Abstract and Applied Analysis

Volume 2013 (2013), Article ID 758730, 16 pages

http://dx.doi.org/10.1155/2013/758730

Research Article

Global Attractors for the Three-Dimensional Viscous Primitive Equations of Large-Scale Atmosphere in Log-Pressure Coordinate

Bo You and Shan Ma

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

Received 4 September 2012; Revised 14 December 2012; Accepted 21 December 2012

Academic Editor: Grzegorz Lukaszewicz

Copyright © 2013 Bo You and Shan Ma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We will first prove the existence of - and -absorbing sets for the three-dimensional primitive equations of large-scale atmosphere in log-pressure coordinate and then prove the existence of -absorbing set by the use of the elliptic regularity theory. Finally, we obtain the existence of - and -global attractors for the three-dimensional viscous primitive equations of large-scale atmosphere in log-pressure coordinate by using the Sobolev compactness embedding theory.

1. Introduction

The fundamental equations governing the motion of the atmosphere consist of the Navier-Stokes equations with Coriolis force and thermodynamics, which account for the buoyancy forces and stratification effects under the Boussinesq approximation. Moreover, due to the shallowness of the atmosphere, that is, the depth of the fluid layer is very small in comparison with the radius of the earth, the vertical large-scale motion in the atmosphere is much smaller than the horizontal one, which in turn leads to modeling the vertical motion by hydrostatic balance ([1–3]). But for large-scale atmospheric flows, it is often necessary to take account of compressibility. This may be done most easily by a change of vertical coordinates to log-pressure coordinate rather than a function of geometric height. As a result, one can obtain the system (1)–(4), which is known as the primitive equations for large-scale atmosphere in log-pressure coordinate.

In recent years, the primitive equations of the atmosphere, the ocean, and the coupled atmosphere-ocean have been extensively studied from the mathematical point of view ([1, 4–13]). The mathematical framework of the primitive equations of the ocean was formulated, and the existence of weak solutions was proved by Lions et al. in [11]. In [1], the authors have studied the primitive equations of large-scale atmosphere in pressure coordinate, where the continuity equation is changed into the incompressible equation, but the thermodynamical equation is very complicated. For the sake of simplicity, the authors have made some approximation of the thermodynamical equation, and in this case, the authors have proved the existence and the uniqueness of strong, local-in-time solutions. By assuming that the initial data is small enough, the authors have studied the existence of strong, global-in-time solutions to the three-dimensional primitive equations of large-scale ocean and obtained the existence of strong, local-in-time solutions to the equations for all initial data in [14]. In [5], the authors have proved the maximum principles of the temperature for the primitive equations of the atmosphere in pressure coordinate. The existence and the uniqueness of strong, global-in-time solutions to the primitive equations in thin domains for a large set of initial datum whose sizes depend inversely on the thickness were established in [9]. In [15], Temam and Ziane have considered the existence of strong, local-in-time solutions for the primitive equations of the atmosphere, the ocean, and the coupled atmosphere-ocean. Asymptotic analysis of the primitive equations under the small depth assumption was established in [10]. In [8], the authors have proved the existence of weak solutions and a trajectory attractors for the moist atmospheric equations in geophysics. The existence and the uniqueness of strong, global-in-time solutions for the three-dimensional viscous primitive equations of large-scale ocean and atmosphere dynamics were established by the authors in [4, 16]. In [6], the authors have considered the long-time dynamics of the primitive equations of large-scale atmosphere and obtained a weakly compact global attractor which captures all the trajectories with respect to the weak topology of . The existence and the uniqueness of strong solutions, the existence of smooth solutions, and the existence of a compact global attractor in for the primitive equations with more regular initial data than that in [6] were established in [7]. In [12], the authors have proved the existence and the uniqueness of -weak, global-in-time solutions when the initial conditions satisfy some regularity. The existence of global attractor in for the primitive equations with initial datum in and the heat source was proved by use of the Aubin-Lions compactness lemma in [17]. In [18], the authors have proved the existence of the global attractor in for the three-dimensional viscous primitive equations of large-scale ocean and atmosphere dynamics. In this paper, we will consider the primitive equations of large-scale atmosphere in log-pressure coordinate, where the continuity equation is transformed into the incompressible equation with the weighted function , which is more complicated than that in pressure coordinate, but we do not need to make any approximation of the thermodynamical equation such that the accuracy of the temperature is higher than that in pressure coordinate which will reveal the real world better. Moreover, the appearance of the weighted function makes the calculations more difficult. In order to make sure of the existence of the absorbing set, we need to add some additional conditions (10) on , which means that the thickness of is very small. In [18], there are not any restrictions on the thickness of to ensure the existence of the absorbing sets, and the estimate of can be obtained directly by taking the inner product (15) with . But in this paper, thanks to the appearance of the weighted function , we have to first estimate in and then in to obtain a priori estimates on in . In this paper, we will prove the existence of -global attractor for the three-dimensional primitive equations of large-scale atmosphere in log-pressure coordinate under the assumptions (10) on .

In this paper, we will consider the following three-dimensional viscous primitive equations of large-scale atmosphere in log-pressure coordinate [3]:

in the domain

where is a bounded domain in with smooth boundary. Here, , is the velocity field, is the temperature, is the geopotential, is the Coriolis parameter, is the vertical unit vector, is a positive constant, and is the heat source. The operators and are given by

where and are positive constants representing the horizontal and the vertical Reynolds numbers, respectively, and and are positive constants which stand for the horizontal and the vertical heat diffusivities, respectively. And is used as a vertical coordinate, where is a constant “scale height” and is a constant reference pressure. For the sake of simplicity, let be the horizontal gradient operator and the horizontal Laplacian.

We denote the different parts of the boundary by

We equip (1)–(4) with the following boundary conditions, with nonslip and nonflux on the side walls and bottom [4]:

where is a positive constant.

In addition, (1)–(8) are supplemented with the following initial conditions:

Additionally, assume that satisfy

This paper is organized as follows. Section 2 is devoted to introduce the mathematical framework and the definition of solutions for (1)–(10) and give some auxiliary lemmas used in the sequel. In Section 3, we will give the existence and the uniqueness of solutions for (1)–(10). Next, some a priori estimates of solutions corresponding to (1)–(10) will be established, which imply the existence of - and -absorbing sets for (1)–(10). In Section 4, we first obtain the existence of the -global attractor for (1)–(10) by compactly, and then get the existence of -absorbing set of (1)–(10) by the elliptic regularity theory. Finally, we prove the existence of the -global attractor for the three-dimensional viscous primitive equations of large-scale atmosphere in log-pressure coordinate by the Sobolev compactness embedding theory.

Throughout this paper, let be positive constants independent of which may be different from line to line.

2. Mathematical Setting of (1)–(10)

2.1. Reformulation of (1)–(9)

Integrating the continuity equation (3) and taking the boundary conditions for into account, we obtain

And then, we consider the vertical momentum equation, that is, the second equation (2). First, we define an unknown function on (i.e., ), where is a constant reference pressure, say

which is the geopotential of the atmosphere on .

Then,

Therefore, (1)–(4) can be rewritten as follows:

with the following boundary conditions:

and the initial data

At last, we will divide (14) into two systems with respect to and [4], where and are defined by

where

and we have .

Therefore, satisfies the following systems:

with the boundary conditions

and satisfies the following systems

with the boundary conditions

2.2. Some Function Spaces

In this subsection, we will quote briefly some notations and function spaces used in this paper.

First of all, we introduce the notations for some function spaces on as follows:

Let (or ) be the weighted Hilbert spaces of (or ) functions or (or )-vector valued functions on and denote the usual -norms by , where (or ). Denote the inner product in by and the norm in by , respectively, given by

for any and .

Similarly, define

for any and , respectively.

Let be the closure of with respect to the following norms:

for any and , respectively, and let .

Next, we recall some results used in the sequel.

Lemma 1 (see [19]). (i) for ,

(ii) for ,

Proposition 2. There exists a positive constant such that

for any .

Lemma 3 (see [4, 6]). Let , , and . Then, we have

(1), (2),(3).

Lemma 4 (see [4], (Minkowski inequality)). Let, be two measure spaces and a measurable function about on , and is a measurable positive function. If for and , then

Finally, we will recall the definition of strong solutions and state it as follows.

Definition 5. Assume that and and let be any fixed positive time. is called a strong solution of (14)–(18) on , if

and they satisfy

for every and , almost every .

3. Some a Priori Estimates for (1)–(10)

We start with the following general existence and uniqueness of solutions which can be obtained by the standard Fatou-Galerkin methods [20–22] and similar methods in [4]. Here, we only state it as follows.

Theorem 6. Assume that and and let be any fixed positive time. Then, there exists a unique strong solution of (14)–(18) on , which satisfies

and depends continuously on the initial data ; that is, the mapping is continuous in .

By Theorem 6, we can define the operator semigroup in as

which is -continuous.

In this section, we will give some a priori estimates which imply the existence of the -absorbing set for (1)–(10). That is, for any bounded subset , there exists such that for any , where is a semigroup on Banach space generated by (1)–(10).

3.1. Estimates of

Taking the inner product of (15) with , using the boundary conditions (17) and using Lemma 3, we deduce that

Employing Hölder’s inequality yields

Thanks to

we obtain

Setting and , we find that

which implies that

Therefore, there exist three positive constants such that

for any . For brevity, we omit writing out explicitly these bounds here, and we also omit writing out other similar bounds in our future discussion for all other uniform a priori estimates.

3.2. Estimates of

Multiplying (14) by , integrating over , and using Lemma 3, we deduce that

Setting and , we obtain

From the classical Gronwall inequality, we have

for all , which implies that there exist three positive constants ,, such that

for all .

3.3. Estimates of

Taking the inner product of (15) with and using Lemma 3, we have

From the inequality , we get

which implies that

Therefore, by virtue of the uniform Gronwall lemma, from (44) and , we deduce that

for any .

3.4. Estimates of

Multiplying (23) by , integrating over , and using Lemma 3, we deduce that

It follows from the Sobolev interpolation inequality and Lemma 4 that

which implies that

where we use the inequality .

We deduce from (54) and (56) that

Therefore, it follows from (44)–(49) and the uniform Gronwall lemma that

for any .

3.5. Estimates of

Taking the inner product of (21) with and combining the boundary conditions (22), we deduce that

By using the Hölder inequality, we get

It follows from (60) and (61) that

In view of (44)–(49) and (59) and the uniform Gronwall lemma, we obtain

for any .

3.6. Estimates of

Denote that . It is clear that satisfies the following equation obtained by differentiating (14) with respect to :

with the boundary conditions

Multiplying (64) by , integrating over , and using Lemma 3, we get

It follows from the Young inequality and the Sobolev embedding theorem that

Since

which implies that

for any .

By virtue of the uniform Gronwall lemma, from (43), (49), and (69), we deduce that

for any .

3.7. Estimates of

Taking the inner product of (14) with and combining the boundary conditions (16), we deduce that

We derive from (72) and the Young inequality that

By virtue of (44)–(49), (69)–(71), and the uniform Gronwall lemma, we get

for any .

3.8. Estimates of

Denote . It is clear that satisfies the following equation obtained by differentiating (15) with respect to :

with the boundary conditions

Multiplying (76) by , integrating over , and using Lemma 3, we deduce that

We deduce from the Young inequality that

It follows from (78) and (79) that

As a result of (44), (53), (69)–(71), and (74), we get

for any .

3.9. Estimates of

Taking the inner product of (15) with and combining the boundary conditions (17), we deduce that

We derive from the Young inequality that

Using (44), (69), (74), (75), and (81), we get

for any .

3.10. Estimates of

Taking the inner product of (64) with , combining the boundary conditions (65), and using Lemma 3, we deduce that

From the Young inequality, we obtain

Thanks to (44), (69), and (71), we have

for any .

3.11. Estimates of

Taking the inner product of (76) with , combining the boundary conditions (77), and using Lemma 3, we deduce that

It follows from the Hölder inequality that

We derive from the Young inequality that

As a consequence of the uniform Gronwall lemma, we get

for any .

3.12. Estimates of

Denote that ,  . It is clear that satisfy the following equations obtained by differentiating (14) and (15) with respect to , respectively,

with the boundary conditions

Multiplying (94) and (95), respectively, by ,  , integrating over , and using Lemma 3, we deduce that

It follows from the Young inequality that

Multiplying (15) by and integrating over , we obtain

Similarly, we have

We derive from the Young inequality that

By use of the uniform Gronwall inequality, we have

for any .

Moreover, we have

for any .

3.13. Estimates of

Multiplying (15) by and integrating over , we obtain

Similarly, we have

It follows from the Young inequality that

By virtue of (48), (69), (74), (85), (89), (93), and (103), we have

for any .

3.14. Estimates of

Taking the inner product of (94) with and combining the boundary conditions (96), we have

Therefore, in view of the Young inequality, we obtain

Employing the uniform Gronwall inequality, we get

for any .

Moreover, we have

Similarly, we can also obtain

for any .

3.15. Estimates of

Denote that ,  . It is clear that ,   satisfy the following equations obtained by differentiating (94) and (95) with respect to , respectively,

with the boundary conditions

Taking the inner product of (114) with and combining the boundary conditions (116), we have

Therefore, in view of the Young inequality, we obtain

Employing the uniform Gronwall inequality, we get

for any .

Similarly, we can also obtain

for any .

From (43), (48), (70), (74), (81), (85) and (108), we get the following results.

Theorem 7. Assume that and . Let be a semigroup generated by the initial boundary problem (14)–(18). Then, there exists an absorbing set in . That is, for any bounded subset of , there exist a positive constant and a positive time which depend on the norm of , such that

for any .

Corollary 8. Assume that and . Let be a semigroup generated by the initial boundary problem (14)–(18). Then, there exist an absorbing set in . That is, for any bounded subset of , there exist a positive constant and a positive time , depending on the norm of , such that

for any .

4. The Existence of the Global Attractors

In this section, we will recall some definitions and lemmas about the global attractor and prove the existence of the global attractor in .

Definition 9 (see [23]). Let be a Banach space and a family operators on . We say that is a norm-to-weak continuous semigroup on , if satisfies that

(i) (the identity), (ii), for all ,(iii) if and in .

Lemma 10 (see [24]). Let , be two Banach spaces and , be the dual spaces of ,, respectively. If is dense in , the injection is continuous and its adjoint is dense. Let be a semigroup on and , respectively, and assume furthermore that is continuous or weak continuous on . Then, is a norm-to-weak continuous semigroup on if and only if maps compact subsets of into bounded sets of .

Definition 11 (see [20]). Let be a Banach space and let be a continuous semigroup on . A subset in is called a global attractor if and only if

(1) is invariant; that is, for all ,(2) is compact in ,(3) attracts each bounded subset in .

Definition 12 (see [20]). Let be a semigroup on Banach space is called uniformly compact, if for any bounded (in ) set , there exists , such that is relatively compact in .

Lemma 13 (see [20, 24]). Let be a Banach space and a semigroup on . Then has a global attractor in provided that the following conditions hold true:

(1) has a bounded absorbing set in , (2) is uniformly compact.

Lemma 14 (see [20, 24, 25]). Let be a norm-to-weak continuous semigroup such that is uniformly compact. If there exists an absorbing set , then has a global attractor and

By virtue of the Rellich-Kondrachov theorem, we obtain the following.

Corollary 15. Assume that ,. Then, the semigroup corresponding to (14)–(18) has a -global attractor .

To prove the existence of the -absorbing set, we need the following lemma.

Lemma 16 (see [7]). Let be a solution to the following Stokes problem:

If ,  , then

Next, we will prove the existence of absorbing set in by the use of Lemma 16, which implies the existence of -global attractor of the initial boundary problem (14)–(18).

Theorem 17. Assume that and . Let be a semigroup generated by the initial boundary problem (14)–(18). Then, there exists an absorbing set in . That is, for any bounded subset of , there exists a positive constant depending on the norm of , such that

for any , where is specified in Theorem 7.

Proof. From the results of a priori estimates in the previous section, we know that there exists a strong solution to (14)–(18), which satisfies for any ,  ,  ,  ,,, where is specified in Theorem 7. Then,

where

Since and , for any , we have

which implies that

Since is a strong solution to (14)–(18) and , it is clear that . Thus,

By the use of Lemma 16, we get

Using the Sobolev embedding theorem , we know that

Now, since and , for any , we have

which implies that

Therefore, we find that

From Lemma 16, we know that

which implies that

Since and , it is clear that . Thus,

From Lemma 16, we deduce that

Employing the elliptic regularity theory, we obtain

This complete the proof of Theorem 17.

Since is compact, from Theorem 7, we know that the semigroup generated by (14)–(18) is compact in . From Theorem 7 and Lemma 10, we deduce that the semigroup generated by (14)–(18) maps a compact set of to be a bounded set of ; that is, the semigroup generated by (14)–(18) is norm to weak continuous in . Furthermore, compactly, and from Theorem 7 and Lemma 13, we immediately get the following result.

Theorem 18. Assume that . Then, the semigroup corresponding to (14)–(18) has a -global attractor .

By using the Rellich-Kondrachov theorem, we can obtain the existence of the -global attractor for .

Appendix

Proof of Proposition 2

If this conclusion is not true, then for each natural number , there exists such that

Let

Hence,

which implies that there exists a subsequence of (still denoted by ) such that

Therefore, we have

which implies that

Combining with , we obtain

this is a contradiction.

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