GLOBAL WELL-POSEDNESS OF STRONG SOLUTIONS TO THE 3D PRIMITIVE EQUATIONS WITH HORIZONTAL EDDY DIFFUSIVITY 4 1 CHONGSHENG CAO, JINKAI LI, AND EDRISS S. TITI 0 2 n Abstract. In this paper, we consider the initial-boundary value problem of the a 3Dprimitive equations for oceanic andatmospheric dynamics with only horizontal J diffusion in the temperature equation. Global well-posedness of strong solutions 6 are established with H2 initial data. ] P A . 1. Introduction h t a The primitive equations derived from the Boussinisq system of incompressible flow m are fundamental models for weather prediction, see, e.g., Lewandowski [12], Majda [ [16], Pedlosky [17], Vallis [21], and Washington and Parkinson [22]. Due to their 1 importance, the primitive equations has been studied analytically by many authors, v see, e.g., [13, 14, 16, 18, 20] and the references therein. 4 In this paper, we consider the viscous primitive equations with only horizontal 3 2 diffusion in the temperature equation: 1 . ∂ v +(v )v +w∂ v + p ∆v +f k v = 0, (1.1) 1 t H z H 0 ·∇ ∇ − × 0 ∂ p+T = 0, (1.2) 4 z 1 v +∂ w = 0, (1.3) H z : ∇ · v ∂ T +(v )T +w∂ T ∆ T = 0, (1.4) t H z H i ·∇ − X where the horizontal velocity v = (v1,v2), the vertical velocity w, the temperature T r a and the pressure p are the unknowns and f is the Coriolis parameter. In this paper, 0 we use the notations = (∂ ,∂ ) and ∆ = ∂2 + ∂2 to denote the horizontal ∇H x y H x y gradient and the horizontal Laplacian, respectively. The dominat horizontal eddy diffusivity in this model is justified by some geophysicists due to the strong horizontal turbulent mixing. In 1990s, Lions, Temam and Wang [13–15] initialed the mathematical studies on the primitive equations, where among other issues they established the global ex- istence of weak solutions. The uniqueness of weak solutions for 2D case was later proven by Bresch, Guill´en-Gonz´alez, Masmoudi and Rodr´ıguez-Bellido [1]; however, Date: January 6, 2014. 1991 Mathematics Subject Classification. AMS 35Q35,76D03, 86A10. Key words and phrases. well-posedness; strong solution; primitive equation. 1 2 CHONGSHENGCAO, JINKAILI, ANDEDRISSS. TITI the uniqueness of weak solutions in the three-dimensional case is still unclear. Lo- cal existence of strong solutions was obtained by Guill´en-Gonz´alez, Masmoudi and Rodr´ıguez-Bellido [8]. Global existence of strong solutions for 2D case was estab- lished by Bresch, Kazhikhov and Lemoine in [2] and Temam and Ziane in [20], while the 3D case was established by Cao and Titi [6]. Global strong solutions for 3D case were also obtained by Kobelkov [9] later by using a different approach, see also the subsequent articles Kukavica and Ziane [10, 11]. The systems considered in all the papers [6, 9–11] are assumed to have diffusion in all directions. It is proven by Cao and Titi [7] that these global existence results still hold true for the system with only vertical diffusion, provided the local in time strong solutions exist. As the complement of [7], local existence results for the system with only vertical diffusion are recently given by Cao, Li and Titi [4] with H2 initial data. Notably, the inviscid primitive equation, with or without coupling to the heat equation has been shown by Cao et al [3] to blow up in finite time (see also [23]). In this paper, we consider the primitive equations with only horizontal diffusion in the temperature equation, i.e. system (1.1)–(1.4). The aim of this paper is to show that the strong solutions exist globally for system (1.1)–(1.4) subject to some initial and boundary conditions. More precisely, we consider the problem in the domain Ω = M ( h,0), with M = (0,1) (0,1), and supplement system (1.1)–(1.4) with 0 × − × the following boundary and initial conditions: v,w and T are periodic in x and y, (1.5) (∂ v,w) = (0,0), T = 1, T = 0, (1.6) z z=−h,0 z=−h z=0 | | | (v,T) = (v ,T ). (1.7) t=0 0 0 | Replacing T and p by T + z and p z2, respectively, then system (1.1)–(1.4) with h − 2h (1.5)–(1.7) is equivalent to the following system ∂ v +(v )v +w∂ v + p ∆v +f k v = 0, (1.8) t H z H 0 ·∇ ∇ − × ∂ p+T = 0, (1.9) z v +∂ w = 0, (1.10) H z ∇ · ∂ T +(v )T +w ∂ T + 1 ∆ T = 0, (1.11) t ·∇H z h − H (cid:0) (cid:1) subject to the boundary and initial conditions v,w,T are periodic in x and y, (1.12) (∂ v,w) = 0, T = 0, (1.13) z z=−h,0 z=−h,0 | | (v,T) = (v ,T ). (1.14) t=0 0 0 | Here, for simplicity, we still use T to denote the initial temperature in (1.14), though 0 it is now different from that in (1.7). Notice that the periodic subspace , given by H := (v,w,p,T) v,w,p and Tare spatially periodic in all three variables H { | STRONG SOLUTIONS TO THE 3D PRIMITIVE EQUATIONS 3 and even, odd, even and odd in z variable, respectively , } is invariant under the dynamics system (1.8)–(1.11). That is if the initial data satisfy the properties stated in the definition of , then, as we will see later (see Theorem H 1.1), the solutions to system (1.8)–(1.11) will obey the same symmetry as the initial data. This motivated us to consider the following system ∂ v +(v )v +w∂ v + p ∆v +f k v = 0, (1.15) t H z H 0 ·∇ ∇ − × ∂ p+T = 0, (1.16) z v +∂ w = 0, (1.17) H z ∇ · ∂ T +(v )T +w ∂ T + 1 ∆ T = 0, (1.18) t ·∇H z h − H (cid:0) (cid:1) in Ω := M ( h,h), subject to the boundary and initial conditions × − v,w,p and T are periodic in x,y,z, (1.19) v and p are even in z, and w and T are odd in z, (1.20) (v,T) = (v ,T ). (1.21) t=0 0 0 | Onecaneasilycheckthattherestrictiononthesub-domainΩ ofasolution(v,w,p,T) 0 to system (1.15)–(1.21) is a solution to the original system (1.8)–(1.14). Because of this, throughout this paper, we mainly concern on the study of system (1.15)–(1.21) defined on Ω, while the well-posedness results for system (1.8)–(1.14) defined on Ω 0 follow as a corollary of those for system (1.15)–(1.21). For any function φ(x,y,z) defined on Ω, we define functions φ¯ and φ˜ as follows 1 h φ¯(x,y) = φ(x,y,z)dz, φ˜= φ φ¯. 2h Z − −h Using these notations, system (1.15)–(1.21) is equivalent to (see [7] for example) ∂ v ∆v +(v )v z v(x,y,ξ,t)dξ ∂ v t − ·∇H − −h∇H · z (cid:16) (cid:17) R +f k v + p (x,y,t) z T(x,y,ξ,t)dξ = 0, (1.22) 0 × ∇H s − −h (cid:16) (cid:17) R v¯ = 0, (1.23) H ∇ · ∂ T ∆ T +v T z v(x,y,ξ,t)dξ ∂ T + 1 = 0, (1.24) t − H ·∇H − −h∇H · z h (cid:16) (cid:17) R (cid:0) (cid:1) complemented with the following boundary and initial conditions v and T are periodic in x,y,z, (1.25) v and T are even and odd in z, respectively, (1.26) (v,T) = (v ,T ). (1.27) t=0 0 0 | Throughoutthispaper, weuseLq(Ω),Lq(M)andWm,q(Ω),Wm,q(M)todenotethe standard Lebesgue and Sobolev spaces, respectively. For q = 2, we use Hm instead of Wm,2. We use Wm,q(Ω) and Hm to denote the spaces of periodic functions in per per 4 CHONGSHENGCAO, JINKAILI, ANDEDRISSS. TITI Wm,q(Ω) and Hm(Ω), respectively. For simplicity, we use the same notations Lp and Hm to denote the N product spaces (Lp)N and (Hm)N, respectively. We always use u to denote the Lp norm of u. p k k Definitions of strong solution, maximal existence time and global strong solution to system (1.22)–(1.27) are given by the following three definitions, respectively. Definition 1.1. Given a positive number t . Let v H2(Ω) and T H2(Ω) be two 0 0 0 ∈ ∈ periodic functions, such that they are even and odd in z, respectively. A couple (v,T) is called a strong solution to system (1.22)-(1.27) (or equivalently (1.15)–(1.21)) on Ω (0,t ) if 0 × (i) v and T are periodic in x,y,z, and they are even and odd in z, respectively; (ii) v and T have the regularities v L∞(0,t ;H2(Ω)) C([0,t ];H1(Ω)) L2(0,t ;H3(Ω)), 0 0 0 ∈ ∩ ∩ T L∞(0,t ;H2(Ω)) C([0,t ];H1(Ω)), T L2(0,t ;H2(Ω)), 0 0 H 0 ∈ ∩ ∇ ∈ ∂ v L2(0,t ;H1(Ω)), ∂ T L2(0,t ;H1(Ω)); t 0 t 0 ∈ ∈ (iii) v and T satisfies (1.22)–(1.24) a.e. in Ω (0,t ) and the initial condition 0 × (1.27). Definition 1.2. A finite positive number ∗ is called the maximal existence time of T a strong solution (v,T) to system (1.22)–(1.27) if (v,T) is a strong solution to the system on Ω (0,t ) for any t < ∗ and lim ( v 2 + T 2 ) = . × 0 0 T t→T−∗ k kH2 k kH2 ∞ Definition 1.3. A couple (v,T) is called a global strong solution to system (1.22)– (1.27) if it is a strong solution on Ω (0,t ) for any t < . 0 0 × ∞ The main result of this paper is the following global existence result. Theorem 1.1. Suppose that the periodic functions v ,T H2(Ω) are even and odd 0 0 ∈ in z, respectively. Then system (1.22)-(1.27) (or equivalently (1.15)–(1.21)) has a unique global strong solution (v,T). Local existence of strong solutions are obtained by a regularization mechanism. Moreprecisely, weaddthevertical diffusionterm, withadiffusioncoefficient ε > 0, in the temperature equation to obtain a regularized system. We then establish uniform estimates, in ε, for strong solutions of the regularized system, over a short interval of timeindependent ofε, andthentakethelimit, asεgoestozero, toobtainlocalstrong solutionstosystem(1.22)–(1.27). Toobtaintheglobalstrongsolutions, fromthelocal existence results, we need to establish the a priori estimates on the derivatives of the solution, up to the second order. The first crucial estimate is the L6 estimate on v, which has been originally obtained by Cao and Titi in [6, 7]. Next, we establish the estimates on the derivatives. Resulting from the lack of sufficient information on the equation for the vertical velocity w, and the absence of the vertical diffusion in the temperature equation, the treatments of different derivatives will vary. In STRONG SOLUTIONS TO THE 3D PRIMITIVE EQUATIONS 5 particular, when we deal with the derivatives of v of the same order, we first work on the vertical derivatives and then the horizontal ones. The reason for this is due to the fact that we need the estimates on the vertical derivatives to handle the term of the form z vdξ ∂ v, which has ”stronger nonlinearity” than the term of −h∇H · z (cid:16) (cid:17) the form (v R )v. Keeping this in mind and making use of the L6 estimates for H · ∇ v, we successfully obtain the estimates on ∂2v, then on ∂ v and finally on 2v z ∇H z ∇ and 2T. As a result, we obtain the a priori estimates which guarantee the global ∇ existence of strong solutions. As a corollary of Theorem 1.1, we have the following theorem, which states the well-posedness of strong solutions to system (1.8)–(1.14). The strong solutions to system (1.8)–(1.14) are defined in the similar way as before. Theorem 1.2. Let v and T be two functions such that they are periodic in x and y. 0 0 Denote by vext and Text the even and odd extensions in z of v and T , respectively. 0 0 0 0 Suppose that vext,Text H2 (Ω). Then system (1.8)–(1.14) has a unique global 0 0 ∈ per strong solution (v,T). The existence part follows directly by applying Theorem 1.1 with initial data (vext,Text) and restricting the solution on the sub-domain Ω . While the unique- 0 0 0 ness part can be proven in the same way as that for Theorem 1.1. It should be pointed out that, due to the same reason as stated in Remark 1.1 in [4], the condi- tion that vext,Text H2 (Ω) in the above theorem is necessary for the existence of 0 0 ∈ per strong solutions to system (1.8)–(1.14). Therest ofthepaperisorganizedasfollows: inthenext section, section2, weprove the local existence of strong solutions; in section 3, by establishing the necessary a priori estimates, we show that thelocal strong solution canbeextended to bea global one, and thus obtain a global strong solution. Throughout this paper, we use C to denote a general constant which may be different from line to line. 2. Local Existence of Strong Solutions Inthis section, we establish the local existence ofstrong solutions tosystem (1.15)– (1.21), or equivalently system (1.22)–(1.27). We first cite the following proposition on the local existence of strong solutions to the system with full diffusion (see Proposition 2.1 of [4]). Proposition 2.1. Let v H2(Ω) and T H2(Ω) be two periodic functions, 0 0 ∈ ∈ such that they are even and odd in z, respectively. Then for any given ε > 0, there is a t > 0, depending on ε, and a unique strong solutions (v ,T ), with ε ε ε (v ,T ) L∞(0,t ;H2(Ω)) C([0,t ];H1(Ω)) L2(0,t ;H3(Ω)) and (∂ v ,∂ T ) ε ε ε ε ε t ε t ε ∈ ∩ ∩ ∈ L2(0,t ;H1(Ω)), to the following system ε ∂ v ∆v +(v )v z v(x,y,ξ,t)dξ ∂ v t − ·∇H − −h∇H · z (cid:16) (cid:17) R 6 CHONGSHENGCAO, JINKAILI, ANDEDRISSS. TITI +f k v+ p (x,y,t) z T(x,y,ξ,t)dξ = 0, (2.1) 0 × ∇H s − −h (cid:16) (cid:17) R v¯ = 0, (2.2) H ∇ · ∂ T ∆ T ε∂2T +v T z v(x,y,ξ,t)dξ ∂ T + 1 = 0, (2.3) t − H − z ·∇H − −h∇H · z h (cid:16) (cid:17) R (cid:0) (cid:1) subject to the boundary and initial conditions (1.25)–(1.27). The following lemma will be used to obtain a uniform lower bound of the existence time, independent of ε, and the uniform in ε estimates on the local strong solution (v ,T ) obtained in Proposition 2.1. It also plays an important role in proving the ε ε uniqueness of strong solutions. Lemma 2.1. (see [5]) The following inequalities hold true h h f(x,y,z)dz g(x,y,z)h(x,y,z)dz dxdy Z (cid:18)Z (cid:19)(cid:18)Z (cid:19) M −h −h C f 1/2 f 1/2 + f 1/2 g h 1/2 h 1/2 + h 1/2 , ≤ k k2 k k2 k∇H k2 k k2k k2 k k2 k∇H k2 (cid:16) (cid:17) (cid:16) (cid:17) and h h f(x,y,z)dz g(x,y,z)h(x,y,z)dz dxdy Z (cid:18)Z (cid:19)(cid:18)Z (cid:19) M −h −h C f g 1/2 g 1/2 + g 1/2 h 1/2 h 1/2 + h 1/2 , ≤ k k2k k2 k k2 k∇H k2 k k2 k k2 k∇H k2 (cid:16) (cid:17) (cid:16) (cid:17) for every f,g,h such that the right hand sides make sense and are finite. We also need the following lemma on differentiation under the integral sign and integration by parts. Lemma 2.2. (see [4]) Let f and g be two spatial periodic functions such that f L2(0,t ;H3(Ω)), ∂ f L2(0,t ;H1(Ω)), 0 t 0 ∈ ∈ g L2(0,t ;H2(Ω)), ∂ g L2(0,t ;L2(Ω)). 0 t 0 ∈ ∈ Then it follows that d ∆f 2dxdydz = 2 ∂ f ∆fdxdydz, dt Z | | − Z ∇ t ∇ Ω Ω ∂2 f ∆fdxdydz = ∂ ∆f 2dxdydz Z ∇ xi ∇ Z | xi | Ω Ω and d ∂ g 2dxdydz = 2 ∂ g∂2 gdxdydz, dt Z | xi | − Z t xi Ω Ω ∂2 g∂2 gdxdydz = ∂ ∂ g 2dxdydz, Z xi xj Z | xi xj | Ω Ω for a.e. t (0,t ), where xi,xj x,y,z . 0 ∈ ∈ { } STRONG SOLUTIONS TO THE 3D PRIMITIVE EQUATIONS 7 The next lemma is a version of the Aubin-Lions lemma. Lemma 2.3. (Aubin-Lions Lemma, See Simon [19] Corollary 4) Assume that X,B and Y are three Banach spaces, with X ֒ ֒ B ֒ Y. Then it holds that →→ → (i) If F is a bounded subset of Lp(0,T;X) where 1 p < , and ∂F = ∂f f F ≤ ∞ ∂t ∂t| ∈ is bounded in L1(0,T;Y), then F is relatively compact in Lp(0,T;B); (cid:8) (cid:9) (ii) If F is bounded in L∞(0,T;X) and ∂F is bounded in Lr(0,T;Y) where r > 1, ∂t then F is relatively compact in C([0,T];B). Now we provide a lower bound, in dependent of ε, for the existence time and establish the uniform, in ε, estimates for the solution (v ,T ) obtained in Proposition ε ε 2.1. We have the following: Proposition 2.2. The local strong solution (v ,T ) given by Proposition 2.1 can be ε ε established on the interval (0,t ), such that 0 t0 sup ( v 2 + T 2 )+ ( v 2 + T 2 +ε ∂ T 2 )dt C k εkH2 k εkH2 Z k∇ εkH2 k∇H εkH2 k z εkH2 ≤ 0≤t≤t0 0 and t0 ( ∂ v 2 + ∂ T 2 )dt C, Z k t εkH1 k t εkH1 ≤ 0 where t and C are two positive constants independent of ε. 0 Proof. Suppose (0,t∗) is the maximal interval of existence of the local strong solution ε (v ,T ). We are going to show that t∗ > t , for some positive number t independent ε ε ε 0 0 of ε. We focus in our analysis on the interval (0,t∗). Multiplying (2.1) by v and (2.3) ε ε by T , respectively, and summing the resulting equations up, then it follows from ε integrating by parts and using (2.2) that 1 d ( v 2 + T 2)dxdydz + ( v 2 + T 2 +ε ∂ T 2)dxdydz 2dt Z | ε| | ε| Z |∇ ε| |∇H ε| | z | Ω Ω z 1 z = T dξ v + v dξ T dxdydz. Z (cid:20)∇H (cid:18)Z ε (cid:19) ε h (cid:18)Z ∇H · ε (cid:19) ε(cid:21) Ω −h −h Applying the operator to equations (2.1) and (2.3), multiplying the resulting equa- ∇ tions by ∆v and ∆T , respectively, summing these equalities upand integrat- ε ε −∇ −∇ ing over Ω, it follows from integrating by parts and Lemma 2.2 that 1 d ( ∆v 2 + ∆T 2)dxdydz + ( ∆v 2 + ∆T 2 +ε ∂ ∆T 2)dxdydz 2dt Z | ε| | ε| Z |∇ ε| |∇H ε| | z ε| Ω Ω z z = (v )v v dξ ∂ v T dξ : ∆v ε H ε H ε z ε H ε ε Z (cid:26)∇(cid:20) ·∇ −(cid:18)Z ∇ · (cid:19) −∇ (cid:18)Z (cid:19)(cid:21) ∇ Ω −h −h z 1 ∆ v T v dξ ∂ T + ∆T dxdydz. − (cid:20) ε ·∇H ε −(cid:18)Z ∇H · ε (cid:19)(cid:18) z ε h(cid:19)(cid:21) ε(cid:27) −h 8 CHONGSHENGCAO, JINKAILI, ANDEDRISSS. TITI Summing the above two equalities up, then it follows from Lemma 2.1, the H¨older, Young, Sobolev and Poincar´e inequalities that 1 d ( v 2 + ∆v 2 + T 2 + ∆T 2)dxdydz 2dt Z | ε| | ε| | ε| | ε| Ω + ( v 2 + ∆v 2 + T 2 + ∆T 2 +ε ∂ T 2 +ε ∂ ∆T 2)dxdydz ε ε H ε H ε z ε z ε Z |∇ | |∇ | |∇ | |∇ | | | | | Ω z 1 z = T dξ v + v dξ T Z (cid:26)∇H (cid:18)Z ε (cid:19) ε h (cid:18)Z ∇H · ε (cid:19) ε Ω −h −h z z + (v )v v dξ ∂ v T dξ : ∆v ε H ε H ε z ε H ε ε ∇(cid:20) ·∇ −(cid:18)Z ∇ · (cid:19) −∇ (cid:18)Z (cid:19)(cid:21) ∇ −h −h z 1 ∆ v T v dξ ∂ T + ∆T dxdydz − (cid:20) ε ·∇H ε −(cid:18)Z ∇H · ε (cid:19)(cid:18) z ε h(cid:19)(cid:21) ε(cid:27) −h z 1 z = T dξ v + v dξ T + v v Z (cid:26)(cid:18)Z ∇H ε (cid:19) ε h (cid:18)Z ∇H · ε (cid:19) ε (cid:20)∇ ε ·∇H ε Ω −h −h z z +v v v dξ ∂ v v dξ ∂ v ε H ε H ε z ε H ε z ε ·∇ ∇ −∇(cid:18)Z ∇ · (cid:19) −(cid:18)Z ∇ · (cid:19)∇ −h −h z T dξ : ∆v ∆v T +2 v T H ε ε ε H ε ε H ε −∇ ∇(cid:18)Z (cid:19)(cid:21) ∇ −(cid:20) ·∇ ∇ ·∇ ∇ −h z 1 z ∆v dξ (∂ T + ) 2 v dξ ∂ T ∆T dxdydz −(cid:18)Z ∇H · ε (cid:19) z ε h − (cid:18)Z ∇∇H · ε (cid:19)∇ z ε(cid:21) ε(cid:27) −h −h h h C T dξ v + v dξ T + v 2 + v 2v H ε ε ε ε ε ε ε ≤ Z (cid:26)(cid:18)Z |∇ | (cid:19)| | (cid:18)Z |∇ | (cid:19)| | (cid:20)|∇ | | ||∇ | Ω −h −h h h h + 2v dξ ∂ v + v dξ 2v + 2T dξ ∆v ε z ε ε ε ε ε (cid:18)Z |∇ | (cid:19)| | (cid:18)Z |∇ | (cid:19)|∇ | (cid:18)Z |∇ | (cid:19)(cid:21)|∇ | −h −h −h h + ∆v T + v T + ∆v dξ ∆T dxdydz ε H ε ε H ε H ε ε (cid:20)| ||∇ | |∇ ||∇ ∇ | (cid:18)Z |∇ | (cid:19)(cid:21)| |(cid:27) −h h h +C ∆v dξ ∂ T ∆T dξ ε z ε ε Z Z (cid:20)(cid:18)Z |∇ | (cid:19)(cid:18)Z | || | (cid:19) Ω M −h −h h h + 2v dξ 2T 2dξ dxdydz ε ε (cid:18)Z |∇ | (cid:19)(cid:18)Z |∇ | (cid:19)(cid:21) −h −h C T v + v T +( v 2 + v 2v ≤ k∇H εk2k εk2 k∇ εk2k εk2 k∇ εk4 k εk∞k∇ εk2 h + 2v v + 2T ) ∆v +( 2v T ε 3 ε 6 ε 2 ε 2 ε 3 H ε 6 k∇ k k∇ k k∇ k k∇ k k∇ k k∇ k + v T + ∆v ) ∆T +C ∆v ∂ T 1/2 k∇ εk3k∇H∇ εk6 k∇ εk2 k εk2 k∇ εk2k z εk2 i STRONG SOLUTIONS TO THE 3D PRIMITIVE EQUATIONS 9 ( ∂ T 1/2 + ∂ T 1/2) ∆T 1/2( ∆T 1/2 + ∆T 1/2) × k z εk2 k∇H z εk2 k εk2 k εk2 k∇H εk2 +C 2v 2T ( 2T + 2T ) ε 2 ε 2 ε 2 H ε 2 k∇ k k∇ k k∇ k k∇ ∇ k C v 2 + T 2 +( v 2 + ∆v 1/2 ∆v 1/2 v + T 2 ) ∆v ≤ k εkH2 k εkH2 k εkH2 k εk2 k∇ εk2 k εkH2 k εkH2 k∇ εk2 h +( ∆v 1/2 ∆v 1/2 T + v ∆T + ∆v ) k εk2 k∇ εk2 k εkH2 k εkH2k∇H εk2 k∇ εk2 ∆T +C ∆v ( T 2 + T 3/2 ∆T 1/2) ×k εk2 k∇ εk2 k εkH2 k εkH2k∇H εk2 i +C v ( T 2 + T ∆T ) k εkH2 k εkH2 k εkH2k∇H εk2 1 ( ∆v 2 + ∆T 2)+C(1+ v 6 + T 6 ) ≤2 k∇ εk2 k∇H εk2 k εkH2 k εkH2 from which, we obtain, for any t (0,t∗), ∈ ε t sup ( v 2 + T 2 )+ ( v 2 + T 2 +ε ∂ T 2 )ds k εkH2 k εkH2 Z k∇ εkH2 k∇H εkH2 k z εkH2 0≤s≤t 0 t CC +C (1+ v 2 + T 2 )3ds, ≤ 0 Z k εkH2 k εkH2 0 where C = v 2 + T 2 +1. 0 k 0kH2 k 0kH2 Setting t f(t) = sup ( v 2 + T 2 +1)+ ( v 2 + T 2 +ε ∂ T 2 )ds k εkH2 k εkH2 Z k∇ εkH2 k∇H εkH2 k z εkH2 0≤s≤t 0 for t [0,t∗). Then one has ∈ ε t f(t) CC +C (f(s))3ds, t [0,t∗). ≤ 0 Z ∈ ε 0 Set F(t) = t(f(t)3)ds+1, then we have 0 R F′(t) = (f(t))3 C (F(t))3, t [0,t∗), ≤ 1 ∀ ∈ ε where C is a positive constant depending only on h and (v ,T ). This inequality 1 0 0 implies 1 1 F(t) , t [0,t∗) [0, ), ≤ √1 2C t ∀ ∈ ε ∩ 2C 1 1 − and thus t sup ( v 2 + T 2 )+ ( v 2 + T 2 +ε ∂ T 2 )ds k εkH2 k εkH2 Z k∇ εkH2 k∇H εkH2 k z εkH2 0≤s≤t 0 C CC +CF(t) CC + C(C +√2), 0 0 0 ≤ ≤ √1 2C t ≤ 1 − for any t [0,t∗) [0, 1 ]. Recalling that t∗ is the maximal existence time, the above ∈ ε ∩ 4C1 ε inequality implies that t∗ > 1 . Thus we can take t = 1 . ε 4C1 0 4C1 10 CHONGSHENGCAO, JINKAILI, ANDEDRISSS. TITI Thanks to the estimates we have just proved, one can use the same argument as in the last paragraph of the proof of Proposition 3.1 in [4] to obtain the estimates on ∂ v and ∂ T , and thus we omit the details here. This completes the proof. (cid:3) t ε t ε Now we can prove the local well-posedness of strong solutions to system (1.15)– (1.21), or equivalently system (1.22)–(1.27). Proposition 2.3. Let v H2(Ω) and T H2(Ω) be two periodic functions, such 0 0 ∈ ∈ that they are even and odd in z, respectively. Then system (1.22)–(1.27) has a unique strong solution (v,T) in Ω (0,t ), where t is the same positive time stated in 0 0 × Proposition 2.2. Moreover, the strong solution depends continuously on the initial data. Proof. By Proposition 2.1 and Proposition 2.2, for any given ε > 0, system (2.1)– (2.3), subject to the boundary and initial conditions (1.25)–(1.27), has a unique strong solution (v ,T ) in Ω (0,t ) such that ε ε 0 × t0 sup ( v 2 + T 2 )+ ( v 2 + T 2 +ε ∂ T 2 )dt C k εkH2 k εkH2 Z k∇ εkH2 k∇H εkH2 k z εkH2 ≤ 0≤t≤t0 0 and t0 ( ∂ v 2 + ∂ T 2 )dt C, Z k t εkH1 k t εkH1 ≤ 0 where C is a constant independent of ε. On account of these estimates and applying Lemma 2.3, there is a subsequence, still denoted by (v ,T ) , and (v,T), such that ε ε { } (v ,T ) (v,T), in C([0,t ];H1(Ω)), ε ε 0 → ( v , T ) ( v, T), in L2(0,t ;H1(Ω)), ε H ε H 0 ∇ ∇ → ∇ ∇ (v ,T )⇀∗ (v,T), in L∞(0,t ;H2(Ω)), ε ε 0 ( v , T ) ⇀ ( v, T), in L2(0,t ;H2(Ω)), ε H ε H 0 ∇ ∇ ∇ ∇ (∂ v ,∂ T ) ⇀ (∂ v,∂ T), in L2(Ω (0,t )), t ε t ε t t 0 × ∗ where ⇀ and ⇀ are the weak and weak- convergence, respectively. Thanks to ∗ these convergence, one can easily show that (v,T) is a strong solution to system (1.22)–(1.27), or equivalently to system (1.15)–(1.21). The continuous dependence on the initial data, in particular the uniqueness, are straightforward consequence of (cid:3) Proposition 2.4 (see Corollary 2.1, below). For the continuous dependence on the initial data, the solutions are not required to have as high regularities as stated in Definition 1.1. In fact, we have the following: Proposition 2.4. Let (v ,T ) and (v ,T ) be two spatially periodic functions, satis- 1 1 2 2 fying the following regularity properties (v ,T ) L∞(0,t ;H1(Ω)) C([0,t ];L2(Ω)), i i 0 0 ∈ ∩ (∂ v ,∂ T ,δ∂2v ) L2(Ω (0,t )), ( v , T ) L2(0,t ;H1(Ω)), t i t i z i ∈ × 0 ∇H i ∇H i ∈ 0