THE PRIMITIVE EQUATIONS ON THE LARGE SCALE OCEAN UNDER THE SMALL DEPTH HYPOTHESIS

In this article we study the global existence of strong solutions of the Primitive Equations (PEs) for the large scale ocean under the small depth hypothesis. The small depth hypothesis implies that the domain Mε occupied by the ocean is a thin domain, its thickness parameter ε is the aspect ratio between its vertical and horizontal scales. Using and generalizing the methods developed in [23],[24], we establish the global existence of strong solutions for initial data and volume and boundary ’forces’, which belong to large sets in their respective phase spaces, provided ε is sufficiently small. Our proof of the existence results for the PEs is based on precise estimates of the dependence of a number of classical constants on the thickness ε of the domain. The extension of the results to the atmosphere or the coupled ocean and atmosphere or to other relevant boundary conditions will appear elsewhere.

1. Introduction.In this article we are concerned with the Primitive Equations (PEs for brevity) governing the motion and state of the ocean in a three dimensional thin domain M ε (thin in a sense to be made precise later on).The primary purpose of this article is to establish the existence and uniqueness of global strong solutions of the PEs, i.e. for arbitrary time, for a broad class of data which includes the most physically relevant ones.
The PEs of the ocean are derived from the Navier-Stokes equations, with Coriolis force, coupled with the thermodynamic equation and the diffusion equation for the salinity by taking into account both the Boussinesq and hydrostatic approximations; see e.g.[7], [19], [12].In a series of articles by Lions, Temam and Wang [11,12,13,14,15], the PEs of the ocean, atmosphere and their interaction, are studied from the mathematical viewpoint (existence of weak solutions, long time behavior, etc.), as well as from the numerical viewpoint.In [2] Temam and Ewald derived the maximum principles of the PEs of atmosphere by changing the temperature variable to a potential temperature variable.However, despite these advances in the study of the PEs, the mathematical theory for the PEs is far from being complete.For example, the uniqueness of weak solutions and the global existence of strong solutions for the PEs remain unsolved.This article presents a step in this direction 98 CHANGBING HU, ROGER TEMAM, MOHAMMED ZIANE taking advantage of the small shape ratio of the ocean, the vertical dimension being much smaller than the horizontal ones.
As it is well-known (see e.g.[7], [19], [11,12,13,14,15]), there are essentially two characteristics used in simplifying the general hydrodynamics and thermodynamics equations of both the atmosphere and ocean.The first one is that, for large scale geophysical flows, the ratio between the vertical and horizontal scales is very small, and thus the vertical momentum equation of the atmosphere or the ocean can be well approximated by the hydrostatic equation; this hypothesis leads to the PEs; one of our objectives in this article is to further explore the mathematical consequences of this hypothesis.Another important characteristics is the rotating effect, i. e., the Coriolis force, see e.g.[1], [16], [17] and the references therein.In this article we will treat the case where the Rossby number is of order, O(1), concentrating on the small depth characteristic.
In the mathematical literature, there is a large amount of work devoted to the study of partial differential equations in thin domains, see e.g.[4,5], [8], [9], [18], [20,21], and [23,24].In this article, we will consider the PEs in thin domains following the methods developed by two of the authors in [23,24].We choose to work with the PEs in the actual domains M ε .Compared to the work in [23,24], the PEs possess some specific difficulties to circumvent, namely: (i) the physical domain has corners and a non-flat bottom topography, (ii) the non-local constraint (incompressibility condition) and the integral expression of the vertical velocity in PEs, which leads to a strong non-linear term, i.e.
(iii) non-homogeneous boundary conditions and the mixed boundary condition on the surface of the ocean which are not considered in [23].
The principal linear part of the PEs is a (stationary elliptic) Stokes type problem which we call the GFD-Stokes problem.In [26], [27], one of the authors, established the H 2 regularity of the solutions of the GFD-Stokes problem under various boundary conditions.In the companion article [6], we extend this result to the more realistic boundary conditions considered in this article.In order to overcome the difficulties (i)-(iii), we also generalize the anisotropic Sobolev-type inequalities from [23] to fit in our problem, (see [6]), and devote great efforts to the study of the strong nonlinear term, see (1.1).
In this article, we obtain the global existence of the strong solutions for the PEs with a large class of initial data and boundary conditions (acting as external forcing in the PEs), provided ε is small enough; see the main theorem in Section 2.
Let us mention a very recent work on PEs in [3], in which the local existence of the strong solutions of the PEs is proved for domains of the class studied in [26].Our results in this paper also imply the local existence of strong solutions in the case ε is of order 1 (see Remark 4.3).
We present an outline of this article: in Section 2 we briefly recall the mathematical setting of the PEs, the corresponding boundary conditions and their nondimensional form, their weak formulation and the main result of this article.In Section 3 we recall some inequalities on v, T, S from [6].In Section 4 we derive a priori energy estimates for the PEs which will guarantee the large time existence of solutions.In Section 5 we prove the existence of the strong solutions of the PEs PRMITIVE EQUATIONS IN THIN DOMAINS 99 with homogenized boundary conditions.In Section 6 we shift the solution obtained in Section 5 to its original counterpart, and prove the main theorem.
2. The primitive equations of the ocean and their mathematical setting.2.1.Brief reminder of the PEs.We first recall the full set of equations which describe the motion and state of the ocean, obtained by taking into account both the Boussinesq and hydrostatic approximations, and which are called the Primitive Equations of the large scale ocean, PEs for brevity, (see e.g.[19], [7], [11], [12]): ) ) Here g is the gravity acceleration, µ, ν are the effective molecular dissipations in the horizontal and vertical directions, µ T , ν T , µ S , ν S reflect similarly the heat and salinity diffusions; ρ 0 , T 0 and S 0 are reference values for the density, the temperature and the salinity; Ω is the angular velocity of the earth, θ (0 ≤ θ ≤ π) is the colatitude of the earth.Q T is the heating source mainly through radiation (sun and clouds).The unknown functions are the horizontal velocity v and vertical velocity w, the temperature T , the salinity S, the density function of the ocean ρ and the pressure p.Throughout this article, we use ∇, , div to denote the two dimensional gradient, Laplacian and divergence operators on the horizontal plane, and use ∇ 3 , 3 and div 3 for the corresponding 3D differential operators.The domain occupied by the ocean is M h ⊂ R 3 : where Γ i is the interface between the ocean and the atmosphere, and h : Γ i → R is the depth function, which is positive and sufficiently smooth.Throughout this article we assume that Γ i ⊂ R 2 is convex and smooth.The boundary of , where Γ b is the bottom of the ocean, and Γ l is the lateral boundary of the ocean.
The boundary conditions for the PEs read (see e.g.[12]): At the interface of the ocean and atmosphere , Γ i : (2.7) At the bottom of the ocean and at the lateral boundary, Remark 2.1.With the same technique as those developed in this article , we could treat the PEs in thin domains with other boundary conditions.As indicated in Section 2, one advantage of the above-mentioned boundary conditions is that a Poincaré-type inequality holds for v, T, S (with small constants for v and T ).Other boundary conditions for which the Poincaré inequality does not hold necessitate different methods, and will be considered elsewhere.Now we briefly recall the non dimensional form of the PEs, following [12].However, and departing here from [12], we want to introduce and emphasize in this work the role of the small depth assumption for the domain occupied by the ocean.To this end, let U, T 0 , ρ 0 be reference values respectively of the horizontal velocity, temperature and density.We also consider L to be the reference value of the horizontal length and H to be the reference value of the vertical length.Let ε = H/L be the small parameter which will be used throughout this article.Then we set ) ) Here Re i (i=1,2) are Reynolds type numbers, and Rt i and Rs i (i=1,2) are the nondimensional eddy diffusion coefficients; Ro is the Rossby number, which compares the angular velocity of the earth to the dynamical behavior of the ocean.In the present work, we will keep the Rossby number constant.For the Coriolis force we could as well consider the β − approximation, see e.g.[19]: where f 0 = 2 cos θ 0 , β 0 = 2 a sin θ 0 , a is the radius of the earth, θ 0 is the mean colatitude.Then, by easy computation, we obtain the following non-dimensional form of the PEs, in which all the primes have been dropped: ) ) Introducing and then dropping the primes, we denote the non-dimensional domain occupied by the ocean, by where ∂M ε = Γ i ∪Γ b ∪Γ l is now the boundary of the domain in the non-dimensional space.Throughout this article , we will assume that h is independent of ε, where ε is the small parameter representing the thickness of the domain.In the final result we will assume that h is a positive constant but in several parts of the article, we will make the following assumptions concerning h: there exist positive constants h, h, h 1 , h 2 such that Then we can write the boundary conditions of the non-dimensional problem as follows: where α v , g v , α T , g T are given functions, α v ≥ 0, α T ≥ 0. Remark 2.2.In [12], by taking a different scale in the vertical direction, the authors obtained the non-dimensional PEs on a domain independent of the small parameter ε.However the parameter ε is hidden in the definition of some physical constants such as Re 2 , Rt 2 , and Rs 2 , see [12] for the details.The influence of this small parameter was not taken into account in [12], and, on the contrary, we want here to emphasize the role of ε 1.
The vertical velocity w and the pressure p are two diagnostic variables that we determine by integration of (2.14) and (2.15), taking into account the boundary conditions.Hence, for w, we obtain Remark 2.3.This formulation of the PEs was first introduced in [11], [12].In this new formulation, p s plays the role of the Lagrange multiplier corresponding to the non-local constraint (2.22).In [25], the author worked directly on the original PEs with anisotropic Sobolev spaces for the vertical velocity, and proved in particular the existence of strong solutions for the PEs for small time under the constant surface pressure assumption, p s = P .
Remark 2.4.In the following argument, we show that the average of the salinity S over M ε is zero if the initial salinity has zero average.i. e.,

Mε
S dx = 0. (2.24) Indeed integrating (2.17), we obtain Using the interface and boundary conditions (2.20), we can deduce easily that the second and third integrals vanish and hence Therefore the average value of S is constant and is denoted by S: This simply expresses the conservation of the total amount of salt in the sea water.In (2.17) we actually replace S by S − S. We can easily see that this translation does not change the equation and the boundary conditions for S.After this translation, we have the natural constraint (2.24) for S. Relation (2.24) makes the Poincaré inequality available for S, see (3.17).
2.2.Mathematical setting of the PEs.In this section we first define the function spaces suitable for the mathematical setting of the problem, see [12] for the details.We denote by H s (M ε ), s ∈ R, the usual Sobolev spaces constructed on L 2 (M ε ), and We also denote by , the space of infinitely differentiable functions with compact support in M ε .Motivated by the boundary condition for the velocity field in the PEs , we define We now define the function spaces for the unknown function u = (v, T, S).
Then we define The norms and inner products for the spaces H and H i , (i = 1, 2) are the L 2 ones, denoted by (•, •) ε and | • | ε .The inner products and norms for the spaces V 1 are given by For V 2 and V 3 we take the usual inner products and norms, i. e., For simplicity, we use also (•, •) ε and | • | ε to denote the norm and inner product in H, ((•, •)) ε and • ε in V , i. e. for the later, ε , for u = (v, T, S), u 1 = (v 1 , T 1 , S 1 ).Remark 2.5.Here we propose different definitions for the inner products and norms for V 1 , V 2 and V 3 based on the observation that the functions v and S in V 1 and V 3 respectively satisfy a Poincaré inequality; this will be made clear in Section 2. For T in V 2 , the classical Poincaré inequality does not hold.Now by the Riesz representation theorem, we can identify the dual space H of H (H i , i = 1, 2, 3) with H (respectively H i with H i , i = 1, 2, 3), i. e., H = H.Then we have where the two inclusions are compact and injective.The following lemma characterizes the relationship between H 1 and its orthogonal complement. Then

27)
where n h is the horizontal unit normal (in the horizontal plane) on the boundary of Γ i .
Proof.Notice first that, in (2.25), we give a slight correction to [12] on the space H 1 , so we prove this lemma in detail.We first prove (2.26); the space in the right hand side of (2.26) is clearly orthogonal to hence v, ∂Ψ/∂z = ∂v/∂z, Ψ = 0, i. e. ∂v/∂z = 0. Now any distribution on M ε which is independent of z defines a distribution on Γ i : for Ψ ∈ C ∞ c (Γ i ) 2 , we define v , Ψ Γi as v, Ψθ 0 Mε , where θ 0 is a real valued C ∞ function with compact support in (− h, 0) and such that +∞ −∞ θ 0 (z)dz = 1.For this definition we observe that v, Ψθ 0 Mε is independent of the choice of θ 0 ; indeed if θ 01 , θ 02 are two such functions, then θ = θ 01 − θ 02 is such that Thus v, Ψ Γi = v, Ψθ 0 Γi = 0 for all such Ψ's and by the classical De Rham Theorem (see e.g.[22]), v is the gradient of a distribution φ on Γ i .Finally, by Deny-Lions theorem (see the same references), the distribution φ has its derivatives in L 2 (Γ i ) (v = gradφ), and thus φ ∈ H 1 (Γ i ).This completes the proof of (2.26).
Then we prove (2.25).First notice, from the classical theory of Navier-Stokes equations, that for v ∈ L 2 (M ε ), with dz is well defined and belongs to H − 1 2 (∂Γ i ); so the constraint appearing in the right hand side of (2.25) makes sense.Denote by Ḣ1 the right hand side of (2.25); the inclusion H 1 ⊂ Ḣ1 is trivial, since V 1 ⊂ Ḣ1 , and Ḣ1 is a closed subspace of L 2 (M ε ).Denote by Ḧ1 the orthogonal complement of H 1 in Ḣ1 , and assume that v ∈ Ḧ1 .Then v ∈ H ⊥ 1 Ḣ1 , and by (2.26), there exists a it should be noticed that v is a 2D vector depending on (x 1 , x 2 , x 3 ), and furthermore (2.28) By the classical elliptic theory, (2.28) implies that grad φ = 0, i. e., v = 0, which yields immediately Ḧ = {0}, and hence H 1 = Ḣ1 ; the proof is complete.For (2.27) we then follow the same methods as in [22], using (2.25) and (2.26).
Throughout this article, we will denote by We now define the bilinear forms a :

3, and the corresponding linear operators
and for u = (v, T, S), and ũ = (ṽ, T , S) in V.
One of the main objectives of [6] is to establish, under suitable hypotheses, the following characterization of the operators A i and their domains, , ∀v ∈ D(A 1 ), , ∀S ∈ D(A 3 ). (2.30) Here P is the orthogonal projection from L 2 (M ε ) onto H 1 ; the regularity of the operator A 1 has been studied in [26] in a different context; one of the main issues in [6] is to derive the similar results for our current boundary conditions and the dependence on ε of the norms of the operators A i .The linear operators A i , i = 1, 2, 3, which are isomorphisms from V i onto V i , can be seen as unbounded, selfadjoint linear operators on H i ; they are positive operators, and they admit compact inverses, so that the fractional powers of A i can be defined; A i , i = 1, 2, 3, especially will be frequently used, and |A 3 provides an equivalent norm on V i , i = 1, 2, 3 respectively.Hereafter we shall alternatively use the following notations, justified by a simple integration by parts, we have We postpone the discussion on the coercivity and continuity of the bilinear forms a i , i = 1, 2, 3 and a until some inequalities on v, T, S are established, see Proposition 4.1 in Section 4.
In relation with the nonlinear terms appearing in the equations, we define the following trilinear functionals and associated operators: and finally for u = (v, T, S), ũ = (ṽ, T , S) and u = (v , T , S ) in V.
These trilinear terms possess the following anti-symmetry properties: which play a crucial role in the proof of the global existence of weak solutions, see [11], [12].It should be noted that the PEs possess some highly nonlinear terms such as w(v) ∂v ∂x3 which need particular attention.Concerning the Coriolis term in the equations, we define a bilinear functional e : H × H → R and the associated linear operator E : H → H by setting It is easy to see that e(v, v) = 0, ∀v ∈ H 1 .We are now in a position to recall the weak formulation of the PEs and to state the main results in this article.

2.3.
Weak formulation of the PEs and the main results.With the above notations, the weak formulation of the PEs is given by: We recall the existence of weak solutions for the PEs which is due to Lions, Temam and Wang in [12]: Theorem 2.2.For any t 1 positive, there exists at least one solution for problem (2.32), u = (v, T, S) (not necessarily unique) defined on (0, t 1 ), and such that where H w is the space H equipped with the weak topology and In this article we are concerned with the global strong solutions (2.32), namely: where u 0 is now given in V .The terms in the right-hand side of (2.32) come by integration by parts from the boundary conditions on v and T .The weak formulation will be sufficient to consider the weak solutions of the PEs.Since we are interested in the strong solutions of the PEs, it is necessary to homogenize the boundary terms, then the integral terms over the boundary will be replaced by some integration over the whole domain M ε ; see (4.3)-(4.6).It should be noted that the uniqueness of the weak solution is an open problem.
Theorem 2.3.(Main result) The assumptions are those above; in particular M ε is convex and the depth h is constant, and let Finally assume that the initial data and boundary conditions satisfy (2.35) Then there exists an ε 0 depending on Re i , Rs i , Rt i , α v , α T , Ro, i = 1, 2 and on the function R 0 , and such that for 0 < ε ≤ ε 0 , there exists a unique global strong solution which satisfy (2.33) , i. e., for any Moreover we have the following estimate on the solution u: , where σ > 1 (independent of ε) will be determined below (see (5.13)).

Remark 2.6
The convexity of M ε and h constant are used in [6] to show (2.29)-(2.30)including the dependence on ε of the norms of the operators A i .However Theorem 2.3 is valid for any domain M ε and any set of boundary conditions for which the analogue of (2.29)-(2.30)has been proven, e.g. the Dirichlet boundary conditions as in [26].This remark extends also to the result in Remark 4.3 (local existence of strong solutions for ε = 1).The extension of the results of [6] (and hence of the results in Theorem 2.3 and Remark 4.3 in this article), to the case where M ε is not convex and h is not constant, will appear elsewhere.Remark 2.7.The assumptions (2.35) are physically relevant, they allow large initial data for which the vertical variation (for v 0 , T 0 , S 0 , Q T ), is O(1).Indeed such functions can be treated as independent of z so that e.g.|A ). Hence v 0 independent of z can have a norm as large as C R 0 (ε)ε − 1 2 , and we observe that R 0 (ε) = ε α− 1 2 , α > 0, is suitable for (2.34).A similar remark holds for functions slowly varying in the z-direction and for other data.Remark 2.8.The hypothesis that Γ i is convex is not natural, and is due to Propositions 2.1 and 2.2 in [6].Note that Theorem 2.3 above and all what follows extend without any modification to all domains M ε for which Propositions 2.1 and 2.2 of [6] is valid.
3. Functional inequalities.In this section we recall from [6] a number of inequalities on the functions v, T, S, satisfying the boundary conditions proposed in the first section.Since we are interested in the nonlinear partial differential equations in thin domains, we need to know the exact dependence, with respect to the thickness of the domain, of the constants appearing the inequalities.As it is well known, in the classical theory of Sobolev spaces, the isotropic feature is emphasized.
In our problem, one of the main features is the natural anisotropy of the domain between different space directions.In [23], the authors derived anisotropic versions of the classical Sobolev-type inequalities, such as the Poincaré inequality, Agmon's inequality and Ladyzhenkaya's inequality, which have proved to be successful in the study of the Navier-Stokes equations in thin domains.In the current work, we need some inequalities with explicit dependence of the constants on the thickness ε of the domain.We present these inequalities without proof, their proof is given in a separate article [6].
Here and after, c 0 will stand for some numerical constant, and C some constant depending on some non-dimensional physical quantities appearing in the PEs, namely, α v , α T , Re i , Rs i , Rt i , i = 1, 2, and Ro.The constant c 0 is independent of the small parameter ε and may vary at each occurrence; C ε is a constant which possibly depend on ε.
First we describe some inequalities that will be used for a linearized version of the primitive equation, the goal of which is to homogenize the boundary conditions and the initial conditions for the velocity and the temperature.For v satisfying the following boundary conditions: • The Poincaré-type inequalities: • The Grisvard-Iooss inequality: (3.5) • The Agmon inequality: • The Ladyzhenskaya inequality: ) Now we recall some inequalities concerning a function T which satisfies the following boundary conditions on ∂M ε : The classical Poincaré inequality does not follow directly.The following inequality will play a role similar to Poincaré's inequality.
• The Poincaré-type inequalities: (3.10) • The Grisvard-Iooss inequality: (3.12) We have some further inequalities on T : and Finally we have the following version of the Ladyzhenskaya inequality for T : We now present some inequalities on a function S satisfying Mε S dx = 0, together with the following boundary condition on ∂M ε : • The Poincaré-type inequalities: . (3.17) • The Grisvard-Iooss inequality: We conclude this section with the Ladyzhenskaya inequality for S: 4. A priori estimates.Before we start to derive a priori estimates for the PEs, we need to homogenize the boundary conditions on v, T .For that purpose, we introduce v , T , that are solutions of the following problems: 3 and 3 The regularity of solutions of the stationary problems associated to (4.1) and (4.2) has been investigated in [6], namely, v ∈ L 2 (0, We set v = v + v , T = T + T and p = p + p , and after dropping the primes, the new system for v , T , S, p reads where The homogenized system is supplemented with the following homogeneous boundary conditions on ∂M ε : and initial data v(•, 0) = 0, T (•, 0) = 0, S(•, 0) = S 0 .
In this section, f v and f T are assumed to be in L 2 (M ε ).In Section 6, we will give estimates on f v and f T needed to prove the main theorem in this article.
Remark 4.1 In [12], the authors assumed that div g v = 0, and they solved an ordinary differential equation to obtain v (the notation is different in [12]).In this work, since v is the solution of the time-dependent GFD-Stokes problem (4.1), it is not necessary to assume this divergence-free condition on g v .
For the readers' convenience, we list in the following proposition some inequalities for the new functions v, T and S satisfying the homogeneous boundary conditions (4.9) .Proposition 4.1.For v ∈ D(A 1 ): ) ) ) For T ∈ D(A 2 ): ) For S ∈ D(A 3 ): Proof.Most of the inequalities presented here follow from the previous section by making the boundary conditions homogeneous.The new one are inequalities (4.14), (4.18) and (4.22); their proofs are given in [6].
The following inequality will also be frequently used , ∀q ∈ L 4 (M ε ).(4.23) We start with the a priori estimates for v.We multiply equation (4.3) by A 1 v, then integrate in x over M ε .Noticing that we are led to d dt In what follows we will write estimates for each term appearing in (4.24).
By Poincaré's inequality (4.10), we have Now we handle the most problematic term mentioned in the Introduction: and with this estimate, the nonlinear term is too strong to be dominated by the linear operator A 1 .We continue the estimate, apply the same technique as for (4.29), and we find In the same way as in (4.30), By the Cauchy-Schwarz inequality, we have Using again the Cauchy-Schwarz inequality, we obtain We collect (4.24)-(4.34)and obtain the final inequality:

.(Local existence of strong solutions)
We note here that inequality (4.35) is enough to establish the global well-posedness for small data and the local well-posedness.Initially the velocity v is zero and therefore its H 1 norm stays small for a small time say on (0, t 1 ).let t 1 > 0, so that c 0 ε 1/2 v ε (t) ≤ 1 on (0, t 1 ).Then we have Therefore, since v = 0 at t = 0, the following inequality hold for t ∈ (0, t 1 ) The integrability of ϕ 1 and ψ 1 follows from the existence of weak solutions and the linearity of the equations satisfied by v .Thus choosing t 1 , so that the right hand side in the above inequality is less than , implies the local existence of strong solutions.
The following lemma specifies the estimates on I 1 , I 2 used in (4.29).
, using the existence of ξ ∈ (a, b) such that f (ξ) = 0, we easily obtain (4.37) for such a function.The lemma then follows by density.for v ∈ D(A 1 ), then Proof.Applying the same technique as in Lemma 4.3, we are led to estimate ∂I2 ∂xi , i = 1, 2, The main task here is to bound the boundary term by suitable integration on v.
Using the boundary condition for v, we obtain .
The remaining part of the proof is easy and will be omitted.
We now derive the a priori estimates for the temperature T .Multiply the equation (4.4) with A 2 T , then integrate over M ε , to obtain

.38)
As we have done for v, we will estimate each term in (4.38) separately.First by Agmon's inequality for v, we obtain 2 and Lemma 4.5 below) In the same fashion, by Lemma Then by Agmon's inequality, Similarly, , then by Hölder's inequality, Lemma 4.2 and Lemma 4.5 below, we have Putting (4.38)-(4.45)together, we find where Now we prove the estimate on (appearing in (4.40)), which we used before.
We deduce from (4.17) ε , so (4.49) is proved.We obtain (4.47) by adding (4.48) and (4.49) together, and the lemma is proved.Now we turn to the a priori estimates for the salinity S. Multiply (4.5) by A 3 S, then integrate both sides over M ε , to obtain 1 2 By the Agmon inequality, we have Applying the same technique as in (4.29), (4.40) and (4.54), we find From (4.52)-(4.56),one is led to with The following lemma yields the desired estimate for appearing in (4.54).
Lemma 4.6.For S ∈ D(A 3 ), Proof.This lemma can be proved in the same fashion as Lemma 4.5 by combining the Ladyzhenskaya inequality in 2-D and the boundary condition for S, we will just give a sketch of the proof.For i = 1, 2, By taking advantage of the boundary condition for S, ∂S ∂x3 | Γi = 0: dz.
The remaining part can be completed by Hölder's inequality and Ladyzhenskaya's inequality.So (4.58) is justified.
We summarize all estimates for v, S, T into the following proposition.
Remark 4.4 Assume c 0 ε v ε is small.The second terms in (4.59) and (4.60) will be positive, the right hand side of (4.59) can be controlled using the inequalities valid for weak solutions and if f v and g v are small, say of order ε, then, the right hand side will be of order ε and therefore v will stay of order ε and the estimates can be closed to obtain the global boundedness of v ε and the integrability (in time) of |A 1 v| ε .Then, without any further assumptions on T and S, their H 1 norms will stay bounded for all time.
5. Existence of global in time strong solutions.In this section we will prove the existence of global in time strong solutions to the PEs under small data depending on the thickness.It is noteworthy that the local/global strong solution for the PEs is so far an open problem due to the lack of the estimates on the vertical component w.This work will give a partial answer to this question when the domain is thin.Recall that R 0 (ε) is a monotone positive function satisfying lim ε→0 εR 2 0 (ε) = 0. (5.1) To proceed, observe first Also we assume ( Notice that all terms in R (ε) appear in the left-hand side of (4.59)-(4.61).Fixed σ > 1 to be specified later on (σ > e 1 4 ), then take ε 1 such that for 0 Set u = (v , T , 0).Then the weak formulation for (4.3)-(4.6)can be rewritten as follows: We are in a position to state the main theorem in this section.
Theorem 5.1.Assume that (5.1)-(5.4)hold.Then there exists a unique strong solution u to the PEs such that u ∈ L ∞ (0, ∞; V ) ∪ L 2 ((0, t 1 ; D(A)), u t ∈ L 2 (0, t 1 ; H), Furthermore the solution u satisfies the following estimate: Proof.We will prove this theorem by using the Galerkin method.Since this method is standard, we will just present an outline of the proof.We divide the proof into 4 steps.
Then the {e j } ∞ j=1 define a complete orthogonal basis in V .Furthermore, let P m the orthogonal projector on the first eigenvalues e 1 , . . ., e m .Now we look for approximate solutions u m = Solving for u m amounts to solving a system of differential equations for the g jm , j = 1, 2, • • • , m.It is easy to see that the above a priori estimates apply as well to u m , namely We are then able to pass to the limit using the standard compactness methods using some additional estimate on (u m ) t .
Step 2. Global solution for the Galerkin Approximation.By the general theory of ODE, the system of differential equations for g jm admits continuous solutions.For ∀σ > 1, there exists t σ 1 > 0 such that |A For any t ∈ [0, t σ 1 ], We rewrite (4.59)-(4.61)as follows (5.9) we have, using the estimates on the linear equation satisfied by v and T , for ε small enough: First we add (5.7) and (5.8) and then use Poincaré's inequalities (4.16) to obtain Taking into consideration the zero initial data for v and T , the Gronwall inequality yields, for t ∈ [0, t σ 1 ) and ε small so that εR .11)For the salinity S m , by using the Poincaré's inequality (4.20), we have from (5.9) The Gronwall inequality yields (5.12) Therefore from (5.10) and (5.12), we obtain Then at t = t σ 1 (5.6) gives We take σ > e 1 4 , and a contradiction occurs, so t σ 1 = ∞.Step 3. Passage to limit and the existence of solution: To pass to the limit, we need to show that {(u m ) t } bounded in L 2 (0, t 1 ; H), ∀t 1 > 0, which can be proved by repeating some of the estimates in the previous section, we omit them.So far we infer that there exists u m ∈ L ∞ (0, ∞; V ) ∩ L 2 (0, t 1 ; D(A)) and subsequence of u m (still denoted by u m ), such that (u m ) t → u t weakly in L 2 (0, t 1 ; H). (5.16) By Aubin compactness lemma, we have u m → u strongly in L 2 (0, t 1 ; V ).Since the PEs bear much similarity with the Navier-Stokes equations, to prove that u is a local strong solution of the PEs, only the vertical convection term needs more care, which is presented below. .Lemma 4.2.Assume u m → u in the sense of (5.14)-(5.16),for any vector function v with components in C((0, t 1 ) × M ε ) and scalar function Q in C((0, t 1 ) × M ε ), then we have Step 4. Uniqueness of the strong solution: To finish the proof, it remains to confirm the uniqueness of the solution obtained.Recall that the uniqueness of strong solutions of the 3-D Navier-Stokes equation is well known; the main difference of the PEs from 3D NS equations is the term w(v).In 3D NS equations, w ∈ H 1 , however all we obtain for the PEs is w ∈ L 2 .Let u 1 = (v 1 , T 1 , S 1 ), u 2 = (v 2 , T 2 , S 2 ) be two solutions of (2.13)-(2.18)satisfying the non-homogeneous boundary condition (4.9).Denote δu = (δv, δT, δS) Thus δu satisfies the following equations: supplemented with the homogeneous boundary condition: (5.22) Take the inner product in H on both side of (5.17) with δv, noticing the skewsymmetry identity for the trilinear term (2.31), we have where By estimates (5.5), we have t 0 G(τ )dτ < ∞.
A direct consequence from the process of above proof is given below.
Corollary 5.2.Assume u = (v, T, S) is the solution obtained in the previous theorem, then there exists a constant λ such that which implies S behaves trivially in the long time.
6. Proof of the main theorem.In this part we will prove the main theorem stated in Section 1. Basically the core part of the proof has been done in the previous section.Recall that we obtain the existence of the solution of the PEs under assumptions in terms of u , T , in this part we will transform the results into the physical boundary condition, i. e., g v , g T .Toward this end, we first present a lemma which gives estimates on v , T in term of g v , g T .Lemma 6.1.Let v , T be solutions of (4.1) (4.2), assume Moreover (v , T ) satisfies the following estimates: Proof.Inequalities (6.2) and (6.3) can be obtained from the weak formulation of (4.1) (4.2) and Lax-Milgram theorem.Inequality (6.4) is the main result in Theorem 3.1 in [6].Inequality (6.5) is just the Grisvard-Iooss inequality for T .Lemma 6.2.f v and f T defined in (4.7) and (4.8) satisfy 2,ε ∂T ∂x 3 Proof.We will use those inequalities developed in previous section to prove (6.6) and (6.7).First by Hölder's inequality and the anisotropic Ladyzhenskaya's inequality, see [23], we find