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A squeezing property and its applications to a description of long time behaviour in the 3D viscous primitive equations

We consider the 3D viscous primitive equations with periodic boundary conditions. These equations arise in the study of ocean dynamics and generate a dynamical system in a Sobolev H^1 type space. Our main result establishes the so-called squeezing property in the Ladyzhenskaya form for this system. As a consequence of this property we prove (i) the finiteness of the fractal dimension of the corresponding global attractor, (ii) the existence of finite number of determining modes, and (iii) ergodicity of a related random kick model. All these results provide a new information concerning long time dynamics of oceanic motions.


Introduction
We deal with the 3D viscous primitive equations which arise in geophysical fluid dynamics for modeling large scale phenomena in oceanic motions. These equations are based on the so-called hydrostatic approximation of the 3D Navier-Stokes equations for velocity field u which also contain a rotational (Coriolis) force and coupled to thermo-and salinity diffusiontransport equations (see, e.g., the survey [26] and the references therein). In this model a small variation of the density ρ of the fluid is taken into account via the buoyancy term only and has the form ρ = ρ 0 − αT + βs, where T is the temperature and s denotes the salinity. If we take into account one of the factors (T or s) or the diffusivities for T and s are equal, then their effect in the dynamics can be represented simply by the density ρ, or equivalently the buoyancy b = ρ 0 − ρ = αT − βs (see [26]). This is why we prefer to deal with the variables u and b. In order to simplify our mathematical presentation we impose on u and b the periodic boundary conditions of the same type as in [24] (see also [26] and the references therein). Thus we consider our problem in the domain O = (0, L 1 ) × (0, L 2 ) × (−L 3 /2, L 3 /2) ⊂ R 3 and denote the spatial variable in O byx = (x, z) = (x 1 , x 2 , z) ∈ O. We use the notation ∇, div and ∆ for the gradient, divergence and Laplace operators in two dimensional variable x = (x 1 , x 2 ). Below we also denote by ∆ x,z the 3D Laplace operator. The notations ∇ x,z and div x,z have a similar meaning.
We consider the following equations for the fluid velocity field u = (v(x, t); w(x, t)) = (v 1 (x, t); v 2 (x, t); w(x, t)),x = (x, z), for the pressure P = P (x, t) and for the buoyancy b = b(x, t): where ν > 0 is the dynamical viscosity, G f and G b are volume sources, and f is the Coriolis parameter. We denote v ⊥ = (−v 2 ; v 1 ). We supplement (1)-(4) with the periodic boundary conditions imposed on (v; w), P and b. As in [24,26] we also assume that v and P are even with respect to z and w and b are odd with respect to z (hence w z=0 = 0 and b z=0 = 0). These requirements and also (2), (3) lead to the following relations for everyx = (x, z) ∈ O. It is important (see [3] for the basic discussion) that the pressure p is independent of the vertical variable z and depends on 2D (horizontal) variable x only. Exploring intensively this observation the authors of the paper [3] have implemented a new effective approach for proving of the global well-posedness for problems like (1)-(4), see the discussion below.
The system in (1)-(4) was studied by many authors for the different types of boundary conditions (see the survey [26] and the references therein). The existence of week solutions [23] and (global) well-posedness of strong solutions were established (see [3] and also [17,18,24]). The existence of a global attractor for the viscous 3D primitive equations was proved in [16] (see also the paper [24] devoted to the periodic case). However, to the best of our knowledge, the question concerning dimension of this attractor is still open. We note that the author of [16] claimed the finiteness of the dimension in 2007 with the reference to a forthcoming paper which is not published yet.
It should be also noted that stochastic perturbations of (1)-(4) were studied in [10,14,15] (see also the references in these publications) and the paper [11] deals with a random kick forcing of the primitive equations.
Our goal is to show that the system in (1)-(4) possesses an additional regularity and satisfies a squeezing property in the Ladyzhenskaya form (see [21,22]). This property implies several important facts about dynamics of the system and, in particular, the finiteness of the fractal dimension of the attractor (see Corollary 3.6) and the existence of finite number of determining modes or functionals (Corollary 3.7). The squeezing property also plays an important role in the study of turbulent behaviour in some classes of dynamical systems with random kicks arising in the fluid dynamics (see, e.g., [19,20]). In this paper we apply the squeezing property established and also the theory developed in [20] to present a result on ergodicity of (1)-(4) forced by random bounded kicks with appropriately chosen frequency (see Corollary 3.9).
As we will see below the main ingredient of our argument is the uniform smoothing stated in Theorem 3.1. There are two ways to obtain this asymptotic regularity property. One method involves appropriate (spatial) multipliers which produce higher kinetic energy type norms in the corresponding energy balance relation. Another method consists of two steps. In the first step the standard phase space estimates for higher time derivatives of solutions are established. Then, on the second step, spatial smoothness of solutions is derived via appropriate elliptic regularity of the corresponding stationary problem. Both methods requires some compatibility conditions involving the initial data and the nonlinearity (see, e.g., [27] for a general presentation). In the periodic case these compatibility relations are satisfied automatically. However the first method is a one-step method and looks simpler. Moreover, it requires minimal hypotheses concerning time regularity of external forces. This why it was used in [24,26] and also in this paper. In the case of other boundary conditions (free type as in [3] or mixed free-Dirichlet as in [18]) we need rather careful analysis of compatibility conditions and elliptic properties of the stationary problem. We plan to analyze this issue in our further publications.
It is commonly recognized (see [3,17,18,24,26]) that the main difficulty arising in the study of the primitive equations in (1)-(4) is related to the hydrodynamical part. The calculations which should be made to obtain appropriate bounds for a buoyancy type variable are either standard or repeat the corresponding argument for the fluid variables in the simplified form. This is why below we state our results for the full problem (1)-(4), but in the proof we concentrate on the fluid equations only. Moreover, for the transparency of our argument we even assume that the buoyancy equation is absent. This corresponds to the case of the zero buoyancy (b(x, t) ≡ 0) which is possible when G b ≡ 0 and b(x, t = 0) = 0.
The paper is organized as follows. Section 2 contains some preliminary material. Here we represent the primitive equations in some standard form, introduce Sobolev type spaces and state the well-known result on well-posedness of strong solutions. In Section 3 we state our main results. The key outcome of Theorem 3.1 is the existence of an absorbing ball lying in an H 2 type space. This allows us not only guarantee the existence of a global attractor but also provides an important step in the proof of the squeezing in Theorem 3.5. In this section we also apply our main results to prove the finiteness of the dimension of the global attractor, the existence of finite number of determining functionals, and ergodicity of a random kicks forced model generated by (1)-(4). In Section 4 we demonstrate the main steps of the proofs of Theorems 3.1 and 3.5 in the case when the buoyancy variable b is absent.

Preliminaries
In this section we rewrite system (1)-(4) in the canonical form, introduce appropriate Sobolev type spaces and quote the result on the strong wellposedness.
Using the representations in (5) we arrive at the system supplied with the conditions: We also need to add initial data for v and b: We denote byḢ s per (O) the Sobolev space of order s consisting of periodic functions such that O f dx = 0. To describe fluid velocity fields we introduce the following spaces with s = 0, 1, 2: We also denote H = V 0 and equip H with L 2 type norm · and denote by (·, ·) the corresponding inner product. The spaces V 1 and V 2 are endowed with the norms · V 1 = ∇ x,z · and · V 2 = ∆ x,z · .
To describe the buoyancy we need the spaces equipped with the standard norms. We also denote W s = V s × E s with the standard (Euclidean) product norms.
As it was already mentioned, starting with [3] the global well-posedness of the equations in (6) and (7) was studied by many authors [17,18,24,26]. In this section we quote the result from [24] (see also [26]) on well-posedness of strong solutions under the periodic boundary conditions.
Moreover, this solution continuously depends on time t and initial data U 0 and generates a dynamical system (S t , W 1 ) with the phase space W 1 and the evolution operator S t defined by the relation S t U 0 = U (t; U 0 ).This evolution operator S t possesses the following Lipschitz property [24,26] do not contain the proofs of continuity of S t U with respect to t and the Lipschitz property in W 1 . However almost the same calculations as in [16] allow us to show these properties. We also note that in the case considered unique global solvability can be also established in smoother classes. [24,26]) the solution U (t) belongs to the class C(R + ; W m ) ∩ L 2 (0, T ; W m+1 ) for every T > 0. This observation is important in our further considerations due to possibility to use smooth approximations of solutions in our calculations.

Remark 2.2 Formally
We conclude this preliminary section with the statement of the following well-known uniform Gronwall lemma.

Results
We start with the following assertion which partially follows from Proposition 2.1 and is based on the calculations given in [24]. 3 . Then the dynamical system (S t , W 1 ) generated by problem (6)-(10) is dissipative and compact 1 . Moreover, there exists R * > 0 such that the ball is absorbing, i.e., for any bounded set B in W 1 there exists t B such that In this ball B there is a forward invariant absorbing set D.
We note that the dissipativity and compactness of (S t , W 1 ) stated in Theorem 3.1 is known from [24] under slightly weaker conditions on G f and G b . However the existence of an absorbing compact set which is bounded in W 2 requires additional hypotheses and additional calculations.

Remark 3.2
In the case when G f ≡ 0 and G b ≡ 0 one can prove (see some argument below in Section 4.1) that This means that in the case of vanishing sources the zero equilibrium state is asymptotically stable.
Using the standard results (see, e.g., one of the monographs [1, 5, 28]) on the existence of global attractors we can derive from Theorem 3.1 the following assertion.

Corollary 3.3 (Global Attractor)
Let the hypotheses of Theorem 3.1 be in force. Then the dynamical system (S t , W 1 ) generated by problem (6)-(10) possesses a compact global attractor A which is a bounded set in W 2 . Moreover, in the case when G b = 0 the subspace V 1 × {0} ⊂ W 1 attracts the trajectories with exponential speed and the attractor A lies in V 2 × {0}.

Remark 3.4
The result on the existence of a global attractor for 3D viscous primitive equations was known before, see [16], and also Remark 2.1 in [24] for the periodic case. Our improvement is that we state the boundedness of A in an H 2 type Sobolev space. One can also show that in the case of sufficiently smooth sources G f and G b the attractor A possesses additional spatial smoothness. Using the method presented in [24] we can even prove Gevrey regularity of the elements from the attractor provided the sources possess appropriate smoothness properties. However we do not pursue these improvements because our main goal is a squeezing property with application to dimension and ergodicity.
In the space W 1 we consider the bilinear form This form generates a positive self-adjoint operator A with a discrete spectrum. Due to periodicity of the problem the corresponding eigenfunctions and eigenvalues can be easily described. We assume that they are reorganized in such way that and, moreover, {e k } is the orthonormal basis in W 0 . We denote by P N the orthoprojector onto Span{e 1 , . . . , e N } and Q N = I − P N . Now we can state our main result.
Theorem 3.5 (Squeezing) Let the hypotheses of Theorem 3.1 be in force. Then the dynamical system (S t , W 1 ) generated by problem (6)-(10) on the forward invariant absorbing set D possesses the properties: • Lipschitz property: For any U, U * ∈ D we have the relation where C D and α D are positive constants.
• Squeezing property: For every T > 0 and 0 < q < 1 there exists for any U and U * from the absorbing forward invariant set D.
The squeezing property stated in Theorem 3.5 allow us to apply Ladyzhenskaya Theorem on the dimension of invariant sets (see [21,22] and also Theorem 8.1 in [5,Chap.1]). This yields the following assertion.
Corollary 3.6 (Finite Dimension) Under the hypotheses of Theorem 3.1 the global attractor A of the dynamical system (S t , W 1 ) generated by problem (6)-(10) has a finite fractal dimension.
This means that the asymptotic dynamics in this model is finite-dimensional and topologically can be represented by a compact set in R d with appropriate d < ∞.
The next outcome of the squeezing property is the existence of a finite number of (asymptotically) determining modes. We note that this notion was introduced in [13] for 2D Navier-Stokes system and was studied by many authors (see the discussion in the recent paper [12]). For a general theory of the determining functionals we refer to [4], see also [5] and [8] for a development of this theory based on the notion of the completeness defect.
Corollary 3.7 (Determining Modes) Let the hypotheses of Theorem 3.1 be in force and N be such that (12) holds for some T > 0 and 0 < q < 1. Then the dynamical system (S t , W 1 ) possesses N determining modes. This means that the relation Proof. The relation in (12) implies that for any U and U * from the absorbing forward invariant set D. Iterating (14) we obtain that for every n = 1, 2, . . ., where U and U * are from D. This inequality and also the Lipschitz property (11) makes it possible to prove that relation (13) implies that S t U − S t U * W 1 → 0 as t → +∞. For some details in the abstract situation we refer to the proof of Theorem 1.3 in [5, Chap.5].
Remark 3.8 Using the same idea as in [8] it is also possible to derive from (15) the existence of other finite families of determining functionals. We also point out the recent paper [25] which contains the proof of the existence of a finite number of determining modes (and also nodes and local volume averages) by another method. Now we consider an example of an application of the results above to ergodicity of a random kicks forced model generated by problem (6)- (10). We deal with the simplest situation and suppose that G f ≡ 0 and G b ≡ 0. Moreover, we consider a model in some fixed ball from W 1 and assume that the frequency of kicks is smaller than a certain critical value (depending on the radius of the ball and the amplitude of the kicks). We plan to remove all these restrictions and discuss further developments in our joint studies with S.Kuksin and A.Shirikyan.
Let {η k } be i.i.d. random variables in W 1 . Assume that the support of the distribution D(η 1 ) in W 1 contains the origin and is bounded, i.e., there exists R kick > 0 such that Here and below we keep the notation B(R) for the ball with the center at 0 of the radius R in the space W 1 .
Let us fix R > R kick and choose T c = T ( R, R kick , D) ≥ 1 such that where D is the forward invariant absorbing set given by Theorem 3.1. This choice of T c is possible due to Remark 3.2 and Theorem 3.1. Now we fix T ≥ T c and define a Markov chain in W 1 by the formula This chain is called a random kicks model generated by (6)-(10) (with G f ≡ 0 and G b ≡ 0). The parameter T −1 has meaning of the frequency of the kicks. We refer to [20] for motivation and physical importance of different types of kick models in the turbulence theory. It is obvious that if U 0 ∈ B( R), then we also have that U k ∈ B( R) for all k. Thus we also have a chain in B( R).
It follows from Proposition 2.1 that Let us fix 0 < q < 1 such that η = qC T, R < 1. Since S T −1 U, S T −1 U * ∈ D, we can apply Theorem 3.5 and choose N such that for all U, U * ∈ B( R). Thus we have the squeezing property on the ball B( R). Applying now Theorem 3.2.5 [20] 2 and the properties of the mapping S t established above we arrive to the following assertion. Corollary 3.9 (Ergodicity) Assume that P N η 1 and Q N η 1 are independent and the distribution of P N η 1 in the (finite-dimensional) space P N W 1 has a density p(U ) with respect to the Lebesgue measure dU on P N W 1 such that where C is a constant. Then for the chain (16) there is a unique invariant probability Borel measure µ on B( R). Moreover, there is 0 < γ < 1 such that F * k λ − µ * L ≤ C( R, λ) · γ k , k = 1, 2, . . . for any probability Borel measure λ on B( R), where the Borel measure F * k λ and the corresponding dual Lipschitz norm · * L are defined in the Appendix.
To apply [20, Theorem 3.2.5] (see Theorem A.1 in the Appendix) we first consider the extension S T of S T from B( R) on the whole space W 1 by the formula One can see that S T maps W 1 into B( R − R kick ). Thus after the first step in (16) we are in B( R) and therefore we have S T = S T for k ≥ 2 in (16). We also note that S T is globally Lipschitz (the proof of the latter property can be found in [9, p.64], for instance) and Conditions (A1)-(A4) of Theorem A.1 follow directly.
For other consequences of the Ladyzhenskaya squeezing property for the model considered we refer to [6].

Proofs
Our arguments are more or less standard and use the methods developed in [3,16,24] (see also the survey [26]). In fact they are some refinement of the calculations known now from [3], see also [16,24,26]). This is why we consider only key steps in the argument. Moreover, we deal with the reduced system which appears in the following way.
If we assume G b ≡ 0 and b 0 = 0, we can take b ≡ 0 as a solution to (7). Thus we arrive to the problem: to find a (horizontal) fluid velocity field v(x, t) = (v 1 (x, t); v 2 (x, t)),x = (x, z), and the pressure p(x, t) satisfying the equation (17) in O × (0, +∞) supplied with conditions (8) and with the initial data v(x, z, 0) = v 0 (x, z). (18) It follows from Proposition 2.1 that for every G f ∈ V 1 and v ∈ V 1 problem (17), (18), (8) has a unique strong solution These solutions generate a dynamical system (S t , V 1 ) in V 1 . Below we prove our main result for this system. Thus we deal with calculations involving the velocity field v, but not the buoyancy. Moreover using approximation procedure and the results from [24,26] we can even assume that all our calculations deal with smooth solutions. We also not that in the case of full system (6)-(10) with G b = 0 the multiplier b in (7) leads to the estimate b(t) ≤ e −c 0 t b 0 for t > 0. Thus using dissipativity stated in Theorem 3.1 we can conclude that for all initial data from a bounded set B ⊂ W 1 . This means that the subspace V 1 × {0} is exponentially attracting and implies that the global attractors for (6)- (10) and (17) are the same.

Preliminary a priori estimates
In order to prove Theorem 3.1 for the system (S t , V 1 ) generated by (17) we need only to check the existence of absorbing ball in the space V 2 . For this we need to repeat and partially refine some calculations in [16,24].

A priori estimate in H
If we multiply (17) by v in H, then we obtain with v(0) ≤ R. Moreover, we can choose R 2 0 = ε + c 1 G f 2 with arbitrary ε > 0, in this case t R also depends on ε.

Splitting
As in [3] one can see that the fieldsv andṽ satisfy the equations

H 1 -estimates
Now we multiply (21) by |ṽ| 4ṽ and integrate over O. In the same way as in [3,16] we obtain d dt This implies that Since ṽ L 6 ≤ c ∇v 2 + ∂ z v 2 , we can apply Lemma 2.3 and relation (19) to obtain that there exists R 1 ≥ R 0 > 0 such that for any R > 0 there exists t * R ≥ t R such that Moreover using (19) with with v(0) ≤ R. Using (22) we can also assume that with v(0) ≤ R. In the case G f ≡ 0 we can change C(R 1 ) into Cε and t * R into t * R,ε . Next we multiply (20) by −∆v in L 2 (T 2 ). As in [3] using the relations This implies (see [3] for some details) that Therefore (22) and (23) give us that there exists R 2 > 0 such that for any As above in the case G f ≡ 0 we can take R 2 2 of the order ε (with t * * R depending on ε). The next step is the estimate for v z . We first note that u ≡ v z solves the problem (25) in the class of periodic (odd in z) functions. Since u ⊥ u = 0 and using the multiplier u after integration by parts we obtain that which implies (for some details we refer to [3]) that Therefore there exists R 3 > 0 such that for any R > 0 there is t with v(0) ≤ R. Moreover, R 2 2 ∼ ε in the case G f ≡ 0. Next we estimate ∇v . For this we multiply (17) by −∆v. As in [3] we obtain for all t ≥ t R with v(0) ≤ R. Moreover one can see from the analysis above that in the case G f ≡ 0 for any 0 < ε < 1 there exists t R,ε > 0 such that The relation in (26) yields that the system (S t , V 1 ) is dissipative in V 1 . In the case G f ≡ 0 by (27) an absorbing ball can be chosen with an arbitrary small radius which implies that the zero equilibrium is asymptotically stable in this case (as it is claimed in Remark 3.2 when the buoyancy is presented).

Completion of the proof of Theorem 3.1
To conclude the proof we need to show uniform smoothing of trajectories. We note that the result for individual trajectories is known from [24] (see also [26]). However in these references the size of the corresponding ball is not controlled and thus we need some refinement of that results. This is why below we mainly follow the line of argument in [24,26].
We multiply (17) by (−∆ x,z ) 2 v to get where w(v) = w(x, t) is defined by (5). We also use the facts that The general structure of the nonlinear terms in (28) is given by the formula Here where k, i, j = 1, 2, D k is a differential operator of order k, and the sign ∼ means that we deal with linear combination of the terms in the right hand side.
Using the embeddings H 1 (O) ⊂ L 6 (O) and H 1/2 (O) ⊂ L 3 (O) (see [29]) and also interpolation one can see that To estimate the second term in (29) we start with the case k = 2: To estimate the term with k = 1 in (29) we use the following lemma (see Proposition 2.2 in [2]).
This lemma yields the following estimate Thus we arrive at the following relation Our next step is an estimate for ∂ z v L 6 . As above let u = ∂ z v. We multiply (25) by |u| 4 u. Using the relations u ⊥ u = 0 and we obtain that In the same way as in [24,26] after integration by parts we have that This implies that Thus we have that Using Lemma 2.3 and the estimate in (26) we obtain that Now returning to (30) and applying Lemma 2.3 allow us to obtain the following assertion.
Proposition 4.2 Let G f ∈ V 1 and ∂ z G f ∈ L 6 (O). Then there exists R * > 0 such that the ball B = {v ∈ V 2 : v V 2 ≤ R * } is absorbing for the dynamical system (S t , V 1 ) generated by problem (17). Moreover, there is a forward invariant absorbing set D in this ball.
Thus the proof of Theorem 3.1 (in the case b 0 ≡ 0 and G b ≡ 0) is complete.

Squeezing estimate
Now we prove Theorem 3.5 (in the case b 0 ≡ 0 and G b ≡ 0). We start with the following assertion.

Proposition 4.3
For any v * , v * * from the absorbing forward invariant set D we have the relation where C D and α D are positive constants. Moreover, for v * (t) =S t v * and v * * (t) =S t v * * we also have that Proof. We use the same type of argument as in [16]. For this we write the equation Then we multiply (31) by (−∆ x,z )u and apply the same argument as in [16]. At the final stage we use the fact that for initial data v * and v * * from D.
Let {λ k } be the eigenvalues of the corresponding Stokes type operator A which arises in problem (17) and {e k } be the corresponding eigenvectors. We can assume that We denote by P N the orthoprojector in H on Span{e 1 , . . . , e N }. Let Q N = I − P N . We multiply (31) by Q N Au and obtain that Now we need the following lemma which follows from the Hölder inequality and from the standard Sobolev embedding (see, e.g., [29]): Lemma 4.4 Let O ⊂ R 3 and u, v be sufficiently smooth functions. Then for any 0 < s, r < 3/2. such that r + s = 3/2.
Using this lemma to estimate the right hand side in (32) on the forward invariant set D we can find that where R * is the same as in the statement of Proposition 4.2. This yields for any ε > 0. This implies that

Using Proposition 4.3 we obtain that
Now for any fixed T > 0 and 0 < q < 1 we can choose ε and N such that Thus we arrive to the squeezing property: For every T > 0 and 0 < q < 1 there is N such that for any v 1 , v 2 from the absorbing forward invariant set D.

A Appendix: Kuksin-Shirikyan ergodicity theorem
In this Appendix for the readers convenience we state Theorem 3.2.5 [20] on the ergodicity of a class of Markov chains. Let H be a separable Banach space. We denote by B(R) the ball with the center at 0 of the radius R in the space H. Let S be a continuous (nonlinear) operator from H into itself. Let {η k } be i.i.d. random variables in H such that η k ≤ r * for some r * > 0. We consider a Markov chain in H defined by the relations U k = F k (U k−1 ) ≡ SU k−1 + η k , k = 1, 2, . . .
We assume that the following hypotheses are satisfied.
(A1) S is a Lipschitz mapping on every ball B(̺) and for any R > r > 0 there exists n * = n(R, r) such that S n B(R) ⊂ B(r) for all n ≥ n * .
(A2) There exists R > 0 such that for every R > 0 we have that A k (R) ⊂ B( R) for all k ≥ k * (R), where the sets A k (R) is defined by the relations (A4) Assume that P η 1 and (I − P )η 1 are independent and the distribution of P η 1 in the (finite-dimensional) space P H has a density p(U ) with respect to the Lebesgue measure dU on P H such that where C is a constant.
We also recall (see, e.g., [20]) that the dual Lipschitz norm in the space of probability Borel measures on H is defined by the relation where L b (H) is the space of bounded Lipschitz functions on H endowed with the norm Now we are in position to state Theorem 3.2.5 [20].
for any probability Borel measure λ on H for which the integral in the righthand side of the relation above is finite (the function k * (R) is defined in (A2)). Here above the probability measure F * k λ is defined on a Borel set Γ by the formula where P k (U 0 , Γ) = P(U k ∈ Γ) is the transition probability for the chain.
We note that the result in Theorem A.1 formally deals with a Markov chain in a whole space. However one can see that the arguments from [20] remain true if we restrict the iteration procedure in (16) on an invariant subset. This observation was already used in [7] to prove ergodicity for random kick-forced 3D Navier-Stokes equations in a thin domain and can be also applied to obtain Corollary 3.9.