he p oblem e i ied by ∂z . Fi s , we o mally de i e (PE)1 espec o z, ob aining ha ∂z sa is ies in D0((0, T)×Ω): ∂ (∂z ) + ∂x(∂z ) + w ∂z(∂z )−νh∂2 xx(∂z )−ν ∂2 zz(∂z ) = ∂z . Knowing and w, he p e ious equa ion is linea and pa abolic, because he p essu e p has disappea ed, so we could expec weak egula i y o ∂z . To his end, we need o s udy a homogeneous sys em, so we conside he auxilia y unc ion ψ=ν ∂z −φ −e wi h φ( ;x, z) = −α1 + z h(x)| ai ( ;x)|− z h(x)β(x) and e( ;x, z) = α| ai ( ;x)| ai ( ;x)1 + z h(x) auxilia y unc ions such ha ψ|∂Ω= 0. Then, ψ e i ies he p oblem: (P)       ∂ ψ+ ∂xψ+w ∂zψ−ν ∂2 xxψ−ν ∂2 zzψ=Fin (0, T)×Ω, ψ= 0 on (0, T)×∂Ω, ψ| =0 =ν ∂z 0−φ| =0 0−e| =0 in Ω, whe e F=G(φ, , w, e, ) + φ ∂xp. A his poin , we ha e 2 p oblems: ge ing an addi ional egula i y o he p essu e p o ob ain weak egula i y o ψ, and iden i ying ψ+φ +ewi h ν ∂z . Once hese p oblems a e sol ed, hen ∂z ∈L2(0, T ;H1(Ω)) ∩L∞(0, T ;L2(Ω)) and in pa icula belongs o L4(0, T;L4(Ω)), so we will be able o conclude weak uniqueness o (PE). 5 Addi ional egula i y o he p essu e. Thanks o ∂zp= 0, we can iden i y pwi h a unc ion psonly de ined on S,ps(x) = p(x, z), h ough he ela ion: ZΩ p(x, z)ϕ(x, z)dx dz =ZS ps(x)hϕi(x)dx ∀ϕ∈L2(Ω). Theo em 5.1 Assume he hypo hesis o he da a o Theo em 4.1. I ( , p)is a weak solu ion o (P E), we ha e: √h ∂xps∈L2(0, T;H−1(S)). Ou line o he p oo : Fo he equa ions o Na ie -S okes ype, he p essu e egula i y is no mally ob ained om he egula i y o he emaining e ms o he equa ion. The e m 139 ∂ p e en s a L2- egula i y in ime o he p essu e. The ac ha h i= 0 on (0, T )×S implies ha ∂ h i= 0 on (0, T )×S, so in eg a ing (PE)1in zwe y o imp o e he egula i y o he p essu e. In a igo ous o m, his e ical in eg a ion co esponds o ake es unc ions independen om zin he mixed a ia ional o mula ion (1). On he o he hand, as he p essu e pis independen om z, i s in eg a ion on zonly adds a ac o h(x) mul iplying p. Mo eo e , o (ϕ, ψ) any es unc ions in (1), ZΩ p∇·(ϕ, ψ)dΩ = ZS ps∂xhϕidx. Then, we choose ϕ=ζ/√hwi h ζ∈C1 0([0, T]; C∞ 0(S)) as a es unc ion (in pa icula , his space is dense in L2(0, T ;H1 0(S))). Conc e ely, we ha e o gi e sense o he e m ZT 0ZS ps( ;x)∂x(√h ζ)( ;x)dx d . To his aim, we p o e ha he o he s e ms om he mixed a ia ional o mula ion a e well-de ined and bounded in unc ion o he L2(0, T ;H1 0(S))-no m o ζ. Addi ional egula i y equi ed o he da a, hypo hesis |h0|/h ≤c/dis (x, ∂S) join ly wi h Ha dy inequali ies and he ac ha ∂ h i= 0 le inish he p oo . 6 Iden i ica ion o ψ+φ +ewi h ν ∂z . Using a Gale kin me hod, he addi ional egula i y o ple us ob ain weak egula - i y o ψ, so ψ∈L2(0, T ;H1(Ω)) ∩L∞(0, T ;L2(Ω)). To ge ∂z ∈L2(0, T;H1(Ω)) ∩ L∞(0, T;L2(Ω)), we p o e ha ψ+φ +e=ν ∂z . The i s idea o ge his esul was o use he uniqueness o weak solu ion o p oblem (P), bu he p oblem was ha we could no assu e he weak egula i y o ∂z (only ∂z ∈L2(0, T;L2(Ω))). Consequen ly, we looked o a new me hod o ou pu pose: We call a=ψ+φ +eand de ine e ∈L2(0, T;H1 0(Ω)) ∩L∞(0, T;L2(Ω)) such ha ν ∂ze =a in Ω and he i= 0 on S. In ac , we can choose: e (x, z) = −1 ν Z0 z a(x, s)ds +1 ν 1 h(x)Z0 −h(x)Z0 z a(x, s)dsdz. The idea is o ob ain uniqueness o bo h eloci ies and e , and hen ∂z =∂ze ∈ L2(0, T;H1(Ω)) ∩L∞(0, T;L2(Ω)). S a ing om he a ia ional o mula ion o ψ, aking χ=Z0 z η(x, s)ds as es unc- ions, whe e η∈ D(Ω) wi h hηi= 0 and aking in o accoun ha ν ∂ze =α| ai |( ai − ) on Γsand ν ∂ze =β on Γb, 140 we can easily deduce ha e e i ies he ollowing a ia ional o mula ion (g FV ): ∀η∈ C1([0, T]; V),                                    Z 0h∂ e , ηiΩ+Z 0ZΩ ( ∂xe +w ∂ze )η +Z 0ZΩ (νh∂xe ∂xη+ν ∂ze ∂zη) + Z 0ZS α| ai |( |Γs− ai )η|Γs +Z 0ZS δ(x) |Γbη|Γb=Z 0ZΩ η +Z 0ZΩ ∂xe +Z0 z ∂x( ∂ze )(x, s)dsη+ν Z 0ZS |Γb∂x[η|Γbh0(x)] . On he o he hand, we know ha sa is ies he ollowing a ia ional o mula ion (FV ): ∀ϕ∈C1([0, T]; V),                                    h ( ), ϕ( )iΩ−Z 0ZΩ (∂ ϕ+ ∂xϕ+w∂zϕ) +Z 0ZΩ (νh∂x ∂xϕ+ν ∂z ∂zϕ) +Z 0ZS α| ai |( |Γs− ai )ϕ|Γs+Z 0ZS δ(x) |Γbϕ|Γb =ZΩ 0ϕ(0) + νhZ 0ZS |Γb∂x[ϕ|Γbh0(x)] + Z 0ZΩ ϕ, Taking in o accoun he weak egula i y o e and ∂ze and a guing by densi y, we can ake e as a es unc ion in (F V ) and as a es unc ion in (g FV ). Sub ac ing bo h exp essions o he ene gy equali y o e and he ene gy inequali y o , we a i e a ([3]): a. e. ∈(0, T ), 1 2k ( )−e ( )k2 L2(Ω) +Z 0νhk∂x( −e ) (s)k2 L2(Ω) +ν k∂z( −e ) (s)k2 L2(Ω)ds ≤Z 0ZΩ ∂xe +Z0 z ∂x( ∂ze ) (x, s)ds(e − )dΩds +νh 2Z 0ZS|e |Γb− |Γb|2h00(x)dxds ≡I+J. (2) No ice ha i e = , hen I= 0 and J= 0. In eg a ing by pa s espec o z, we 141 ew i e Ias: I=Z 0ZΩ{∂ze ∂x( −e )−∂xe ∂z( −e )}Z0 z ( −e )(x, s)dsdΩds ≤min{νh, ν } 4Z 0k −e k2 H1(Ω)ds +C(νh, ν )Z 0k∂xe k2 L2(Ω) +k∂x(∂ze )k4/3 L2(Ω)k −e k2 L2(Ω)ds. We bound Jusing he T ace and In e pola ion Theo y in Hs(Ω)-spaces wi h s∈R: J≤CZ 0kh00kL2(S)k( −e )|Γbk2 L4(S)ds ≤CZ 0kh00kL2(S)k −e k2 H3/4(Ω)ds ≤CZ 0kh00kL2(S)k −e k1/2 L2(Ω)k −e k3/2 H1(Ω)ds ≤min{νh, ν } 4Z 0k −e k2 H1(Ω)ds +C(νh, ν )Z 0kh00k4 L2(S)k −e k2 L2(Ω)ds Then, (2) becomes: k ( )−e ( )k2 L2(Ω) +Z 0νhk∂x( −e ) (s)k2 L2(Ω) +ν k∂z( −e ) (s)k2 L2(Ω)ds ≤C(νh, ν )Z 0k∂ze kL2(Ω)k∂ze kH1(Ω) +k∂xe k2 L2(Ω) +k∂x(∂ze )k4/3 L2(Ω) +kh00k4 L2(S)k −e k2 L2(Ω)ds. Since ∂ze has weak egula i y, we can use he G onwall Lemma and deduce ha e = . 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 ap- p oxima ion in he P imi i e Equa ions o Geophysical luid dynamics. To appea in Siam J. Ma h. Anal. , Vol. 33, No. 4, 847-859. [2] D. B esch, F. 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