Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics

Transcription

1 Annals of athematics, , Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics By Chongsheng Cao and Edriss S. Titi Abstract In this paper we prove the global existence and uniqueness regularity of strong solutions to the three-dimensional viscous primitive equations, which model large scale ocean and atmosphere dynamics. 1. Introduction Large scale dynamics of oceans and atmosphere is governed by the primitive equations which are derived from the Navier-Stokes equations, with rotation, coupled to thermodynamics and salinity diffusion-transport equations, which account for the buoyancy forces and stratification effects under the Boussinesq approximation. oreover, and due to the shallowness of the oceans and the atmosphere, i.e., the depth of the fluid layer is very small in comparison to the radius of the earth, the vertical large scale motion in the oceans and the atmosphere is much smaller than the horizontal one, which in turn leads to modeling the vertical motion by the hydrostatic balance. As a result one obtains the system 1 4, which is known as the primitive equations for ocean and atmosphere dynamics see, e.g., [0, [1, [, [3, [4, [33 and references therein. We observe that in the case of ocean dynamics one has to add the diffusion-transport equation of the salinity to the system 1 4. We omitted it here in order to simplify our mathematical presentation. However, we emphasize that our results are equally valid when the salinity effects are taking into account. Note that the horizontal motion can be further approximated by the geostrophic balance when the Rossby number the ratio of the horizontal acceleration to the Coriolis force is very small. By taking advantage of these assumptions and other geophysical considerations, we have developed and used several intermediate models in numerical studies of weather prediction and long-time climate dynamics see, e.g., [4, [7, [8, [, [3, [5, [8, [9, [30, [31 and references therein. Some of these models have also been the subject of analytical mathematical study see, e.g., [, [3, [5, [6, [9, [11, [1, [13, [15, [16, [17, [6, [7, [33, [34 and references therein.

2 46 CHONGSHENG CAO AND EDRISS S. TITI In this paper we will focus on the ء 3 primitive equations in a cylindrical domain 0 و ح= where ح is a smooth bounded domain in R : ك ك + + ك 1 属 =0 1 ج + ن + + =0 ش + z ك 3 =0 属 z ك + شك شك + ش + ك 4 属 ر = ش ج + where the horizontal velocity field = 1, the three-dimensional velocity = ن unknowns. and the pressure are the ش field 1 属, the temperature is a given heat source. The viscosity ر β + is the Coriolis parameter and 0 ن and the heat diffusion operators ج 1 and ج are given by ك 1 Δ 1 = 1 ج 5 مز 1 مز ك 1 Δ 1 = ج 6 ز 1 ز where مز 1 مز are positive constants representing the horizontal and vertical Reynolds numbers, respectively, and ز 1 ز are positive constants which stand for the horizontal and vertical heat diffusion, respectively. We set yك to be + xك = Δ y to be the horizontal gradient operator and ك x ك = the horizontal Laplacian. We observe that the above system is similar to the Boussinesq system with the equation of vertical motion approximated by ء 3 the hydrostatic balance. We partition the boundary of into: Γ u = } «:«=0} {و = :««} = b Γ 0}. «و حك : «} = s Γ We equip the system 1 4 with the following boundary conditions: winddriven on the top surface and free-slip and non-heat flux on the side walls and bottom see, e.g., [0, [1, [, [4, [5, [8, [9, [30: 10 ك : u on Γ 0= 属 τو = ; ش ش α = 11 ك : b on Γ =0 属 =0 1 on Γ s : =0 شك ك =0 ك ك

3 PRIITIVE EQUATIONS 47 where τ is the wind stress on the ocean surface, is the normal vector to Γ s, and ش is typical temperature distribution of the top surface of the ocean. For simplicity we assume here that τ and ش are time independent. However, the results presented here are equally valid when these quantities are time dependent and satisfy certain bounds in space and time. Due to the boundary conditions 10 1, it is natural to assume that τ and ش satisfy the compatibility boundary conditions: τك 13.حك τ =0 =0 on ك شك.حكon 0= ك 14 In addition, we supply the system with the initial condition: 15 «. 0 = 0 «16 «. 0 ش = 0 «ش In [0, [1 and [33 the authors set up the mathematical framework to study the viscous primitive equations for the atmosphere and ocean circulation. oreover, similar to the ء 3 Navier-Stokes equations, they have shown the global existence of weak solutions, but the question of their uniqueness is still open. The short time existence and uniqueness of strong solutions to the viscous primitive equations model was established in [15 and [33. In [16 the authors proved the global existence and uniqueness of strong solutions to the viscous primitive equations in thin domains for a large set of initial data whose size depends inversely on the thickness of the domain. In this paper we show the global existence, uniqueness and continuous dependence on initial data, i.e. global regularity and well-posedness, of the strong solutions to the domain, viscous primitive equations model 1 16 in a general cylindrical ء 3, and for any initial data. It is worth stressing that the ideas developed in this paper can equally apply to the primitive equations subject to other kinds of boundary conditions. As in the case of ء 3 Navier-Stokes equations the question of uniqueness of the weak solutions to this model is still open.. Preliminaries.1. New Formulation. First, let us reformulate the system 1 16 see also [0, [1 and [33. We integrate the equation 3 in the «direction to obtain z» 属 = و 属 ξ. ξل By virtue of 10 and 11 we have 17 = «属 z ξل ξ

4 48 CHONGSHENG CAO AND EDRISS S. TITI and 18 We denote 19 In particular, 0 0 =0. ξل ξ = ξل ξ و = 1 φ 0.ح ξلξ φ 0 0 ξلξ = 1 h و denotes the barotropic mode. We will denote by 1 = ح in the baroclinic mode, that is the fluctuation about the barotropic mode. Notice that =0. Based on the above and 1 we obtain 3 =0 and 4 =0 =0 ك ك By integrating equation we obtain = «z ح in.حك on. s ξل + ξ ش Substituting 17 and the above relation into equation 1, we reach ك z ك ξل ξ + 5 ك s + z =0. 1 ج + ن + ξل ξ ش Remark 1. Notice that due to the compatibility boundary conditions 13 and 14 one can convert the boundary condition 10 1 to be homogeneous by replacing ش by + z+h 3 /3 ش + ش τ while 3 is still true. ش, 0 = For simplicity and without loss of generality we will assume that τ =0. However, we emphasize that our results are still valid for general τ and provided they are smooth enough. In a forthcoming paper we will study ش the long-time dynamics and global attractors to the primitive equations with. ش general τ and

5 PRIITIVE EQUATIONS 49 Therefore, under the assumption that τ = ش 0 = 0, we have the following new formulation for system 1 16: ك z ك 6 ξل ξ + 1 ج + ك z =0 ن + ξل ξ ش s + شك z شك ξل ξ ش + ش ج + ك 7 ر = 8 9 ك ك =0 ك z=0 Γs =0 =0 z= ك =0 Γs z=0 شα + ش z ك z= ش z ك =0; Γs ش n ك =0; = «0 = 0 ««. 0 ش = 0 «ش.. Properties of and. By taking the average of equations 6 in the «direction, over the interval و 0, and using the boundary conditions 8, we obtain the following equation for the barotropic mode 3 ك + ك [ 1 0 h و z ξل ξ z»لξل ξ ش As a result of, 3 and integration by parts, 33 z ξل ξ ك ك s + 1 ن + 1 مز Δ =0. = + [ +. By subtracting 3 from 6 and using 33 we obtain the following equation for the baroclinic mode 34 ك z ك ξل ξ + 1 ج + ك + + [ + z ξل 1 0 z ξ ش ن +»لξل ξ ش و h =0. Therefore, satisfies the following equations and boundary conditions:

6 50 CHONGSHENG CAO AND EDRISS S. TITI 35 ك ن+[ + [ + + Δ 1 ك مز [ s 0 z 0=»ل ξل ξ ش h و 36 =0 ح in 37 =0 =0 ك ك حك on and satisfies the following equations and boundary conditions: ك z ك 38 ξل ξ + 1 ج + ك ن+[ + [ + + z ξل 1 0 z ξ ش 0=»لξل ξ ش و 39 ك ك =0 ك z=0 Γs =0 =0 z= ك =0. Γs Remark. We recall that by virtue of the maximum principle one is able to show the global well-posedness of the ء 3 viscous Burgers equations see, for instance, [19 and references therein. Such an argument, however, is not valid for the ء 3 Navier-Stokes equations because of the pressure term. Remarkably, the pressure term is absent from equation 38. This fact allows us to obtain a bound for the ج 6 norm of, which is a key estimate in our proof of the global regularity for the system Functional spaces and inequalities. Denote by ج ج ح and [1. respectively -Lebesgue and Sobolev spaces, ج the usual ح m ب m ب Let { φ p = φ «p»ل ل ل 1 p for every φ pج 40.ح pج φ 1 p for every ل ل p φ Now, { ك : Ṽ 1 = ك =0 z=0 =0 z= } ك Γs =0 ك =0 =0 Γs Ṽ = { شك : ش شك =0; z= z=0 =0; شα + } شك. =0 ك Γs We denote by ض 1 and ض the closure spaces of Ṽ1 in ب 1, and Ṽ in ب 1 under ب 1 -topology, respectively.

7 PRIITIVE EQUATIONS 51 Definition 1. Let 0 ض 1 and ض 0 ش, and let T be a fixed positive time. is called a strong solution of 6 31 on the time interval [0 T ifit ش satisfies 6 and 7 in a weak sense, and also [0 T ج 1 ض [ [0 T ب [ ب [ [0 T ج ض [ [0 T ش ل ج [ [0 T 1 ج ل شل. ج [ [0 T 1 ج ل For convenience, we recall the following Sobolev and Ladyzhenskaya s inequalities in R see, e.g., [1, [10, [14, [18: 41 φ L4 0 φ 1/ φ L8 0 φ 3/4 4 L φ 1/ H 1 L 6 φ 1/4 H 1 for every φ 1 ب ح and the following Sobolev and Ladyzhenskaya s inequalities in R 3 see, e.g., [1, [10, [14, [18: 1/ 0 L3 43 H1 0 L6 44 L 1/ H 1 for every ب 1. Here 0 is a positive constant which might depend on the shape of ح and but not on their size. oreover, by 41 we get 45 φ 1 L 1 = φ 3 4 L 4 0 φ 3 L φ 3 H 1 0 φ 6 L 6 φ 4 φ ل ل + φ 1 L 6 for every φ 1 ب.ح Also, we recall the integral version of inkowsky inequality for the ج p spaces, 1. Let 1 R m1 and R m be two measurable sets, where 1 and are positive integers. Suppose that ξ η ن is measurable over 1. Then, [ p 1/p 1/p 46. ηل ξل ξ η ن p ξل ηل ξ η ن A priori estimates In the previous subsections we have reformulated the system 1 16 and ش when obtained the system The two systems are equivalent is a strong solution. The existence of such a strong solution for a short interval of time, whose length depends on the initial data and the other physical parameters of the system 1 16, was established in [15 and [33. Let

8 5 CHONGSHENG CAO AND EDRISS S. TITI be given initial data. In this section we will consider the strong solution 0 ش 0 that corresponds to the initial data in its maximal interval of existence [0 T. Specifically, we will establish a priori upper estimates for various norms of this solution in the interval [0 T. In particular, we will show that if T ڤ then the ب 1 norm of the strong solution is bounded over the interval [0 T. This key observation plays a major role in the proof of global regularity of strong solutions to the system 1 16.,in ش estimates. We take the inner product of equation 7 with ج 3.1., and obtain ج ش ل 1 ل = =0 «ش α z + ش + 1 ش + 1 ز 1 ز z ش»ل ل ل شر After integrating by parts we get z ش 47 ξل ξ ξل ξ شك.»ل ل ل ش شك.0=»ل ل ل ش As a result of the above we conclude ش ل 1 =0 «ش α z + ش + 1 ش + 1 ز 1 ز ل. ش ر»ل ل ل شر = Notice that 48. =0 «ش و + z ش و ش Using 48 and the Cauchy-Schwarz inequality we obtain ش ل ل + 1 ز + 1 ش ز =0 «ش α z + ش. ر و + α ز و By the inequality 48 and thanks to Gronwall inequality the above gives م ش 51 t ر α /و + ز و + 0 ش +h/α h Rt oreover, we have t [ ش 5 ل 0 = «ش α + z ش ز 1 ز 0 ر و + ز و α. 0 ش +

9 PRIITIVE EQUATIONS 53 By taking the inner product of equation 6 with, جin, we reach ل z مز 1 مز ل [ z ك»ل ل ل ξل ξ = z.»ل ل ل ξل ξ ش s + ن + By integration by parts we get [ z 53 By 36 we have و =»ل ل ل s 54 Since 55 then from we have ξل ξ ك =0.»ل ل ل =0. ل ل s و = ل ل s =0 ن ل z مز 1 مز ل z. ش و»ل ل ل ξل ξ ش = By Cauchy-Schwarz and 51 we obtain ل ل مز + 1 مز z و ش 1 مز و ر α /و + ز و + 0 ش 1 مز. Recall that cf., e.g., [14, Vol. I p. 55. By the above and thanks to Gronwall s inequality we get t م و C Re 1 56 [ مز و + ر α /و + ز و + 0 ش 1. oreover, t [ ل z مز 1 مز 0 و ر α /و + ز و + 0 ش 1 مز و +.

10 54 CHONGSHENG CAO AND EDRISS S. TITI Therefore, by 51, 5, 56 and 57 we have t 0 t 0 [ ش + ل z مز 1 مز [ 1 0 = «ش α + z ش + 1 ش ز 1 ز 1 ث ل where و + ر α /و + ز و = 1 ث 59 ر α /و + ز و + 0 ش [ 1 مز و + 1 مز و estimates. Taking the inner product of the equation 38 with ج in ج, we get 6 6 ل »ل ل ل 1 مز ل 6 + 1»ل ل ل zك + 4 z مز { z ك + + ξل ξ = ن+[ + [ z 0 1 ξل ξ ش و h Integrating by parts we get [ z 60 Since 61 z ξل ξ ن 4 =0 }.»ل ل ل 4»لξل ξ ش ك =0.»ل ل ل 4 then by 36 and the boundary condition 8 we also have 6 =0.»ل ل ل 4 Thus, by 60 6, 6 6 ل مز ل مز»ل ل ل + 4»ل ل ل z ك + 4 z

11 = { + z ξل 1 0 z ξ ش و PRIITIVE EQUATIONS 55 }.»ل ل ل 4»لξل ξ ش Notice that by integration by parts and boundary condition 8, [ [ + z 0 z»ل ل ل 4»لξل ξ ش 1 ξل ξ ش و [ = k j ك xk 4 j z ξل 1 0 z ξ ش و 4»لξل ξ ش.»ل ل ل Therefore, by Cauchy-Schwarz inequality and Hölder inequality we obtain 6 6 ل »ل ل ل 1 مز ل 6 + 1»ل ل ل zك + 4 z مز [ 0»ل 5 ل ل [ 0 0 +»ل»ل 4 ل ل [ 0 + ش»ل 4 ل ل [ 1/ [ 0 [ ش»ل 4 0»ل 0»ل 4 1/ 0»ل 6»ل 4 1/ 0 1/»ل ل ل 4 L /4 + ل ل»ل 1/ 0»ل 4 0 ل ل 1/»ل 4 1/ ل ل ل ل ل ل 6»ل 1/4

12 56 CHONGSHENG CAO AND EDRISS S. TITI 1/ 0 1/4»ل ل ل 4 ل ل 4»ل 1/ 0 L4 ش +»ل ل ل 4 1/4»ل. 4 ل ل By using inkowsky inequality 46, we get 0 1/ 0 1/»ل.»ل 1 ل ل 6 ل ل By 45,. 6 ل ل + ل ل 4 ل ل 6 0 ل ل 1 Thus, by Cauchy-Schwarz inequality, / 1/ 3 ل ل 6»ل L 6 6 +»ل ل ل 4 Similarly, by 46 and 4, 64 and /»ل 4 ل ل 0 1/ ل ل 8»ل 3 L 6 L + L»ل /4»ل ل ل Therefore, by and 41, 6 6 ل مز ل مز 0 0 3/ 6 3/ L 6 1/4 ل ل 8»ل 1/ 1/ + 1/»ل ل ل + 4»ل ل ل z ك + 4 z L + 1/. L L 6.»ل

13 1/ 1/ 3/»ل ل ل / ش + 1/ ش 3/ 6 PRIITIVE EQUATIONS 57»ل ل ل 4 1/ + 1/ 3/4 + 1/ 1/ 1/ 6 6 1/»ل ل ل 4. Thanks to the Young and the Cauchy-Schwarz inequalities, 6 6 ل + 1»ل ل ل مز ل + 1»ل ل ل z ك + 4 z مز ش ش By 58 and Gronwall inequality, we get t »ل ل ل 4 1 مز ث»ل ل ل 4 z مز where 67 [ t K 1 م= 6 ث. 1 ث H 1 Taking the inner product of the equation 7 with ش 4 ش in ج, and by 7, we get 6 6 ش ل 1 + 5»ل ل ل 4 ش ش 1 ز ل =0 6 6 «ش α +»ل ل ل 4 ش z ش ز»ل ل ل ش 4 ش ر = z شك.»ل ل ل ش 4 ش ξل ξ ش By integration by parts and 36, z شك =0.»ل ل ل ش 4 ش ξل ξ ش 68 As a result of the above we conclude 6 6 ش ل »ل ل ل 4 ش ش»ل ل ل 4 ش z ش 1 ز ل 6 ز 6. 5 ش 6 ر»ل ل ل ش 4 ش ر = 6 =0 6 «ش α +

14 58 CHONGSHENG CAO AND EDRISS S. TITI By Gronwall, again, 69. H1 0 ش + H1 ر 6 ش estimates. 1 ب estimates. First, we observe that since is a strong solution on the interval [0 T then Δ ج [0 T ج.ح Consequently, and by virtue of 36, Δ ج [0 T 1/ ب حك see, e.g., [10, [3. oreover, and thanks to 36 and 37, we have Δ حك 0on = see, e.g., [35. This observation implies also that the Stokes operator in the domain,ح subject to the boundary conditions 37, is equal to the Δ operator. As a result of the above and 36 we apply a generalized version of the Stokes theorem see, e.g., [10, [3 to conclude:.0= ل ل Δ s By taking the inner product of equation 35 with Δ in ج,ح and applying 36 and the above, we reach ل 1 ل = مز Δ { + [ + }. ل ل Δ ن + ل ل Δ Following similar steps as in the proof of ء Navier-Stokes equations cf. e.g., [10, [3 one obtains ل ل Δ 1/ Δ 3/. Applying the Cauchy-Schwarz and Hölder inequalities, we get 0 ل ل Δ + ل ل Δ»ل [ 0 1/ 0 1/»ل»ل ل ل Δ [ 0 1/4»ل ل ل [ 1/ 0 1/4 [»ل ل ل 1/4. Δ»ل ل ل 4 1/ ل ل Δ

15 PRIITIVE EQUATIONS 59 Thus, by Young s and Cauchy-Schwarz inequalities, ل ل + 1 Δ مز »ل ل ل 4 By 58, 66 and thanks tothe Gronwall inequality, we obtain + 1 t 70 ث ل Δ 1 0 مز where [ Kt 1 م= ث ث+ 1 ث + 0 H z estimates. Since = z, it is clear that satisfies ك z ك 7 ξل ξ + 1 ج + ك =0. ش ن + + Taking the inner product of the equation 7 with in ج and using the boundary condition 8, we get ل ل z ك + 1 مز 1 مز z = From integration by parts we get z 73 Since ξل ξ ش ن + ξل ξ ك»ل ل ل.»ل ل ل ك.0=»ل ل ل 74 =0 ن then by 73 and 74 we have ل 1 z ك مز 1 مز ل»ل ل ل ش = ش +»ل ل ل

16 60 CHONGSHENG CAO AND EDRISS S. TITI ش / 6 3/. ش + By Young s inequality and Cauchy-Schwarz inequality, we have ل ل مز + 1 مز ش z ك By 58, 66, 70, and Gronwall inequality, 75 where 76 z مز t z مز ش + t 0 z ث ل zz [ K t+k /3 م= z ث 6 tt. 1 ث + 0 H estimates. By taking the inner product of equation 6 with Δ in ج, we reach ل Δ + 1 z مز 1 مز ل [ z ك ξل ξ = z ξل ξ ش s + ن+»ل ل ل Δ [ 0 0»ل ل ل Δ»ل ش + z»ل + L6 L3 Δ 0 0. Δ ش + ل ل»ل z Δ»ل + Notice that by applying Proposition. in [5 with ن = =Δ and ه = z, we get 0 0 ل ل»ل z Δ»ل 1/ z 1/ z 1/ Δ 3/. As a result and by 43 and 44, we obtain ل 1 ل مز Δ + 1 z مز L6 + 1/ z 1/ 1/ Δ 3/. Δ ش و +

17 PRIITIVE EQUATIONS 61 Thus, by Young s inequality and Cauchy-Schwarz inequality, ل ل مز Δ + 1 مز z. ش + z 4 L 6 + By 58, 66, 70, 75 and thanks to Gronwall inequality, we obtain t Δ + 77 V ث ل z مز 1 مز 0 where [ K/3 6 t t+k 1t K م= V ث 78 zt. 1 ث + 0 H 1 H ش estimates. Taking the inner product of equation 7 with ش شΔ zz in ج, we get 1 ل =0 «ش α z + ش + ش ل شΔ ز 1 ز 1 ز [ z = ش ξل»ل ل ل zz ش + شΔ ر + ش [ 0 0 +»ل zش 1 =0 «ش α + +»ل ل ل zz ش + شΔ ر z ش ل ل»ل zz ش + شΔ z ش 1/ شΔ 3 ش 6 zz ش + z ش + + 1/ Δ 1/ 1/ z ش شΔ 3/ zz ش + z ش + 1/ ز zz ش شΔ ر + zz ش + z ش + [ شΔ 1/ ش 6 + 1/ Δ 1/ 1/ z ش 3/ zz ش + z ش + ر + شΔ 1/ zz ش + z ش +. By Young s inequality and Cauchy-Schwarz inequality, «ش α z + ش + ش ل =0 ل شΔ zش 1 =0 «ش α + zz ش + ز 1 ز 1 ز ز Δ ش z ش + ر +. By 66, 77, and Gronwall inequality, we get

18 6 CHONGSHENG CAO AND EDRISS S. TITI 79 where 80 + =0 «ش α z + ش + ش ز 1 ز t 0 [ 1 1 ز شΔ zش =0 «ش α + ر + H 1 0 ش [ t K 6 t t+k V م = t ث 1 ز zz ش. t ث ل 4. Existence and uniqueness of the strong solutions In this section we will use the a priori estimates to show the global existence and uniqueness, i.e. global regularity, of strong solutions to the system Theorem. Let ب ر 1, 0 ض 0 ش, 1 ض and T ڤ 0 be given. Then there exists a unique strong solution ش of the system 6 31 on the interval [0 T which depends continuously on the initial data. Proof. As indicated earlier the short time existence of the strong solution was established in [15 and [33. Let ش be the strong solution corresponding to the initial data ش 0 0 with maximal interval of existence [0 T. If we assume that T ڤ then it is clear that lim sup H1 + ش H1 =. t T Otherwise, the solution can be extended beyond the time T. However, the above contradicts the a priori estimates 75, 77 and 79. Therefore T =, and the solution ش exists globally in time. Next, we show the continuous dependence on the initial data and the the uniqueness of the strong solutions. Let ش 1 1 and ش be two strong solutions of the system 6 31 with corresponding pressures s 1 and s, =, respectively. Denote by 0 ش 0 and 1 0 ش 1 0 data and initial 1 s = s 1 s and θ = ش 1 ش. It is clear that ك ج + ك z z ك ξل ξ 1 ك ξل ξ z =0 ξل ξ θ s + ن+ θك z θك 8 ξل ξ 1 ش + θ θ + 1 ج + ك z شك ξل ξ 0=

19 83 84 PRIITIVE EQUATIONS = «. 0 ش 1 0 ش = 0 «θ By taking the inner product of equation 81 with in ج, and equation 8 with θ in ج, we get and ل 1 z مز 1 مز ل [ z + 1 = z ξل ξ [ z s + ن ك ك ξل ξ 1 θ ξ ξل θ ل θ + 1 θ z + α θ«=0 ز 1 ز ل [ z ش + θ 1 = z ξل ξ»ل ل ل»ل ل ل θك ξل ξ 1 شك.»ل ل ل θ By integration by parts, and the boundary conditions 8 and 9, we get z ك =0»ل ل ل ξل ξ z θك =0.»ل ل ل ξل θ ξ 1 θ 1 86 Since 87 and by 85, 86 and 87 we have ل 1 z مز 1 مز ل +»ل ل ل = then θ ل 1 ل ن =0 z + 1 θ + 1 θ z + α θ«=0 ز 1 ز z +»ل ل ل θ ش = ك ξل ξ»ل ل ل شك ξل ξ.»ل ل ل θ

20 64 CHONGSHENG CAO AND EDRISS S. TITI Notice that 88 89»ل ل ل 6 3 1/ 3/»ل ل ل θ ش 6 θ 3 θ 1/ ش θ 1/. oreover, z ك ξل ξ»ل ل ل 0 0 ل ل»ل z ك»ل 0 0 1/ 0 1/ ل ل»ل»ل z ك»ل 1 0»ل ل ل 0 ل ل»ل z ك By Cauchy-Schwarz inequality, /»ل ل ل. ل ل»ل By using inkowsky inequality 46 and 41, we obtain 0 1/ 0 1/ 91»ل»ل 4 ل ل ل ل and /»ل z ك ل ل 0 1 4»ل 0 1/ ل ل z 4 ك»ل. z ك z ك»ل z ك z ك.

21 PRIITIVE EQUATIONS 65 Similarly, 93 z شك ξل ξ»ل ل ل θ 1/ ش z ك 1/ ش z ك θ 1/ θ 1/. Therefore, by estimates 88 93, we reach 1 + ل θ ل + 1 θ z + α θ«=0 ز مز + 1 مز + 1 z 1 ز θ z 1/ ك + z 1/ ك 3/ 1/ θ 1/ ش + θ 1/ 1/ ش z ك + 1/ ش z ك θ 1/ θ 1/. By Young s inequality, we get ل ل ش z ك ش z ك + z ك z ك + 4 ش + 4 θ +. Thanks to Gronwall inequality, =0 θ + =0 θ + { t z ك z ك + 4 ش + 4 exp 0 } ش z ك ش z ك +. ل Since ش is a strong solution, θ + =0 θ + =0 exp{ ث V ث + t ث + z ث V + ث t }. The above inequality proves the continuous dependence of the solutions on the θ = 0, we have = 0 = θ = 0 = initial data; in particular, when = 0, for all 0. Therefore, the strong solution is unique. Acknowledgments. We are thankful to the anonymous referee for the useful comments and suggestions. This work was supported in part by NSF grants No. DS and DS , the AOF Fellowship of the Israeli Council of Higher Education, and by the USA Department of Energy, under contract number W-7405-ENG-36 and ASCR Program in Applied athematical Sciences.

22 66 CHONGSHENG CAO AND EDRISS S. TITI Florida International University, iami, FL address: Dept. of athematics and Dept. of echanical and Aerospace Engineering, University of California, Irvine, CA and Dept. of Computer Science and Applied athematics, Weizmann Institute of Science, Rehovot, Israel addresses: References [1 R. A. Adams, Sobolev Spaces, Academic Press, New York, [ A. Babin, A. ahalov, and B. Nicolaenko, Fast singular oscillating limits and global regularity for the 3D primitive equations of geophysics, ath. odel Numer. Anal , 01. [3 V. Barcilon, P. Constantin, and E. S. Titi, Existence of solutions to the Stommel- Charney model of the Gulf Stream, SIA J. ath. Anal , [4 G. Browning, A. Kasahara, and H.-O. Kreiss, Initialization of the primitive equations by the bounded derivative method, J. Atmospheric Sci , [5 C. Cao and E. S. Titi, Global well-posedness and finite-dimensional global attractor for a 3-D planetary geostrophic viscous model, Comm. Pure Appl. ath , [6 C. Cao, E. S. Titi, and. Ziane, A horizontal hyper-diffusion three-dimensional thermocline planetary geostrophic model: well-posedness and long time behaviour, Nonlinearity , [7 J. G. Charney, The use of the primitive equations of motion in numerical prediction, Tellus , 6. [8, The gulf stream as an inertial boundary layer, Proc. Nat. Acad. Sci. U.S.A , [9 J.-Y. Chemin, B. Desjardins, I. Gallagher, and E. Grenier, Anisotropy and dispersion in rotating fluids, in Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, Studies in athematics and its Applications 31 00, [10 P. Constantin and C. Foias, Navier-Stokes Equations, The University of Chicago Press, Chicago, IL, [11 P. Constantin, A. ajda, and E. Tabak, Formation of strong fronts in the -D quasigeostrophic thermal active scalar, Nonlinearity , [1, Singular front formation in a model for quasigeostrophic flow, Phys. Fluids , [13 P. F. Embid and A. J. ajda, Low Froude number limiting dynamics for stably stratified flow with small or finite Rossby numbers, Geophys. Astrophys. Fluid Dynam , [14 G. P. Galdi, An Introduction to the athematical Theory of the Navier-Stokes Equations, Vol. I& II, Springer-Verlag, New York, [15 F. Guillén-González, N. asmoudi, and. A. Rodríguez-Bellido, Anisotropic estimates and strong solutions of the primitive equations, Differential and Integral Equations ,

23 PRIITIVE EQUATIONS 67 [16 C. Hu, R. Temam and. Ziane, The primitive equations on the large scale ocean under the small depth hypothesis, Discrete Contin. Dynam. Systems 9 003, [17 D. A. Jones, athematical analysis of geophysical balance models, J. Differential Eq , 1 6. [18 O. A. Ladyžhenskaya, The Boundary Value Problems of athematical Physics, Springer- Verlag, New York, [19 O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural ceva, Linear and Quasilinear Equations of Parabolic Type, A.. S., Providence, R.I [0 J. L. Lions, R. Temam, and S. Wang, New formulations of the primitive equations of atmosphere and applications, Nonlinearity 5 199, [1, On the equations of the large scale ocean, Nonlinearity 5 199, [ J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, [3 L. F. Richardson, Weather Prediction by Numerical Process, Cambridge Univ. Press, Cambridge 19 reprint, Dover, New York, [4 R. Salmon, Lectures on Geophysical Fluid Dynamics, Oxford Univ. Press, New York, [5 R. Samelson, Coastal boundary conditions and the baroclinic structure of wind-driven continental shelf currents, J. Physical Oceanography , [6 R. Samelson, R. Temam, and S. Wang, Some mathematical properties of the planetary geostrophic equations for large scale ocean circulation, Applicable Analysis , [7, Remarks on the planetary geostrophic model of gyre scale ocean circulation, Differential and Integral Equations , [8 R. Samelson and G. Vallis, A simple friction and diffusion scheme for planetary geostrophic basin models, J. Physical Oceanography , [9, Large-scale circulation with small diapycnal diffusion: The two-thermocline limit, J. arine Res , [30 D. Seidov, An intermediate model for large-scale ocean circulation studies, Dynamics of Atmospheres and Oceans , [31 H. Stommel, The westward intensification of wind-driven ocean currents, Trans. Amer. Geophys. Union , [3 R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland Publ. Co., Amsterdam, [33 R. Temam and. Ziane, Some mathematical problems in geophysical fluid dynamics, in Handbook of athematical Fluid Dynamics, Vol. III, , North Holland Publ. Co., Amsterdam, 004. [34 G. Wolansky, Existence, uniqueness, and stability of stationary barotropic flow with forcing and dissipation, Comm. Pure Appl. ath , [35. Ziane, On the two-dimensional Navier-Stokes equations with the free boundary condition, J. Appl. ath. & Optimization , Received arch, 005 Revised November 14, 005