Abstract and Applied Analysis

Volume 2013 (2013), Article ID 691615, 12 pages

http://dx.doi.org/10.1155/2013/691615

Research Article

Pullback Attractor for Nonautonomous Primitive Equations of Large-Scale Ocean and Atmosphere Dynamics

Kun Li¹ and Fang Li²

¹Department of Basic, Henan Mechanical and Electrical Engineering College, Xinxiang 453003, China

²Department of Mathematics, Nanjing University, Nanjing 210093, China

Received 9 April 2013; Accepted 18 June 2013

Academic Editor: Grzegorz Lukaszewicz

Copyright © 2013 Kun Li and Fang Li. 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 consider the existence of -pullback attractor for nonautonomous primitive equations of large-scale ocean and atmosphere dynamics in a three-dimensional bounded cylindrical domain by verifying pullback condition.

1. Introduction

This paper is concerned with the existence of pullback attractor for the following nonautonomous primitive equations of large-scale ocean and atmosphere dynamics:

with the following boundary conditions:

and the initial data

in the three-dimensional bounded cylindrical domain

where is a bounded domain in with smooth boundary and is a positive constant, the unknown functions denote the horizontal velocity, the temperature, and the pressure at , respectively, is the Coriolis parameter, Ro is the Rossby number which measures the significant influence of the Earth’s rotation to the dynamical behaviour of the ocean, is a given heat source, denotes the lateral boundary of is the normal vector to , and is a positive constant related with the turbulent heating on the surface of the ocean. The viscosity and the heat diffusion operators and are given by

where are positive constants representing the horizontal and vertical Reynolds numbers, respectively, and are positive constants which stand for the horizontal and vertical eddy diffusivity, respectively. For the sake of simplicity, let be the horizontal gradient operator, and let be the horizontal Laplacian.

Nonautonomous equations appear in many applications in the natural sciences, and so they are of great importance and interest. The long-time behavior of solutions of such equations has been studied extensively in recent years (see [1–12], etc.). The first attempt was to extend the notion of global attractor to the nonautonomous case, leading to the concept of the so-called uniform attractor (see [13]). It is remarkable that the conditions ensuring the existence of the uniform attractor is parallel those for the autonomous case. However, one disadvantage of the uniform attractor is that it does not need to be “invariant,” unlike the global attractor for autonomous systems. Moreover, it is well known that the trajectories may be unbounded for many nonautonomous systems when the time tends to infinity, and there does not exist a uniform attractor for these systems. In order to overcome this drawback, the concept of pullback attractor has been introduced for the nonautonomous case. The theory of pullback attractors has been developed for both nonautonomous and random dynamical systems, and it has been shown to be very useful in the understanding of the dynamics of nonautonomous dynamical systems (see [6]).

Large-scale dynamics of oceans and atmosphere is governed by the primitive equations which are derived from the Navier-Stokes equations with rotation, coupled to thermodynamics and salinity diffusion-transport equations, which account for the buoyancy forces and stratification effects under the Boussinesq approximation. Moreover, due to the shallowness of the oceans and atmosphere, that is, the depth of the fluid layer is very small in comparison to the radius of the Earth, the vertical large-scale motion in the oceans and atmosphere is much smaller than the horizontal one, which in turn leads to modeling the vertical motion by the hydrostatic balance. As a result, one can obtain the system (1)–(4), which is known as the primitive equations for large-scale oceans and atmosphere dynamics (see [14–18]). In the case of ocean dynamics, we know that one has to add the diffusion-transport equation of the salinity to the system (1)–(4), but we will omit it here in order to simplify our mathematical presentation. However, we emphasize that our results are equally valid when the salinity effects are taken into account.

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 (see [14–16, 19–26], etc.). 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 [15]. In [16], the authors proved the existence and the uniqueness of local in time strong solutions of the primitive equations of the ocean. The global existence of strong solutions of the three-dimensional primitive equations of large-scale ocean, by assuming that the initial data are small enough, and the local existence of strong solutions of the three-dimensional primitive equations of large-scale ocean for all initial data were proved by Guillén-González et al. in [27]. In [19], the authors proved the maximum principles for the primitive equations of the atmosphere. The existence and uniqueness of strong solutions, global in time, to the primitive equations in thin domains for a large set of initial data whose sizes depend inversely on the thickness were established in [23]. In [28], Temam and Ziane considered the local existence of strong 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 [24]. In [22], the authors proved the existence of weak solutions and trajectory attractors for the moist atmospheric equations in geophysics. The existence and uniqueness of global strong solutions for the initial boundary value problem of the three-dimensional viscous primitive equations of large-scale ocean were established by the authors in [14]. In [20], the authors 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 -weak topology. Under the assumption of the initial data , the existence of compact global attractor in for the primitive equations of large-scale atmosphere was established by the elliptic regularity theory in [21]. In [26], the authors proved the global existence and uniqueness of -weak solutions when the initial conditions satisfy some regularity. The existence of global attractors in and for three-dimensional viscous primitive equations of large-scale atmosphere in log-pressure coordinates was established in [29]. In [30], the authors proved the existence of -global attractor for the primitive equations of large-scale atmosphere and ocean dynamics by use of the Aubin-Lions compactness theorem. Attractor is an important concept in the study of infinite-dimensional dynamical systems; however, the existence of pullback attractor for the three-dimensional nonautonomous viscous primitive equations of large-scale atmosphere and ocean dynamics remains unsolvable. In this paper, we prove the existence of pullback attractor for nonautonomous equations (1)–(5) by verifying pullback condition.

This paper is organized as follows. In Section 2, we introduce the mathematical framework of the system (1)–(5) and recall some auxiliary lemmas used to give some a priori estimates of strong solutions for (1)–(5) in Section 3, which are used to obtain the existence of -pullback absorbing set. Finally, in Section 4, we are devoted to proving the existence of -pullback attractor for the three dimensional viscous primitive equations of large-scale ocean and atmosphere dynamics by verifying pullback condition.

Throughout this paper, let be a Banach space endowed with the norm , let be the -norm of , and let be positive constants, which may be different from line to line.

2. Preliminaries

2.1. Functional Spaces and Some Lemmas

To study problem (1)–(5), we introduce some function spaces. Let

Denote the closure of by with respect to the following norms, respectively, defined as follows:

for any , and let the closure of with respect to the norm .

Now, we recall some lemmas used in the sequel.

Lemma 1 (see [13, 31]). Let be three positive locally integrable functions on , and for some and all , the following inequalities hold:

Then,

for all .

Lemma 2 (see [14]) (Minkowski inequality). Let , be two measure spaces and be a measurable function about on . If for .. , , and , then

2.2. New Formulation

In this subsection, we divide (1) into two systems with respect to and defined by

Taking the average of (1) in the direction over the interval and using the boundary conditions (3), we obtain

with the boundary conditions

Subtracting (14) from (1), we get

with the boundary conditions

3. Some a Priori Estimates of Strong Solutions

3.1. The Well Posedness of Strong Solutions

We start with the following general existence and uniqueness of solutions which can be obtained by the standard Fatou-Galerkin methods [31] and the similar methods established in [14]. Here, we only state the result as follows.

Theorem 3. Suppose that . Then for any , any initial data , and any , there exists a unique solution of (1)–(5). Furthermore, the solution is continuous with respect to the initial data in .

By Theorem 3, we can define a family of continuous processes in by

where is the solution of (1)–(5) with initial data . That is, a family of mappings satisfies

3.2. Some a Priori Estimates about Strong Solutions

In this subsection, assume that and ; we give some a priori estimates of strong solutions which imply the existence of pullback absorbing set.

3.2.1. Estimate of

Taking the inner product of (2) with in and combining the boundary conditions (4), we get

Integrating by parts and using the boundary conditions (3), we obtain

Since

it is implied that

It follows from (20) and (23) that

Using (23) again, we get

Therefore, we deduce from (24) and (25) that

Integrating (26) from to , we obtain

where we use the inequality

That is,

Let , for any ; integrating (24) from to and combining with (29), we obtain

3.2.2. Estimates of

Multiplying (1) by and integrating by parts over and using the boundary conditions (3), we deduce

That is,

It follows from that

Thanks to (32) and (33), we get

Integrating (34) over and using (23) and (27), we have

where

That is,

Let , for any ; integrating (32) from to and combining with (23), (30), and (37), we obtain

3.2.3. Estimates of

Taking the inner product of (2) with in and using the boundary conditions (4), we have

Using the Young inequality, we obtain

Integrating (40) from to and combining with (27) and , we get

Therefore, we have

3.2.4. Estimates of

Multiplying (16) by and integrating by parts over , we deduce

It follows from interpolation inequality and Lemma 2 that

which implies that

We deduce from (44) and (46) that

Repeating the similar process with the previously mentioned, we have

where we use the inequality .

We deduce from (43) and (47)–(48) that

which implies that

Therefore, it follows from (35)–(38), (42), and Lemma 1 that there exists a positive constant independent of and a constant 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.

3.2.5. Estimates of   

Taking the inner product of (14) with and combining the boundary conditions (15), we deduce

We give the estimates of each term of the right-hand side of (53) as follows:

It follows from (53)–(54) that

In view of (27), (35)–(38), (52), and Lemma 1, we obtain

for any .

3.2.6. Estimates of   

Denote . It is clear that satisfies the following equation obtained by differentiating (1) with respect to :

with the boundary conditions

Multiplying (57) by , integrating over , and using the boundary conditions (58), we get

We estimate the right-hand side of (59) term by term as follows:

It follows from (59)–(60) that

Therefore, we have

It is shown in [30] that

which implies that

for any .

By virtue of Lemma 1, from (23), (27)–(29), and (35), we deduce

for any .

3.2.7. Estimates of   

Taking the inner product of (1) with and combining the boundary conditions (3), we deduce

We estimate each term in the right-hand side of (67) as follows

We derive from (67)–(68) that

Therefore, we have

By virtue of (27), (35), (64)–(66), and Lemma 1, we get

for any .

3.2.8. Estimates of

Multiplying (2) by , integrating over , and combining the boundary conditions (4), we deduce

We give the estimates of each term in the right-hand side of (73) as follows

We derive from (73)–(74) that

Using (27), (64), (71), (72), and Lemma 1, we get

for any .

4. The Existence of the Pullback Attractor

In this section, we recall some definitions and lemmas about pullback attractor and prove the existence of -pullback attractor.

Definition 4 (see [13, 32]). Let be a complete metric space. A two-parameter family of mappings is said to be a continuous process in if

(i) for any ,(ii), (iii), if in .

Definition 5 (see [11, 13, 33]). Let be two Banach spaces. Then, the family is said to be a -pullback attractor for , if

(i) is closed in and compact in for any ,(ii) for any (invariance property),(iii)it pullback attracts every bounded subsets of in the topology of ((,)-pullback attracting), that is, for any nonempty class of bounded subsets ,

for any .

Definition 6 (see [31, 33]). The process is said to be pullback asymptotically compact if for any and any nonempty class of bounded subsets , any sequence , and any sequence , the sequence is relatively compact in .

Lemma 7 (see [33]). Let be a continuous process such that is pullback asymptotically compact. If there exists a family of pullback absorbing sets , then has a unique pullback attractor and

Definition 8 (see [33]). A process is said to satisfy pullback condition (PDC), if for any fixed , any bounded subset of a Banach space and any , there exist a time and a finite-dimensional subspace of such that

(i) is bounded, (ii) for any and any ,

where is a bounded projection and Id is the identity.

Lemma 9 (see [33]). A process satisfying pullback condition (PDC) is pullback asymptotically compact.

From (29), (37), (65), (71), and (76), we get the following theorem.

Theorem 10. Assume that satisfies

Then, for any , there exists a -pullback absorbing set for the processes associated with (1)–(5).

Thanks to the knowledge of functional analysis, we know that the operator with domain

is a positive self-adjoint operator with compact inverse. Therefore, the space possesses an orthonormal basis of eigenfunctions of the operator such that

where

Let , and let be the orthogonal projection onto .

In the following, we prove that the process associated with (1)–(5) is pullback asymptotically compact in by verifying the pullback condition.

Theorem 11. Assume that satisfies

Then, for any , the process associated with (1)–(5) is pullback asymptotically compact in .

Proof. Let be a bounded subset of , where , and .

Taking the inner product of (1) with , we have

From the Young inequality, we obtain

By virtue of Lemma 1, we get

where

Therefore, we obtain

for any , and any , if is sufficiently large.

Similarly, we get

for any , and any , if is sufficiently large. Hence,

for any , and any , if .

From Theorems 10 and 11 and Lemma 7, we obtain the following main Theorem.

Theorem 12. Assume that satisfies

Then, for any , the process associated with the solutions of (1)–(5) possesses a -pullback attractor .

Acknowledgment

The authors of this paper would like to express their sincere thanks to the reviewer for valuable comments and suggestions.

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