CHONGSHENG CAO AND EDRISS S. TITI
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1 GLOBAL WELL-POSEDNESS OF THE THREE-DIENSIONAL VISCOUS PRIITIVE EQUATIONS OF LARGE SCALE OCEAN AND ATOSPHERE DYNAICS arxiv:math.ap/538 v 16 Nov 5 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. SC Subject Classifications: 35Q35, 657, 86-8,86A1. Keywords: Primitive equations, Boussinesq equations, Navier Stokes equations, global regularity. 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 diffusiontransport 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 horiontal 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., [,[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 emphasie that our results are equally valid when the salinity effects are taking into account. Let us remark that the horiontal motion can be further approximated by the geostrophic balance when the Rossby number the ratio of the horiontal acceleration to the Coriolis force is very small. By taking advantage of these assumptions and other geophysical considerations several intermediate models have been developed and used in numerical studies of weather prediction and long-time climate dynamics see, e.g., [4, [7, [8, [, [3, [5, [8, [9, [3, [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. In this paper we will focus on the 3D primitive equations in a cylindrical domain =,, Date: Submitted: arch, 5. Revised: November 14, 5. 1
2 C. CAO AND E.S. TITI where is a smooth bounded domain in R : + v v + w t + p + f k v + L 1 v = 1 p + T = v + w = 3 T + v T + w T t + L T = Q 4 where the horiontal velocity field v = v 1, v, the three-dimensional velocity field v 1, v, w, the temperature T and the pressure p are the unknowns. f = f β + y is the Coriolis parameter, Q is a given heat source. The viscosity and the heat diffusion operators L 1 and L are given by L 1 = 1 1 Re 1 Re, 5 L = 1 1 Rt 1 Rt, 6 where Re 1, Re are positive constants representing the horiontal and vertical Reynolds numbers, respectively, and Rt 1, Rt are positive constants which stand for the horiontal and vertical heat diffusivity, respectively. We set = x, y to be the horiontal gradient operator and = x + y to be the horiontal Laplacian. We observe that the above system is similar to the 3D Boussinesq system with the equation of vertical motion is approximated by the hydrostatic balance. We partition the boundary of into: Γ u = {x, y, : = }, 7 Γ b = {x, y, : = }, 8 Γ s = {x, y, : x, y, }. 9 We equip the system 1 4 with the following boundary conditions with wind driven on the top surface and non-slip and non-heat flux on the side walls and bottom see, e.g., [, [1, [, [4, [5, [8,[9, [3: on Γ u : = h τ, w =, T = αt T ; 1 on Γ b : T =, w =, = ; 11 on Γ s : v n =, n n =, T =, n 1 where τx, y is the wind stress on ocean surface, n is the normal vector to Γ s, and T x, y is typical temperature distribution of the top surface of the ocean. For simplicity we assume here that τ and T 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 1 1, it is natural to assume that τ and T satisfy the compatibility boundary conditions: τ n =, T n τ n =, on. 13 n = on. 14
3 PRIITIVE EQUATIONS 3 In addition, we supply the system with the initial condition: vx, y,, = v x, y,. 15 Tx, y,, = T x, y,. 16 In [, [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 3D 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 sie 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 3D viscous primitive equations model 1 16 in general cylindrical domain,, 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 kids of boundary conditions. As in the case of 3D 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 [, [1 and [33. We integrate the equation 3 in the direction to obtain By virtue of 1 and 11 we have and We denote by In particular, wx, y,, t = wx, y,, t We will denote the fluctuation by Notice that wx, y,, t = vx, y, ξ, tdξ. vx, y, ξ, tdξ, 17 vx, y, ξ, tdξ = vx, y, ξ, tdξ =. 18 φx, y = 1 h vx, y = 1 h Based on the above and 1 we obtain and v n =, φx, y, ξdξ, x, y. 19 vx, y, ξdξ, in. ṽ = v v. 1 ṽ =. v =, in, 3 n =, on. 4 n
4 4 C. CAO AND E.S. TITI By integrating equation we obtain px, y,, t = Tx, y, ξ, tdξ + p s x, y, t. Substitute 17 and the above relation into equation 1 we reach + v v vx, y, ξ, tdξ t + p s x, y, t Tx, y, ξ, tdξ + f k v + L 1 v =. 5 Remark 1. Notice that due to the compatibility boundary conditions 13 and 14 one can convert the boundary condition 1 1 to be homogeneous by replacing v, T by v + +h 3 /3 τ, T + T while 3 is still true. For simplicity and without loss generality we will assume that τ =, T =. However, we emphasie that our results are still valid for general τ and T provided they are smooth enough. In a forthcoming paper we will study the long-time dynamics and global attractors to the primitive equations with general τ and T. Therefore, under the assumption that τ =, T =, we have the following new formulation for system 1 16: t + L 1v + v v vx, y, ξ, tdξ + p s x, y, t Tx, y, ξ, tdξ + f k v =, 6 T T t + L T + v T vx, y, ξ, tdξ = Q, 7 =, = =, v n Γs =, = n n =, 8 Γs T + αt = = ; T = = ; n T Γs =, 9 vx, y,, = v x, y,, 3 Tx, y,, = T x, y, Properties of v and ṽ. By taking the average of equations 6 in the direction, over the interval,, and using the boundary conditions 8, we obtain [ 1 + v v vx, y, ξ, tdξ t + p sx, y, t Tx, y, ξ, tdξd h +f k v 1 v =. Re 1 As a result of, 3 and integration by parts we have v v vx, y, ξ, tdξ = v v + [ṽ ṽ + ṽ ṽ. 33 3
5 PRIITIVE EQUATIONS 5 By subtracting 3 from 6 and using 33 we get ṽ ṽ t + L 1ṽ + ṽ ṽ ṽx, y, ξ, tdξ + ṽ v + v ṽ [ṽ ṽ + ṽ ṽ Tx, y, ξ, tdξ 1 Tx, y, ξ, tdξd + f h k ṽ =. 34 Therefore, v satisfies the following equations and boundary conditions: t 1 v + v v + [ṽ ṽ + ṽ ṽ + f Re k v 1 [ + p s x, y, t 1 Tx, y, ξ, t dξ d =, 35 h v =, in, 36 v n =, n =, n on, 37 and ṽ satisfies the following equations and boundary conditions: ṽ ṽ t + L 1ṽ + ṽ ṽ ṽx, y, ξ, tdξ + ṽ v + v ṽ [ṽ ṽ + ṽ ṽ + f k ṽ Tx, y, ξ, tdξ 1 Tx, y, ξ, tdξd =, 38 h ṽ ṽ =, ṽ = =, ṽ n Γs =, = n n =,. 39 Γs Remark. We recall that by virtue of the maximum principle one is able to show the global well-posedness of the 3D viscous Burgers equations see, for instance, [19 and references therein. Such an argument, however, is not valid for the 3D 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 L 6 norm of ṽ, which is a key estimate in our proof of the global regularity for the system Functional spaces and Inequalities. Let us denote by L, L and H m, H m the usual L Lebesgue and Sobolev spaces, respectively [1. We denote by φ p = { φx, y, p dxdyd 1 p, for every φ L p φx, y p dxdy 1 p, for every φ L p. 4 Let Ṽ 1 = Ṽ = { v C : } =, = =, v n Γs =, = n n =, v =, Γs { T C : T αt } T = ; = = + T = ; n =. Γs We denote by V 1 and V be the closure spaces of Ṽ1 in H 1, and Ṽ in H 1 under H 1 topology, respectively.
6 6 C. CAO AND E.S. TITI Definition 1. Let v V 1 and T V, and let T be a fixed positive time. v, T is called a strong solution of 6 31 on the time interval [, T if it satisfies 6 and 7 in weak sense, and also v C[, T, V 1 L [, T, H, T C[, T, V L [, T, H, dv dt L1 [, T, L, dt L 1 [, T, L. dt For convenience, we recall the following Sobolev and Ladyhenskaya s inequalities in R see, e.g., [1, [1, [14, [18 φ L4 C φ 1/ L φ 1/ H 1, 41 φ L 8 C φ 3/4 L 6 φ 1/4 H 1, 4 for every φ H 1, and the following Sobolev and Ladyhenskaya s inequalities in R 3 see, e.g., [1, [1, [14, [18 ψ L3 C u 1/ L u 1/ H 1, 43 u L 6 C u H 1, 44 for every u H 1. Here C is a positive constant which might depend on the shape of and but not on their sie. oreover, by 41 we get φ 1 L 1 = φ 3 4 L 4 C φ 3 L φ 3 H 1 C φ 6 L 6 φ 4 φ dxdy + φ 1 L 6, 45 for every φ H 1. Also, we recall the integral version of inkowsky inequality for the L p spaces, p 1. Let 1 R m1 and R m be two measurable sets, where m 1 and m are two positive integers. Suppose that fξ, η is measurable over 1. Then, [ 1 fξ, η dη p 1/p dξ 1/p fξ, η p dξ dη A Priori estimates In the previous subsections we have reformulated the system 1 16 and obtained the system The two systems are equivalent when v, T 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 v, T be a given initial data. In this section we will consider the strong solution that corresponds to this initial data in its maximal interval of existence [, T. Specifically, we will establish a priori upper estimates for various norms of this solution in the interval [, T. In particular, we will show that if T < then the H 1 norm of the strong solution is bounded over the interval [, T. This key observation plays a major role in the proof of global regularity of strong solutions to the system 1 16.
7 PRIITIVE EQUATIONS L estimates. We take the inner product of equation 7 with T, in L, and obtain 1 d T + 1 T + 1 dt Rt 1 = QT dxdyd v T T + α T = Rt vx, y, ξ, tdξ After integrating by parts we get T v T vx, y, ξ, tdξ T As a result of the above we conclude 1 d T + 1 T dt Rt + 1 T 1 Rt + α T = = QT dxdyd Q T. Notice that Using 48 and the Cauchy Schwar inequality we obtain d T dt T dxdyd. T dxdyd =. 47 T h T + h T = Rt 1 T + 1 Rt T + α T = 49 h Rt + h α Q. 5 By the inequality 48 and thanks to Gronwall inequality the above gives T e t h Rt +h/α T + h Rt + h/α Q, 51 oreover, we have t [ 1 Ts + 1 T s + α T = s ds Rt 1 Rt h Rt + h α Q t + e By taking the inner product of equation 6 with v, in L, we reach 1 d v + 1 v + 1 dt Re [ 1 = v v + f k v + p s t h Rt +h/α T + h Rt + h/α Q, 5 Re v vx, y, ξ, tdξ Tx, y, ξ, tdξ v dxdyd v dxdyd. By integration by parts we get [ v v vx, y, ξ, tdξ v dxdyd =. 53 By 36 we have p s v dxdyd = h p s v dxdy = p s v dxdy =. 54
8 8 C. CAO AND E.S. TITI Since then from we have f k v v =, 55 1 d v + 1 v dt Re + 1 v 1 Re = Tx, y, ξ, t dξ v dxdyd h T v. By Cauchy Schwar and 51 we obtain Recall that cf., e.g., [14 Vol. I p d v + 1 v + 1 v dt Re 1 Re h Re 1 T h Re 1 T + h Rt + h/α Q. v C v. By the above and thanks to Gronwall inequality we get v t e C Re 1 h v + ṽ +C h Re [ 1 T + h Rt + h/α Q. 56 oreover, [ 1 vs + 1 v s ds Re 1 Re h Re 1 T + h Rt + h/α Q t t + e C Re 1 h v + ṽ +C h Re [ 1 T + h Rt + h/α Q. 57 t Therefore, by 51, 5, 56 and 57 we have t [ 1 vt + vs + 1 v s ds Re 1 Re t [ 1 + Tt + Ts + 1 T s + α T = s ds K 1 t, 58 Rt 1 Rt where K 1 t = h Rt + h/α Q ; t + h v + ṽ C h Re 1 + h Re 1 t [ T + h Rt + h/α Q L 6 estimates. Taking the inner product of the equation 38 with ṽ 4 ṽ in L, we get 1 d ṽ ṽ ṽ 4 + ṽ ṽ dxdyd + 1 ṽ ṽ 4 + ṽ ṽ dxdyd 6 dt Re 1 Re { ṽ = ṽ ṽ ṽx, y, ξ, tdξ + ṽ v + v ṽ [ṽ ṽ + ṽ ṽ +f k ṽ Tx, y, ξ, tdξ 1 } Tx, y, ξ, tdξd ṽ 4 ṽ dxdyd. h
9 PRIITIVE EQUATIONS 9 By integration by parts we get [ ṽ ṽ ṽx, y, ξ, tdξ ṽ ṽ 4 ṽ dxdyd =. 6 Since f k ṽ ṽ 4 ṽ =, 61 then by 36 and the boundary condition 8 we also have v ṽ ṽ 4 ṽ dxdyd =. 6 Thus, by 6 6 we have 1 d ṽ dt = + 1 Re 1 { ṽ v ṽ ṽ + ṽ ṽ Tx, y, ξ, tdξ 1 h ṽ ṽ 4 + ṽ ṽ dxdyd + 1 ṽ ṽ 4 + ṽ ṽ dxdyd Re } Tx, y, ξ, tdξd ṽ 4 ṽ dxdyd. Notice that by integration by parts and boundary condition 8 we have [ ṽ v [ṽ ṽ + ṽ ṽ = Tx, y, ξ, tdξd ṽ 4 ṽ dxdyd Tx, y, ξ, tdξ 1 h [ ṽ v ṽ 4 ṽ + ṽ ṽ 4 ṽ v ṽ k ṽ j xk ṽ 4 ṽ j Tx, y, ξ, tdξ 1 h Tx, y, ξ, tdξd ṽ 4 ṽ dxdyd.
10 1 C. CAO AND E.S. TITI Therefore, by Cauchy Schwar inequality and Hölder inequality we obtain 1 d ṽ ṽ ṽ 4 + ṽ ṽ dxdyd + 1 ṽ ṽ 4 + ṽ ṽ dxdyd 6 dt Re 1 Re [ C v ṽ ṽ 5 d dxdy +C +C [ ṽ d [ T [ C v [ +C +C [ T ṽ ṽ 4 d ṽ ṽ 4 d ṽ d ṽ ṽ 4 d ṽ ṽ 4 d dxdy 1/ ṽ ṽ 4 d 1/ 1/ C v L 4 ṽ ṽ 4 dxdyd 4 1/4 +C ṽ d dxdy 1/ +C T L 4 ṽ ṽ 4 dxdyd dxdy ṽ d1/ 6 dxdy 1/ ṽ d1/ 4 dxdy ṽ d1/ 4 dxdy ṽ 6 d dxdy 1/ ṽ ṽ 4 dxdyd By using inkowsky inequality 46, we get 1/ ṽ d dxdy 6 C ṽ 4 d 1/4 ṽ 4 d dxdy 1/4. ṽ 1 dxdy 1/ d. By 45, ṽ 1 dxdy C ṽ 6 dxdy ṽ 4 ṽ dxdy + ṽ dxdy 6. Thus, by Cauchy Schwar inequality we obtain ṽ 6 d dxdy 1/ C ṽ 3 Similarly, by 46 and 4, we also get 1/ ṽ d dxdy 4 C C L 6 dxdy 1/4 1/ ṽ 4 ṽ dxdyd + ṽ 6 L ṽ 8 dxdy 1/ d ṽ 3 L 6 ṽ L + ṽ L d C ṽ 3 6 ṽ + ṽ, 64
11 PRIITIVE EQUATIONS 11 and C 4 1/4 ṽ d dxdy C ṽ 3/ L 6 Therefore, by and 41, we reach 1 d ṽ dt Re 1 C v 1/ v 1/ ṽ 3/ 6 ṽ 1/ L + ṽ 1/ L ṽ ṽ 4 + ṽ ṽ dxdyd + 1 Re +C ṽ 3 6 ṽ + ṽ +C T 1/ T 1/ ṽ 3/ 6 ṽ ṽ 4 dxdyd ṽ ṽ 4 dxdyd ṽ 1/ + ṽ 1/ ṽ 8 dxdy 1/4 d d C ṽ 3/ 6 3/4 + C v 1/ v 1/ ṽ 6 6 1/ ṽ 1/ + ṽ 1/. 65 ṽ ṽ 4 + ṽ ṽ dxdyd ṽ ṽ 4 dxdyd 1/. Thanks to the Young s and the Cauchy Schwar inequalities we have d ṽ ṽ ṽ 4 + ṽ ṽ dxdyd + 1 ṽ ṽ 4 + ṽ ṽ dxdyd dt Re 1 Re C v v ṽ C ṽ 6 6 ṽ + C T T + C ṽ ṽ 6 6. By 58 and Gronwall inequality, we get t 1 ṽt ṽ ṽ 4 dxdyd + 1 Re 1 Re where ṽ ṽ 4 dxdyd K 6 t, 66 [ K 6 t = e K 1 t v 6 H 1 + K 1t. 67 Taking the inner product of the equation 7 with T 4 T in L, and by 7, we get 1 d T T T 4 dxdyd + 5 T T 4 dxdyd + α T = dt Rt 1 Rt T = Q T 4 T dxdyd v T vx, y, ξ, tdξ T 4 T dxdyd. By integration by parts and 36 we get T v T vx, y, ξ, tdξ T 4 T dxdyd =. 68 As a result of the above we conclude 1 d T T T 4 dxdyd + 5 T T 4 dxdyd + α T = dt Rt 1 Rt = Q T 4 T dxdyd Q 6 T 5 6. By Gronwall, again, we get Tt 6 Q H 1 t + T H 1. 69
12 1 C. CAO AND E.S. TITI 3.3. H 1 estimates v estimates. First, let us observe that since v is a strong solution on the interval [, T then v L [, T, L. Consequently, and by virtue of 36, v n L [, T, H 1/ see, e.g., [1, [3. oreover, and thanks to 36 and 37, we have v n = on see, e.g., [35. This observation implies also that the Stokes operator in the domain, subject to the boundary conditions 37, is equal to the operator. As a result of the above and 36 we apply a generalied version of the Stokes theorem see, e.g., [1, [3 to conclude: p s x, y, t vx, y, tdxdy =. By taking the inner product of equation 35 with v in L, and applying 36 and the above, we reach 1 d v + 1 { } v dt Re = v v + [ṽ ṽ + ṽ ṽ v dxdy + f k v v dxdy. 1 Following similar steps as in the proof of D Navier Stokes equations cf. e.g., [1, [3 one obtains v v v dxdy C v 1/ v v 3/. Applying the Cauchy Schwar and Hölder inequalities, we get ṽ ṽ + ṽ ṽ v dxdy C ṽ ṽ d v dxdy [ 1/ 1/ C ṽ ṽ d ṽ d v dxdy [ 1/4 [ 1/4 [ C ṽ ṽ d dxdy ṽ d dxdy C ṽ 1/ 1/4 ṽ 4 ṽ dxdyd v. Thus, by Young s and Cauchy Schwar inequalities, we have d v dt 1/ v dxdy + 1 v Re C v v 4 + C ṽ + C ṽ 4 ṽ dxdyd + C v. 1 By 58, 66 and thanks to Gronwall inequality and we obtain where v + 1 Re 1 t v ds K t, 7 [ K t = e K 1 t v H 1 + K 1t + K 6 t. 71
13 PRIITIVE EQUATIONS v estimates. Denote by u = v. It is clear that u satisfies u u t + L 1u + v u vx, y, ξ, tdξ +u v vu + f k u T =. 7 Taking the inner product of the equation 7 with u in L and using the boundary condition 8, we get 1 d u + 1 u dt Re = v u Re u vx, y, ξ, tdξ u v vu + f k u T u dxdyd u u dxdyd. By integration by parts we get u v u vx, y, ξ, tdξ u dxdyd =. 73 Since then by 73 and 74 we have f k u u =, 74 1 d u + 1 u + 1 u dt Re 1 Re = u v vu T u dxdyd C v u u dxdyd + T u C v 6 u 3 u + T u C v 6 u 1/ u 3/ + T u. By Young s inequality and Cauchy Schwar inequality, we have d u dt + 1 Re 1 u + 1 Re u C v 4 6 u + C T C v 4 + ṽ 4 6 u + C T. By 58, 66, 7, and Gronwall inequality, we get where v + 1 Re 1 t v s + 1 Re t v s ds K t, 75 [ K t = e K t+k/3 6 tt v H 1 + K 1t. 76
14 14 C. CAO AND E.S. TITI v estimates. By taking the inner product of equation 6 with v in L, we reach 1 d v + 1 v dt Re + 1 v 1 Re [ = v v vx, y, ξ, tdξ +f k v + p s Tx, y, ξ, tdξ [ C v v + v d ṽ + C v L 6 v L 3 v + C v dxdyd T d v dxdyd v d v v d dxdy + C T v. Notice that by applying the Proposition. in [5 with u = v, f = v and g = v, we get v d v v d dxdy C v 1/ v 1/ v 1/ v 3/. As a result and by 43 and 44, we obtain 1 d v dt C + 1 Re 1 v + 1 v Re v L 6 + v 1/ v 1/ v 1/ v 3/ + h T v. Thus, by Young s inequality and Cauchy Schwar inequality, we have d v dt C + 1 Re 1 v + 1 v 4 L 6 + v v v Re v + C T. By 58, 66, 7, 75 and thanks to Gronwall inequality, we obtain v + t 1 vs Re + 1 v s ds K V t, 77 1 Re where [ K V t = e K/3 6 t t+k 1t K t v H 1 + K 1t. 78
15 PRIITIVE EQUATIONS T H 1 estimates. Taking the inner product of the equation 7 with T T in L, we get 1 d T + T + α T = dt T T + α T = 1 + T Rt 1 Rt 1 Rt Rt [ = v T v dξ T Q [ T + T dxdyd C v T + Q T + T dxdyd + 1/ [ v d T T + T d dxdy C v 6 T 3 T + T + T +C v 1/ v 1/ T 1/ T + T + T 3/ + Q T + T + T 1/ [ T C v 6 T 1/ + v 1/ v 1/ T 1/ + T + T 3/ + Q T + T + T 1/. By Young s inequality and Cauchy Schwar inequality we have d T + T + α T = dt T Rt T 1 Rt 1 Rt + α T = 1 + T Rt C v v v T + T + C Q. By 66, 77, and Gronwall inequality, we get where T + T + α T = t [ 1 + T + Rt 1 1 Rt Rt T + α T = 1 T ds K t, 79 Rt [ K t = e K 6 t t+k V t T H 1 + Q 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 Q H 1, v V 1, T V and T >, be given. Then there exists a unique strong solution v, T of the system 6 31 on the interval [, T which depends continuously on the initial data. Proof. As we have indicated earlier the short time existence of the strong solution was established in [15 and [33. Let v, T be the strong solution corresponding to the initial data v, T with maximal interval of existence [, T. If we assume that T < then it is clear that lim sup v H 1 + T H 1 =. 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 v, T exists globally in time.
16 16 C. CAO AND E.S. TITI Next, we show the continuous dependence on the initial data and the the uniqueness of the strong solutions. Let v 1, T 1 and v, T be two strong solutions of the system 6 31 with corresponding pressures p s 1 and p s, and initial data v 1, T 1 and v, T, respectively. Denote by u = v 1 v, q s = p s 1 p s and θ = T 1 T. It is clear that u u t + L 1u + v 1 u + u v v 1 x, y, ξ, tdξ ux, y, ξ, tdξ +f k u + q s θx, y, ξ, tdξ =, 81 θ t + L θ + v 1 θ + u T v 1 x, y, ξ, tdξ ux, y,, t = v 1 v, θx, y,, = T 1 T. θ ux, y, ξ, tdξ T =, 8 By taking the inner product of equation 81 with with u in L, and equation 8 with θ, in L we get 1 d u + 1 u dt Re + 1 u 1 Re [ = v 1 u + u v [ f k u + q s and v 1 x, y, ξ, tdξ θx, y, ξ, tdξ u dxdyd, u 1 d θ + 1 θ + 1 θ + α θ = dt Rt 1 Rt [ θ = v 1 θ + u T v 1 x, y, ξ, tdξ By integration by parts, and the boundary conditions 8 and 9, we get u v 1 u v 1 x, y, ξ, tdξ θ v 1 θ v 1 x, y, ξ, tdξ Since Then by 85, 86 and 87 we have 1 d u + 1 u dt Re + 1 u 1 Re = u v u dxdyd + ux, y, ξ, tdξ u dxdyd T ux, y, ξ, tdξ θ dxdyd. u dxdyd =, 85 θ dxdyd =. 86 f k u u =, 87 ux, y, ξ, tdξ u dxdyd.
17 PRIITIVE EQUATIONS 17 and 1 d θ + 1 θ dt Rt + 1 θ 1 Rt + α θ = = u T θ dxdyd + ux, y, ξ, tdξ T θ dxdyd. Notice that u v u dxdyd v u 3 u 6 C v u 1/ u T θ dxdyd v θ 3 u 6 C T θ 1/ oreover, ux, y, ξ, tdξ u dxdyd u d v d u d dxdy 1 By Cauchy Schwar inequality, we get 1/ u d u d1/ dxdy v d dxdy u 3/ θ 1/, 88 u. 89 v u d dxdy 1 4 u d dxdy 1 4. u d dxdy 1/ C u. 9 By using inkowsky inequality 46 and 41, we obtain C u d 1/ 1/ dxdy C u dxdy 4 d u u d C u u, 91 and C 1/ v d dxdy C v 4 dxdy 1/ d v v d C v v. 9 Similarly, we have ux, y, ξ, tdξ T θ dxdyd C u T 1/ T 1/ θ 1/ θ 1/. 93
18 18 C. CAO AND E.S. TITI Therefore, by estimates 88 93, we reach 1 d u + θ + 1 u + 1 u + 1 θ + 1 θ + α θ = dt Re 1 Re Rt 1 Rt C v + v 1/ v 1/ u 1/ u 3/ +C T θ 1/ θ 1/ u + C u T 1/ T 1/ θ 1/ θ 1/. By Young s inequality, we get d u C v 4 dt + T 4 + v v + T T u + θ. Thanks to Gronwall inequality, we obtain ut + θt ut = + θt = { t } exp C v s 4 + T s 4 + v s v s + T s T s ds. Since v, T is a strong solution, we have ut + θt ut = + θt = exp{c K V t + K t t + K K V + K t }. The above inequality proves the continuous dependence of the solutions on the initial data, and in particular, when ut = = θt = =, we have ut = θt =, for all t. 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 the NSF grants No. DS 4794 and DS 54619, the AOF Fellowship of the Israeli Council of Higher Education, and by the USA Department of Energy, under contract number W 745 ENG 36 and ASCR Program in Applied athematical Sciences. 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. 34, 1. [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, H.-O. Kreiss, Initialiation 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. 56 3, [6 C. Cao, E.S. Titi and. Ziane, A horiontal hyper diffusion 3 D thermocline planetary geostrophic model: wellposedness and long time behavior, Nonlinearity 17 4, [7 J.G. Charney, The use of the primitive equations of motion in numerical prediction, Tellus , 6. [8 J.G. Charney, The gulf stream as an inertial boundary layer, Proc. Nat. Acad. Sci. U.S.A , [9 J.-Y. Chemin, B. Desjardines, I. Gallagher and E. Grenier, Anisotropy and dispersion in rotating fluids, Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, Studies in athematics and its Applications, 31, [1 P. Constantin and C. Foias, Navier-Stokes Equations, The University of Chicago Press, [11 P. Constantin, A. ajda and E. Tabak, Formation of strong fronts in the -D quasigeostrophic thermal active scalar, Nonlinearity , [1 P. Constantin, A. ajda and E. Tabak, Singular front formation in a model for quasigeostrophic flow, Phys. Fluids , 9 11.
19 PRIITIVE EQUATIONS 19 [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 , 1 5. [14 G.P. Galdi, An Introduction to the athematical Theory of the Navier-Stokes Equations, Vol. I & II, Springer-Verlag, [15 F. Guillén Gonáe, N. asmoudi and.a. Rodrígue-Bellido, Anisotropic estimates and strong solutions of the Primitive Equations, Differential and Intergral Equations 14 no. 1 1, [16 C. Hu, R. Temam and. Ziane, The primitive equations on the large scale ocean under the small depth hypothesis, Discrete Contin. Dyn. Syst. 9 3, [17 D.A. Jones, athematical analysis of geophysical balance models, J. Diff. Eq. 179, 1 6. [18 O.A. Ladyhenskaya, The Boundary Value Problems of athematical Physics, Springer-Verlag, [19 O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural ceva, Linear and Quasilinear Equations of Parabolic Type, American athematical Society, Providence, R.I., [ J.L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of atmosphere and applications, Nonlinearity 5 199, [1 J.L. Lions, R. Temam and S. Wang, 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 University Press, Cambridge 19 reprint, Dover, New York, [4 R. Salmon, Lectures on Geophysical Fluid Dynamics, Oxford University Press, New York, Oxford, [5 R. Samelson, Coastal boundary conditions and the baroclinic structure of wind driven continental shelf currents, Journal of 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 R. Samelson, R. Temam and S. Wang, Remarks on the planetary geostrophic model of gyre scale ocean circulation, Differential and Integral Equation, 13, [8 R. Samelson and G. Vallis, A simple friction and diffusion scheme for planetary geostrophic basin models, J. Phys. Oceanogr., , [9 R. Samelson and G. Vallis, Large-scale circulation with small diapycnal diffusion: The two-thermocline limit, J. arine Res., , [3 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, [33 R. Temam and. Ziane, Some mathematical problems in geophysical fluid dynamics, Handbook of athematical Fluid Dynamics, 3. [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. & Optimiation, , C. Cao Department of athematics, Florida International University, University Park, iami, FL 33199, USA. address: caoc@fiu.edu E.S. Titi Department of athematics, and Department of echanical and Aerospace Engineering, University of California, Irvine, CA , USA. Also: Department of Computer Science and Applied athematics, Weimann Institute of Science, Rehovot 761, Israel. address: etiti@math.uci.edu address: edriss.titi@weimann.ac.il
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