Global Well-Posedness of the Three-Dimensional Viscous Primitive Equations of Large Scale Ocean and Atmosphere Dynamics

Annals of Mathematics, 166 (2007), 245–267

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 rota- tion, coupled to thermodynamics and salinity diffusion-transport equations, which account for the buoyancy forces and stratification effects under the Boussinesq approximation. Moreover, and due to the shallowness of the oceans and the atmosphere, i.e., the depth of the fluid layer is very small in compar- ison 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., [20], [21], [22], [23], [24], [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 ac- celeration 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], [22], [23], [25], [28], [29], [30], [31] and references therein). Some of these models have also been the subject of analytical mathematical study (see, e.g., [2], [3], [5], [6], [9], [11], [12], [13], [15], [16], [17], [26], [27], [33], [34] and references therein). 246 CHONGSHENG CAO AND EDRISS S. TITI

In this paper we will focus on the 3D primitive equations in a cylindrical domain

Ω=M × (−h, 0),

where M is a smooth bounded domain in R2: ∂v ∂v (1) +(v ·∇)v + w + ∇p + fk × v + L v =0, ∂t ∂z 1 (2) ∂zp + T =0,

(3) ∇·v + ∂zw =0, ∂T ∂T (4) + v ·∇T + w + L T = Q, ∂t ∂z 2 where the horizontal velocity field v =(v1,v2), the three-dimensional velocity field (v1,v2,w), the temperature T and the pressure p are the unknowns. f = f0(β + y) is the Coriolis parameter and Q is a given heat source. The viscosity and the heat diffusion operators L1 and L2 are given by 2 − 1 − 1 ∂ (5) L1 = Δ 2 , Re1 Re2 ∂z 2 − 1 − 1 ∂ (6) L2 = Δ 2 , Rt1 Rt2 ∂z where Re1,Re2 are positive constants representing the horizontal and vertical Reynolds numbers, respectively, and Rt1,Rt2 are positive constants which stand for the horizontal and vertical heat diffusion, respectively. We set 2 2 =(∂x,∂y) to be the horizontal gradient operator and Δ = ∂x + ∂y to be the horizontal Laplacian. We observe that the above system is similar to the 3D Boussinesq system with the equation of vertical motion approximated by the hydrostatic balance. We partition the boundary of Ω into:

(7) Γu = {(x, y, z) ∈ Ω:z =0},

(8) Γb = {(x, y, z) ∈ Ω:z = −h},

(9) Γs = {(x, y, z) ∈ Ω:(x, y) ∈ ∂M, −h ≤ z ≤ 0}. We equip the system (1)–(4) with the following boundary conditions: wind- driven on the top surface and free-slip and non-heat ﬂux on the side walls and bottom (see, e.g., [20], [21], [22], [24], [25], [28], [29], [30]):

∂v ∂T ∗ (10) on Γ : = hτ, w =0, = −α(T − T ); u ∂z ∂z ∂v ∂T (11) on Γ : =0,w=0, =0; b ∂z ∂z ∂v ∂T (12) on Γ : v · n =0, × n =0, =0, s ∂n ∂n PRIMITIVE EQUATIONS 247 where τ(x, y) is the wind stress on the ocean surface, n is the normal vector ∗ to Γs, 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 (10)–(12), it is natural to assume that τ and T ∗ satisfy the compatibility boundary conditions: ∂τ (13) τ · n =0, × n =0, on ∂M. ∂n ∂T∗ (14) =0 on∂M. ∂n In addition, we supply the system with the initial condition:

(15) v(x, y, z, 0) = v0(x, y, z).

(16) T (x, y, z, 0) = T0(x, y, z). In [20], [21] and [33] the authors set up the mathematical framework to study the viscous primitive equations for the atmosphere and ocean circulation. Moreover, 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 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 3D viscous primitive equations model (1)–(16) in a general cylindrical domain, Ω, 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 3D Navier-Stokes equations the question of uniqueness of the weak solutions to this model is still open.

2. Preliminaries

2.1. New Formulation. First, let us reformulate the system (1)–(16) (see also [20], [21] and [33]). We integrate the equation (3) in the z direction to obtain z w(x, y, z, t)=w(x, y, −h, t) − ∇·v(x, y, ξ, t)dξ. −h By virtue of (10) and (11) we have z (17) w(x, y, z, t)=− ∇·v(x, y, ξ, t)dξ, −h 248 CHONGSHENG CAO AND EDRISS S. TITI and 0 0 (18) ∇·v(x, y, ξ, t)dξ = ∇· v(x, y, ξ, t)dξ =0. −h −h We denote 1 0 (19) φ(x, y)= φ(x, y, ξ)dξ, ∀ (x, y) ∈ M. h −h In particular, 1 0 (20) v(x, y)= v(x, y, ξ)dξ, in M, h −h denotes the barotropic mode. We will denote by (21) v = v − v, the baroclinic mode, that is the ﬂuctuation about the barotropic mode. Notice that (22) v =0. Based on the above and (12) we obtain (23) ∇·v =0, in M, and ∂v (24) v · n =0, × n =0, on ∂M. ∂n By integrating equation (2) we obtain z p(x, y, z, t)=− T (x, y, ξ, t)dξ + ps(x, y, t). −h Substituting (17) and the above relation into equation (1), we reach ∂v z ∂v (25) +(v ·∇)v − ∇·v(x, y, ξ, t)dξ ∂t −h ∂z z + ∇ps(x, y, t) −∇ T (x, y, ξ, t)dξ + fk × v + L1v =0. −h Remark 1. Notice that due to the compatibility boundary conditions (13) and (14) one can convert the boundary condition (10)–(12) to be homoge- (z+h)2−h3/3 ∗ neous by replacing (v, T)by(v + 2 τ,T + T ) while (23) is still true. For simplicity and without loss of generality we will assume that τ =0,T ∗ =0. However, we emphasize 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 ∗. PRIMITIVE EQUATIONS 249

Therefore, under the assumption that τ =0,T∗ = 0, we have the following new formulation for system (1)–(16): ∂v z ∂v (26) + L1v +(v ·∇)v − ∇·v(x, y, ξ, t)dξ ∂t −h ∂z z + ∇ps(x, y, t) −∇ T (x, y, ξ, t)dξ + fk × v =0, −h ∂T z ∂T (27) + L2T + v ·∇T − ∇·v(x, y, ξ, t)dξ = Q, ∂t −h ∂z ∂v ∂v ∂v (28) =0, =0,v· n| =0, × n =0, ∂z ∂z Γs ∂n z=0 z=−h Γs (29) (∂ T + αT )| =0; ∂ T | =0; ∂ T | =0, z z=0 z z=−h n Γs

(30) v(x, y, z, 0) = v0(x, y, z),

(31) T (x, y, z, 0) = T0(x, y, z).

2.2. Properties of v and v. By taking the average of equations (26) in the z direction, over the interval (−h, 0), and using the boundary conditions (28), we obtain the following equation for the barotropic mode ∂v z ∂v (32) + (v ·∇)v − ∇·v(x, y, ξ, t)dξ + ∇ps(x, y, t) ∂t − ∂z h 1 0 z 1 −∇ T (x, y, ξ, t)dξdz + fk × v − Δv =0. h −h −h Re1 As a result of (22), (23) and integration by parts, (33) z ∂v (v ·∇)v − ∇·v(x, y, ξ, t)dξ =(v ·∇)v + [(v ·∇)v +(∇·v) v]. −h ∂z By subtracting (32) from (26) and using (33) we obtain the following equation for the baroclinic mode (34) ∂v z ∂v + L1v +(v ·∇)v − ∇·v(x, y, ξ, t)dξ ∂t −h ∂z +(v ·∇)v +(v ·∇)v − [(v ·∇)v +(∇·v) v] z 1 0 z −∇ T (x, y, ξ, t)dξ − T (x, y, ξ, t)dξdz + fk × v =0. −h h −h −h Therefore, v satisﬁes the following equations and boundary conditions: 250 CHONGSHENG CAO AND EDRISS S. TITI ∂v 1 (35) − Δv +(v ·∇)v + [(v ·∇)v +(∇·v) v]+fk × v ∂t Re1 1 0 z + ∇ ps(x, y, t) − T (x, y, ξ, t) dξ dz =0, h −h −h (36) ∇·v =0, in M, ∂v (37) v · n =0, × n =0, on ∂M, ∂n and v satisﬁes the following equations and boundary conditions: ∂v z ∂v (38) + L1v +(v ·∇)v − ∇·v(x, y, ξ, t)dξ ∂t −h ∂z +(v ·∇)v +(v ·∇)v − [(v ·∇)v +(∇·v) v]+fk × v z 1 0 z −∇ T (x, y, ξ, t)dξ − T (x, y, ξ, t)dξdz =0, −h h −h −h ∂v ∂v ∂v (39) =0, =0, v · n| =0, × n =0. ∂z ∂z Γs ∂n z=0 z=−h Γs Remark 2. 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 L6 norm of v, which is a key estimate in our proof of the global regularity for the system (1)–(16). 2.3. Functional spaces and inequalities. Denote by L2(Ω),L2(M) and Hm(Ω),Hm(M) the usual L2-Lebesgue and Sobolev spaces, respectively ([1]). Let 1 | |p p ∈ p Ω φ(x, y, z) dxdydz , for every φ L (Ω), (40) φ p = 1 | |p p ∈ p M φ(x, y) dxdy , for every φ L (M). Now, ∞ ∂v ∂v V1 = v ∈ C (Ω) : =0, =0, ∂z ∂z − z=0 z= h ∂v v · n| =0, × n =0, ∇·v =0 , Γs ∂n Γ s ∞ ∂T ∂T ∂T V2 = T ∈ C (Ω) : =0; + αT =0; =0 . ∂z ∂z ∂n z=−h z=0 Γs 1 1 We denote by V1 and V2 the closure spaces of V1 in H (Ω), and V2 in H (Ω) under H1-topology, respectively. PRIMITIVE EQUATIONS 251

Definition 1. Let v0 ∈ V1 and T0 ∈ V2, and let T be a fixed positive time. (v, T) is called a strong solution of (26)–(31) on the time interval [0, T ]ifit satisfies (26) and (27) in a weak sense, and also 2 2 v ∈ C([0, T ],V1) ∩ L ([0, T ],H (Ω)), 2 2 T ∈ C([0, T ],V2) ∩ L ([0, T ],H (Ω)), dv ∈ L1([0, T ],L2(Ω)), dt dT ∈ L1([0, T ],L2(Ω)). dt For convenience, we recall the following Sobolev and Ladyzhenskaya’s inequalities in R2 (see, e.g., [1], [10], [14], [18]): ≤ 1/2 1/2 (41) φ L4(M) C0 φ L2 φ H1(M), ≤ 3/4 1/4 (42) φ L8(M) C0 φ L6(M) φ H1(M), for every φ ∈ H1(M), and the following Sobolev and Ladyzhenskaya’s inequalities in R3 (see, e.g., [1], [10], [14], [18]): ≤ 1/2 1/2 (43) u L3(Ω) C0 u L2(Ω) u H1(Ω), ≤ (44) u L6(Ω) C0 u H1(Ω), 1 for every u ∈ H (Ω). Here C0 is a positive constant which might depend on the shape of M and Ω but not on their size. Moreover, by (41) we get 12 3 4 3 2 3 2 (45) φ 12 = |φ| 4 ≤ C |φ| 2 |φ| 1 L (M) L (M) 0 L (M) H (M) ≤ 6 | |4 |∇ |2 12 C0 φ L6(M) φ φ dxdy + φ L6(M), M for every φ ∈ H1(M). Also, we recall the integral version of Minkowsky in- p m m equality for the L spaces, p ≥ 1. Let Ω1 ⊂ R 1 and Ω2 ⊂ R 2 be two measurable sets, where m1 and m2 are positive integers. Suppose that f(ξ,η) is measurable over Ω1 × Ω2. Then, p 1/p 1/p (46) |f(ξ,η)|dη dξ ≤ |f(ξ,η)|pdξ dη. Ω1 Ω2 Ω2 Ω1

3. A priori estimates

In the previous subsections we have reformulated the system (1)–(16) and obtained the system (26)–(31). 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 252 CHONGSHENG CAO AND EDRISS S. TITI

(v0,T0) be given initial data. In this section we will consider the strong solution 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∗ < ∞ then the H1 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).

3.1. L2 estimates. We take the inner product of equation (27) with T ,in L2(Ω), and obtain 2 1 d T 2 1 ∇ 2 1 2 2 + T 2 + Tz 2 + α T (z =0) 2 2 dt Rt1 Rt2 z ∂T = QT dxdydz − v ·∇T − ∇·v(x, y, ξ, t)dξ T dxdydz. Ω Ω −h ∂z After integrating by parts we get z ∂T (47) − v ·∇T − ∇·v(x, y, ξ, t)dξ T dxdydz =0. Ω −h ∂z As a result of the above we conclude 2 1 d T 2 1 ∇ 2 1 2 2 + T 2 + Tz 2 + α T (z =0) 2 2 dt Rt 1 Rt2

= QT dxdydz ≤Q2 T 2. Ω Notice that 2 ≤ 2 2 2 (48) T 2 2h Tz 2 +2h T (z =0) 2. Using (48) and the Cauchy-Schwarz inequality we obtain 2 d T 2 2 ∇ 2 1 2 2 (49) + T 2 + Tz 2 + α T (z =0) 2 dt Rt1 Rt2 h (50) ≤ 2(h2 Rt + )Q2. 2 α 2 By the inequality (48) and thanks to Gronwall inequality the above gives

− t 2 ≤ 2(h2 Rt +h/α) 2 2 2 2 (51) T 2 e 2 T0 2 +(2h Rt2 +2h/α) Q 2, Moreover, we have t 1 1 (52) ∇T (s)2 + T (s)2 + αT (z = 0)(s)2 ds Rt 2 Rt z 2 2 0 1 2 h ≤ 2 h2 Rt + Q2 t + T 2. 2 α 2 0 2 PRIMITIVE EQUATIONS 253

By taking the inner product of equation (26) with v,inL2(Ω), we reach 2 1 d v 2 1 ∇ 2 1 2 + v 2 + vz 2 2 dt Re 1 Re 2 z ∂v = − (v ·∇)v − ∇·v(x, y, ξ, t)dξ · v dxdydz − ∂z Ω h z + fk × v + ∇ps −∇ T (x, y, ξ, t)dξ · v dxdydz. Ω −h By integration by parts we get z ∂v (53) (v ·∇)v − ∇·v(x, y, ξ, t)dξ · v dxdydz =0. Ω −h ∂z By (36) we have

(54) ∇ps · v dxdydz = h ∇ps · v dxdy = −h ps(∇·v) dxdy =0. Ω M Ω Since (55) (fk × v) · v =0, then from (53)–(55) we have 2 1 d v 2 1 ∇ 2 1 2 + v 2 + vz 2 2 dt Re1 Re2 z = − T (x, y, ξ, t) dξ(∇·v) dxdydz ≤ hT 2 ∇v2. Ω −h By Cauchy-Schwarz and (51) we obtain 2 d v 2 1 ∇ 2 1 2 + v 2 + vz 2 dt Re1 Re2 ≤ 2 2 ≤ 2 2 2 2 2 h Re1 T 2 h Re1 T0 2 +(2h Rt2 +2h/α) Q 2 . Recall that (cf., e.g., [14, Vol. I p. 55]) 2 ≤ ∇ 2 v 2 CM v 2. By the above and thanks to Gronwall’s inequality we get − t 2 ≤ C Re1 2 2 (56) v 2 e M hv0 2 + v0 2 2 2 2 2 2 2 +CM h Re1 T0 2 +(2h Rt2 +2h/α) Q 2 . Moreover, t 1 ∇ 2 1 2 (57) v(s) 2 + vz(s) 2 ds 0 Re1 Re2 ≤ 2 2 2 2 2 2 2 h Re1 T0 2 +(2h Rt2 +2h/α) Q 2 t + h v0 2 + v0 2 . 254 CHONGSHENG CAO AND EDRISS S. TITI

Therefore, by (51), (52), (56) and (57) we have

(58) t 1 1 v(t)2 + ∇v(s)2 + v (s)2 ds + T (t)2 2 Re 2 Re z 2 2 0 1 2 t 1 ∇ 2 1 2 2 ≤ + T (s) 2 + Tz(s) 2 + α T (z = 0)(s) 2 ds K1(t), 0 Rt1 Rt2 where 2 2 2 2 (59) K1(t)=2(h Rt2 + h/α) Q 2 t + hv0 2 + v0 2 2 2 2 2 2 2 2 + 1+CM h Re1 + h Re1 t T0 2 +(2h Rt2 +2h/α) Q 2 .

3.2. L6 estimates. Taking the inner product of the equation (38) with |v|4v in L2(Ω), we get 6 1 dv 1 2 6 + |∇v|2|v|4 + ∇|v|2 |v|2 dxdydz 6 dt Re1 Ω 1 2 4 22 2 + |vz| |v| + ∂z|v| |v| dxdydz Re 2 Ω z ∂v = − (v ·∇)v − ∇·v(x, y, ξ, t)dξ +(v ·∇)v +(v ·∇)v Ω −h ∂z −[(v ·∇)v +(∇·v) v]+fk × v z 1 0 z −∇ T (x, y, ξ, t)dξ − T (x, y, ξ, t)dξdz ·|v|4v dxdydz. −h h −h −h Integrating by parts we get z ∂v (60) − (v ·∇)v − ∇·v(x, y, ξ, t)dξ ·|v|4v dxdydz =0. Ω −h ∂z Since (61) fk × v ·|v|4v =0, then by (36) and the boundary condition (28) we also have (62) (v ·∇)v ·|v|4v dxdydz =0. Ω Thus, by (60)–(62), 6 1 dv 1 2 6 + |∇v|2|v|4 + ∇|v|2 |v|2 dxdydz 6 dt Re 1 Ω 1 2 4 22 2 + |vz| |v| + ∂z|v| |v| dxdydz Re2 Ω PRIMITIVE EQUATIONS 255 = − (v ·∇)v − (v ·∇)v +(∇·v) v Ω z 1 0 z −∇ T (x, y, ξ, t)dξ − T (x, y, ξ, t)dξdz ·|v|4v dxdydz. −h h −h −h

Notice that by integration by parts and boundary condition (28), − (v ·∇)v − [(v ·∇)v +(∇·v) v] Ω z 1 0 z −∇ T (x, y, ξ, t)dξ − T (x, y, ξ, t)dξdz ·|v|4v dxdydz − h − − h h h ∇· ·||4 ·∇ ||4 · − kj ||4j = ( v) v v v +(v )( v v) v v v ∂xk ( v v ) Ω z 1 0 z − T (x, y, ξ, t)dξ − T (x, y, ξ, t)dξdz ∇·(|v|4v) dxdydz. −h h −h −h

Therefore, by Cauchy-Schwarz inequality and H¨older inequality we obtain 6 1 dv 1 2 6 + |∇v|2|v|4 + ∇|v|2 |v|2 dxdydz 6 dt Re 1 Ω 1 2 + |v |2|v|4 + ∂ |v|2 |v|2 dxdydz Re z z 2 Ω 0 ≤ C |v| |∇v||v|5 dz dxdy M −h 0 0 +C |v|2 dz |∇v||v|4 dz dxdy M −h −h 0 +C |T | |∇v||v|4 dz dxdy M −h 0 1/2 0 1/2 ≤ C |v| |∇v|2 |v|4 dz |v|6 dz dxdy M −h −h 0 0 1/2 0 1/2 +C |v|2 dz |∇v|2 |v|4 dz |v|4 dz dxdy M −h −h −h 0 1/2 0 1/2 +C |T | |∇v|2 |v|4 dz |v|4 dz dxdy M −h −h 1/4 1/2 0 2 ≤ |∇|2 ||4 ||6 C v L4(M) v v dxdydz v dz dxdy Ω M −h 1/4 0 4 +C |v|2 dz dxdy M −h 256 CHONGSHENG CAO AND EDRISS S. TITI 1/4 1/2 0 2 × |∇v|2 |v|4 dxdydz |v|4 dz dxdy Ω M −h 1/4 1/2 0 2 | | |∇|2 ||4 ||4 +C T L4(M) v v dxdydz v dz dxdy . Ω M −h By using Minkowsky inequality (46), we get 1/2 0 2 0 1/2 |v|6 dz dxdy ≤ C |v|12 dxdy dz. M −h −h M By (45), 2 12 6 4 2 6 |v| dxdy ≤ C0 |v| dxdy |v| |∇v| dxdy + |v| dxdy . M M M M Thus, by Cauchy-Schwarz inequality, (63) 1/2 0 2 1/2 ||6 ≤ 3 ||4|∇|2 6 v dz dxdy C v L6(Ω) v v dxdydz + v L6(Ω). M −h Ω Similarly, by (46) and (42), (64) 1/2 0 2 0 1/2 |v|4 dz dxdy ≤ C |v|8 dxdy dz M −h −h M 0 ≤ 3 ∇ ≤ 3 ∇ C v L6(M) v L2(M) + v L2(M) dz C v 6 ( v 2 + v 2) , −h and (65) 1/4 0 4 0 1/4 |v|2 dz dxdy ≤ C |v|8 dxdy dz M −h −h M 0 ≤ 3/2 ∇1/2 1/2 C v L6(M) v L2(M) + v L2(M) dz −h ≤ 3/2 ∇1/2 1/2 C v 6 v 2 + v 2 . Therefore, by (63)–(65) and (41), 6 1 dv 1 2 6 + |∇v|2|v|4 + ∇|v|2 |v|2 dxdydz 6 dt Re1 Ω 1 2 4 22 2 + |vz| |v| + ∂z|v| |v| dxdydz Re2 Ω PRIMITIVE EQUATIONS 257 3/4 ≤ 1/2 ∇ 1/23/2 |∇|2 ||4 1/2 ∇ 1/26 C v 2 v 2 v 6 v v dxdydz + C v 2 v 2 v 6 Ω 1/2 3 ∇ |∇|2 ||4 +C v 6 ( v 2 + v 2) v v dxdydz Ω 1/2 1/2 ∇ 1/23/2 ∇1/2 1/2 |∇|2 ||4 +C T 2 T 2 v 6 v 2 + v 2 v v dxdydz . Ω Thanks to the Young and the Cauchy-Schwarz inequalities, 6 dv 1 2 6 + |∇v|2|v|4 + ∇|v|2 |v|2 dxdydz dt Re1 Ω 1 2 4 22 2 + |vz| |v| + ∂z|v| |v| dxdydz Re2 Ω ≤ 2 ∇ 26 6∇2 2 ∇ 2 26 C v 2 v 2 v 6 + C v 6 v 2 + C T 2 T 2 + C v 2 v 6. By (58) and Gronwall inequality, we get t 6 1 |∇|2||4 (66) v(t) 6 + v v dxdydz 0 Re1 Ω 1 2 4 + |vz| |v| dxdydz ≤ K6(t), Re2 Ω where 2 K1 (t) 6 2 (67) K6(t)=e v0 H1(Ω) + K1 (t) . Taking the inner product of the equation (27) with |T |4T in L2(Ω), and by (27), we get 1 dT 6 5 6 + |∇T |2|T |4 dxdydz 6 dt Rt1 Ω 5 | |2| |4 6 + Tz T dxdydz + α T (z =0) 6 Rt2 Ω = Q|T |4T dxdydz Ω z ∂T − v ·∇T − ∇·v(x, y, ξ, t)dξ |T |4T dxdydz. Ω −h ∂z By integration by parts and (36), z ∂T (68) − v ·∇T − ∇·v(x, y, ξ, t)dξ |T |4T dxdydz =0. Ω −h ∂z As a result of the above we conclude 6 1 d T 6 5 2 4 5 2 4 + |∇T | |T | dxdydz + |Tz| |T | dxdydz 6 dt Rt1 Ω Rt2 Ω 6 | |4 ≤ 5 + α T (z =0) 6 = Q T T dxdydz Q 6 T 6. Ω 258 CHONGSHENG CAO AND EDRISS S. TITI

By Gronwall, again, ≤ (69) T (t) 6 Q H1(Ω) t + T0 H1(Ω).

3.3. H1 estimates.

3.3.1. ∇v2 estimates. First, we observe that since v is a strong solution on the interval [0, T∗) then Δv ∈ L2([0, T∗),L2(M)). Consequently, and by − virtue of (36), Δv ·n ∈ L2([0, T∗),H 1/2(∂M)) (see, e.g., [10], [32]). Moreover, and thanks to (36) and (37), we have Δv · n =0on∂M (see, e.g., [35]). This observation implies also that the Stokes operator in the domain M, 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], [32]) to conclude:

∇ps(x, y, t) · Δv(x, y, t)dxdy =0. M By taking the inner product of equation (35) with −Δv in L2(M), and applying (36) and the above, we reach 1 d∇v2 1 2 + Δv2 2 dt Re 2 1 = (v ·∇)v + [(v ·∇)v +(∇·v) v] · Δv dxdy + fk × v · Δv dxdy. M M Following similar steps as in the proof of 2D Navier-Stokes equations (cf. e.g., [10], [32]) one obtains ·∇ · ≤ 1/2∇ 3/2 (v )v Δv dxdy C v 2 v 2 Δv 2 . M Applying the Cauchy-Schwarz and H¨older inequalities, we get 0 (v ·∇)v +(∇·v) v · Δv dxdy ≤ C |v||∇v| dz |Δv| dxdy M M −h 0 1/2 0 1/2 ≤ C |v|2 |∇v| dz |∇v| dz |Δv| dxdy M −h −h 1/4 0 2 ≤ C |v|2 |∇v| dz dxdy M −h 1/4 0 2 1/2 × |∇v| dz dxdy |Δv|2 dxdy M −h M 1/4 ≤ ∇1/2 ||4|∇|2 C v 2 v v dxdydz Δv 2. Ω PRIMITIVE EQUATIONS 259

Thus, by Young’s and Cauchy-Schwarz inequalities, ∇ 2 d v 2 1 2 ≤ 2∇ 4 + Δv 2 C v 2 v 2 dt Re1 ∇2 ||4|∇|2 2 +C v 2 + C v v dxdydz + C v 2. Ω By (58), (66) and thanks tothe Gronwall inequality, we obtain t ∇ 2 1 | |2 ≤ (70) v 2 + Δv 2 ds K2(t), Re1 0 where 2 K1 (t) 2 (71) K2(t)=e v0 H1(Ω) + K1(t)+K6(t) .

3.3.2. vz2 estimates. Since u = vz, it is clear that u satisﬁes ∂u z ∂u (72) + L1u +(v ·∇)u − ∇·v(x, y, ξ, t)dξ ∂t −h ∂z +(u ·∇)v − (∇·v)u + fk × u −∇T =0. Taking the inner product of the equation (72) with u in L2 and using the boundary condition (28), we get 2 1 d u 2 1 ∇ 2 1 2 + u 2 + ∂zu 2 2 dt Re1 Re2 z ∂u = − (v ·∇)u − ∇·v(x, y, ξ, t)dξ · u dxdydz − ∂z Ω h − (u ·∇)v − (∇·v)u + fk × u −∇T · u dxdydz. Ω From integration by parts we get z ∂u (73) − (v ·∇)u − ∇·v(x, y, ξ, t)dξ · u dxdydz =0. Ω −h ∂z Since (74) (fk × u) · u =0, then by (73) and (74) we have 2 1 d u 2 1 ∇ 2 1 2 + u 2 + ∂zu 2 2 dt Re1 Re2 = − ((u ·∇)v − (∇·v)u −∇T ) · u dxdydz Ω

≤ C (|v|) |u||∇u| dxdydz + T 2 ∇u2 Ω 260 CHONGSHENG CAO AND EDRISS S. TITI

≤ Cv6 u3∇u2 + T 2 ∇u2 ≤ 1/2∇ 3/2 ∇ C v 6 u 2 u 2 + T 2 u 2. By Young’s inequality and Cauchy-Schwarz inequality, we have 2 d u 2 1 ∇ 2 1 2 ≤ 4 2 2 + u 2 + ∂zu 2 C v 6 u 2 + C T 2 dt Re1 Re2 ≤ ∇ 4 4 2 2 C v 2 + v 6 u 2 + C T 2. By (58), (66), (70), and Gronwall inequality, t t 2 1 ∇ 2 1 2 ≤ (75) vz 2 + vz(s) 2 + vzz(s) 2 ds Kz(t), Re1 0 Re2 0 where 2 2/3 (K2 (t)+K6 (t))t 2 (76) Kz(t)=e v0 H1(Ω) + K1(t) .

3.3.3. ∇v2 estimates. By taking the inner product of equation (26) with −Δv in L2(Ω), we reach ∇ 2 1 d v 2 1 2 1 ∇ 2 + Δv 2 + vz 2 2 dt Re1 Re2 z ∂v = − (v ·∇)v − ∇·v(x, y, ξ, t)dξ − ∂z Ω h z +fk × v + ∇ps −∇ T (x, y, ξ, t)dξ · Δv dxdydz −h 0 0 ≤ C |v||∇v| + |∇v| dz|vz| + |∇T | dz |Δv| dxdydz Ω −h −h

≤ Cv 6 ∇v 3 Δv L (Ω) L (Ω) 2 0 0 +C |∇v| dz |vz||Δv| dz dxdy + C∇T 2 Δv2. M −h −h

Notice that by applying Proposition 2.2 in [5] with u = v, f =Δv and g = vz, we get 0 0 |∇v| dz |vz||Δv| dz dxdy M −h −h ≤ ∇ 1/2 1/2∇ 1/2 3/2 C v 2 vz 2 vz 2 Δv 2 . As a result and by (43) and (44), we obtain ∇ 2 1 d v 2 1 2 1 ∇ 2 + Δv 2 + vz 2 2 dt Re1 Re2 ≤ ∇ 1/2 1/2 ∇ 1/2 3/2 ∇ C v L6(Ω) + v 2 vz 2 v 2 Δv 2 + h T 2 Δv 2. PRIMITIVE EQUATIONS 261

Thus, by Young’s inequality and Cauchy-Schwarz inequality,

∇ 2 d v 2 1 2 1 ∇ 2 + Δv 2 + vz 2 dt Re1 Re2 ≤ 4 ∇ 2 2 ∇ 2 ∇ 2 C v L6(Ω) + v 2 vz 2 v 2 + C T 2. By (58), (66), (70), (75) and thanks to Gronwall inequality, we obtain t ∇ 2 1 2 1 ∇ 2 ≤ (77) v 2 + Δv(s) 2 + vz(s) 2 ds KV (t), 0 Re1 Re2 where 2/3 K6 (t) t+K1(t) Kz (t) 2 (78) KV (t)=e v0 H1(Ω) + K1(t) .

3.3.4. T H1 estimates. Taking the inner product of equation (27) with 2 −ΔT − Tzz in L (Ω), we get ∇ 2 2 ∇ 2 1 d T 2 + Tz 2 + α T (z =0) 2 2 dt 1 1 1 1 + ΔT 2 + + ∇T 2 + α∇T (z =0)2 + T 2 Rt 2 Rt Rt z 2 2 Rt zz 2 1 1 2 2 z = v ·∇T − ∇·vdξ Tz − Q [ΔT + Tzz] dxdydz Ω −h

≤ C (|v||∇T | + |Q|) |ΔT + Tzz| dxdydz Ω 0 0 + |∇v| dz |Tz||ΔT + Tzz| dz dxdy − − M h h ≤ ∇ 2 ∇ 2 2 1/2 C v 6 T 3 ΔT 2 + Tz 2 + Tzz 2 ∇ 1/2 1/2 1/2 2 ∇ 2 2 3/2 +C v Δv Tz ΔT 2 + Tz 2 + Tzz 2 2 2 2 2 ∇ 2 2 1/2 + Q 2 ΔT 2 + Tz 2 + Tzz 2 ≤ ∇ 1/2 ∇ 1/2 1/2 1/2 2 ∇ 2 2 3/2 C v 6 T 2 + v 2 Δv 2 Tz 2 ΔT 2 + Tz 2 + Tzz 2 2 ∇ 2 2 1/2 + Q 2 ΔT 2 + Tz 2 + Tzz 2 .

By Young’s inequality and Cauchy-Schwarz inequality, ∇ 2 2 ∇ 2 d T 2 + Tz 2 + α T (z =0) 2 dt 1 2 1 1 ∇ 2 ∇ 2 1 2 + ΔT 2 + + Tz 2 + α T (z =0) 2 + Tzz 2 Rt 1 Rt1 Rt2 Rt2 ≤ 4 ∇ 2 2 ∇ 2 2 2 C v 6 + v 2 Δv 2 T 2 + Tz 2 + C Q 2. By (66), (77), and Gronwall inequality, we get 262 CHONGSHENG CAO AND EDRISS S. TITI t ∇ 2 2 ∇ 2 1 2 (79) T 2 + Tz 2 + α T (z =0) 2 + ΔT 2 0 Rt1 1 1 ∇ 2 ∇ 2 1 2 ≤ + + Tz 2 + α T (z =0) 2 Tzz 2 ds Kt, Rt1 Rt2 Rt2 where 2 2 K6 (t) t+KV (t) 2 2 (80) Kt = e T0 H1(Ω) + Q 2 .

4. Existence and uniqueness of the strong solutions

In this section we will use the a priori estimates (58)–(79) to show the global existence and uniqueness, i.e. global regularity, of strong solutions to the system (26)–(31).

1 Theorem 2. Let Q ∈ H (Ω), v0 ∈ V1, T0 ∈ V2 and T > 0, be given. Then there exists a unique strong solution (v, T) of the system (26)–(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 (v, T) be the strong solution corresponding to the initial data (v0,T0) with maximal interval of existence [0, T∗). If we assume that T∗ < ∞ then it is clear that ∞ lim sup v H1(Ω) + T H1(Ω) = . − 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. Next, we show the continuous dependence on the initial data and the the uniqueness of the strong solutions. Let (v1,T1) and (v2,T2) be two strong solutions of the system (26)–(31) with corresponding pressures (ps)1 and (ps)2, and initial data ((v0)1, (T0)1) and ((v0)2, (T0)2), respectively. Denote by u = v1 − v2,qs =(ps)1 − (ps)2 and θ = T1 − T2. It is clear that ∂u (81) + L1u +(v1 ·∇)u +(u ·∇)v2 ∂t z z ∂u ∂v2 − ∇·v1(x, y, ξ, t)dξ − ∇·u(x, y, ξ, t)dξ − ∂z − ∂z h h z +fk × u + ∇qs −∇ θ(x, y, ξ, t)dξ =0, −h ∂θ z ∂θ (82) + L2θ + v1 ·∇θ + u ·∇T2 − ∇·v1(x, y, ξ, t)dξ ∂t − ∂z h z ∂T − ∇·u(x, y, ξ, t)dξ 2 =0, −h ∂z PRIMITIVE EQUATIONS 263

(83) u(x, y, z, t)=(v0)1 − (v0)2,

(84) θ(x, y, z, 0) = (T0)1 − (T0)2. By taking the inner product of equation (81) with u in L2(Ω), and equation (82) with θ in L2(Ω), we get 2 1 d u 2 1 ∇ 2 1 2 + u 2 + uz 2 2 dt Re1 Re2 z ∂u = − (v1 ·∇)u +(u ·∇)v2 − ∇·v1(x, y, ξ, t)dξ − ∂z Ω h z ∂v − ∇·u(x, y, ξ, t)dξ 2 · u dxdydz − ∂z h z − fk × u + ∇qs −∇ θ(x, y, ξ, t)dξ · u dxdydz, Ω −h and 2 1 d θ 2 1 ∇ 2 1 2 2 + θ 2 + θz 2 + α θ(z =0) 2 2 dt Rt1 Rt2 z ∂θ = − v1 ·∇θ + u ·∇T2 − ∇·v1(x, y, ξ, t)dξ − ∂z Ω h z ∂T − ∇·u(x, y, ξ, t)dξ 2 θ dxdydz. −h ∂z By integration by parts, and the boundary conditions (28) and (29), we get z ∂u (85) − (v1 ·∇)u − ∇·v1(x, y, ξ, t)dξ · u dxdydz =0, − ∂z Ω h z ∂θ (86) − v1 ·∇θ − ∇·v1(x, y, ξ, t)dξ · θ dxdydz =0. Ω −h ∂z Since (87) fk × u · u =0, and by (85), (86) and (87) we have 2 1 d u 2 1 ∇ 2 1 2 + u 2 + uz 2 2 dt Re1 Re2 z ∂v2 = − (u ·∇)v2 · u dxdydz + ∇·u(x, y, ξ, t)dξ · u dxdydz, Ω Ω −h ∂z then 2 1 d θ 2 1 ∇ 2 1 2 2 + θ 2 + θz 2 + α θ(z =0) 2 2 dt Rt1 Rt2 z ∂T2 = − (u ·∇)T2θ dxdydz + ∇·u(x, y, ξ, t)dξ θ dxdydz. Ω Ω −h ∂z 264 CHONGSHENG CAO AND EDRISS S. TITI

Notice that (88) (u ·∇)v2 · u dxdydz ≤∇v22u3u6 Ω ≤ ∇ 1/2∇ 3/2 C v2 2 u 2 u 2 , (89) (u ·∇)T2θ dxdydz ≤∇v22θ3u6 Ω ≤ ∇ 1/2∇ 1/2∇ C T2 2 θ 2 θ 2 u 2. Moreover, z ∂v2 ∇·u(x, y, ξ, t)dξ · u dxdydz − ∂z Ω h 0 0 ≤ |∇u| dz |∂zv2||u| dz dxdy M −h −h 0 0 1/2 0 1/2 2 2 ≤ |∇u| dz |∂zv2| dz |u| dz dxdy M −h −h −h 1 0 2 2 ≤ |∇u| dz dxdy M −h 1 1 0 2 4 0 2 4 2 2 × |∂zv2| dz dxdy |u| dz dxdy . M −h M −h By Cauchy-Schwarz inequality, 1/2 0 2 (90) |∇u| dz dxdy ≤ C∇u2. M −h By using Minkowsky inequality (46) and (41), we obtain 1/2 0 2 0 1/2 (91) |u|2 dz dxdy ≤ C |u|4 dxdy dz M −h −h M 0 ≤ C |u||∇u| dz ≤ Cu2∇u2, −h and 1/2 0 2 0 1/2 2 4 (92) |∂zv2| dz dxdy ≤ C |∂zv2| dxdy dz M −h −h M 0 ≤ C |∂zv2||∇∂zv2| dz ≤ C∂zv22∇∂zv22. −h PRIMITIVE EQUATIONS 265

Similarly, z ∂T2 (93) ∇·u(x, y, ξ, t)dξ θ dxdydz Ω −h ∂z ≤ ∇ 1/2∇ 1/2 1/2∇ 1/2 C u 2 ∂zT2 2 ∂zT2 2 θ 2 θ 2 . Therefore, by estimates (88)–(93), we reach d u2 + θ2 1 2 2 1 ∇ 2 1 2 1 ∇ 2 + u 2 + uz 2 + θ 2 2 dt Re1 Re2 Rt1 1 2 2 + θz 2 + α θ(z =0) 2 Rt 2 ≤ ∇ 1/2∇ 1/2 1/2∇ 3/2 C v2 2 + ∂zv2 2 ∂zv2 2 u u 2 ∇ 1/2∇ 1/2∇ +C T2 2 θ 2 θ 2 u 2 ∇ 1/2∇ 1/2 1/2∇ 1/2 +C u 2 ∂zT2 2 ∂zT2 2 θ 2 θ 2 . By Young’s inequality, we get

2 d u 2 ≤ ∇ 4 ∇ 4 2∇ 2 2∇ 2 C v2 2 + T2 2 + ∂zv2 2 ∂zv2 2 + ∂zT2 2 ∂zT2 2 dt × 2 2 u 2 + θ 2 . Thanks to Gronwall inequality, u(t)2 + θ(t)2 ≤ u(t =0)2 + θ(t =0)2 2 2 2 2 t × ∇ 4 ∇ 4 2∇ 2 exp C v2(s) 2 + T2(s) 2 + ∂zv2(s) 2 ∂zv2(s) 2 0 2∇ 2 + ∂zT2(s) 2 ∂zT2(s) 2 ds .

Since (v2,T2) is a strong solution, 2 2 u(t) 2 + θ(t) 2 ≤ 2 2 { 2 2 2 } u(t =0) 2 + θ(t =0) 2 exp C KV t + Kt t + KzKV + Kt . The above inequality proves the continuous dependence of the solutions on the initial data; in particular, when u(t =0)=θ(t =0)=0,wehaveu(t)=θ(t) = 0, for all t ≥ 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. DMS-0204794 and DMS-0504619, the MAOF 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 Math- ematical Sciences. 266 CHONGSHENG CAO AND EDRISS S. TITI

Florida International University, Miami, FL E-mail address: caoc@ﬁu.edu Dept. of Mathematics and Dept. of Mechanical and Aerospace Engineering, University of California, Irvine, CA and Dept. of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel E-mail addresses: [email protected], [email protected]

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