- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

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

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

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

^{1}Department of Basic, Henan Mechanical and Electrical Engineering College, Xinxiang 453003, China^{2}Department 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

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.

#### References

- C. S. Cao and E. S. Titi, “Global well-posedness and finite-dimensional global attractor for a 3-D planetary geostrophic viscous model,”
*Communications on Pure and Applied Mathematics*, vol. 56, no. 2, pp. 198–233, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. S. Cao, E. S. Titi, and M. Ziane, “A ‘horizontal’ hyper-diffusion three-dimensional thermocline planetary geostrophic model: well-posedness and long-time behaviour,”
*Nonlinearity*, vol. 17, no. 5, pp. 1749–1776, 2004. View at Publisher · View at Google Scholar · View at MathSciNet - D. N. Cheban, P. E. Kloeden, and B. Schmalfuß, “The relationship between pullback, forwards and global attractors of nonautonomous dynamical systems,”
*Nonlinear Dynamics and Systems Theory*, vol. 2, pp. 9–28, 2002. View at Zentralblatt MATH · View at MathSciNet - H. Crauel and F. Flandoli, “Attractors for random dynamical systems,”
*Probability Theory and Related Fields*, vol. 100, no. 3, pp. 365–393, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Crauel, A. Debussche, and F. Flandoli, “Random attractors,”
*Journal of Dynamics and Differential Equations*, vol. 9, no. 2, pp. 307–341, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Caraballo, G. Łukaszewicz, and J. Real, “Pullback attractors for asymptotically compact non-autonomous dynamical systems,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 64, no. 3, pp. 484–498, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. E. Kloeden and B. Schmalfuß, “Nonautonomous systems, cocycle attractors and variable time-step discretization,”
*Numerical Algorithms*, vol. 14, no. 1–3, pp. 141–152, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. E. Kloeden and D. J. Stonier, “Cocycle attractors in nonautonomously perturbed differential equations,”
*Dynamics of Continuous, Discrete and Impulsive Systems*, vol. 4, no. 2, pp. 211–226, 1998. View at Zentralblatt MATH · View at MathSciNet - G. Łukaszewicz, “On pullback attractors in ${H}_{0}^{1}$ for nonautonomous reaction-diffusion equations,”
*International Journal of Bifurcation and Chaos*, vol. 20, no. 9, pp. 2637–2644, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. S. Lu, H. Q. Wu, and C. K. Zhong, “Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces,”
*Discrete and Continuous Dynamical Systems A*, vol. 13, no. 3, pp. 701–719, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Schmalfuß, “Attractors for non-autonomous dynamical systems,” in
*Proceedings of the International Conference on Differential Equations (Equadiff '99)*, B. Fiedler, K. Gröer, and J. Sprekels, Eds., pp. 684–689, World Scientific, Berlin, Germany, 2000. - L. Yang, M. H. Yang, and P. E. Kloeden, “Pullback attractors for non-autonomous quasi-linear parabolic equations with dynamical boundary conditions,”
*Discrete and Continuous Dynamical Systems B*, vol. 17, no. 7, pp. 2635–2651, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. V. Chepyzhov and M. I. Vishik,
*Attractors for Equations of Mathematical Physics*, vol. 49 of*American Mathematical Society Colloquium Publications*, American Mathematical Society, Providence, RI, USA, 2002. View at MathSciNet - C. S. Cao and E. S. Titi, “Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics,”
*Annals of Mathematics*, vol. 166, no. 1, pp. 245–267, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. L. Lions, R. Temam, and S. Wang, “New formulations of the primitive equations of atmosphere and applications,”
*Nonlinearity*, vol. 5, no. 2, pp. 237–288, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. L. Lions, R. Temam, and S. Wang, “On the equations of the large-scale ocean,”
*Nonlinearity*, vol. 5, no. 5, pp. 1007–1053, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Pedlosky,
*Geophysical Fluid Dynamics*, Springer, New York, NY, USA, 1987. - G. K. Vallis,
*Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation*, Cambridge University Press, Cambridge, UK, 2006. - B. D. Ewald and R. Témam, “Maximum principles for the primitive equations of the atmosphere,”
*Discrete and Continuous Dynamical Systems A*, vol. 7, no. 2, pp. 343–362, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. L. Guo and D. W. Huang, “On the existence of atmospheric attractors,”
*Science in China D*, vol. 51, no. 3, pp. 469–480, 2008. View at Publisher · View at Google Scholar · View at Scopus - B. L. Guo and D. W. Huang, “On the 3D viscous primitive equations of the large-scale atmosphere,”
*Acta Mathematica Scientia B*, vol. 29, no. 4, pp. 846–866, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. L. Guo and D. W. Huang, “Existence of weak solutions and trajectory attractors for the moist atmospheric equations in geophysics,”
*Journal of Mathematical Physics*, vol. 47, no. 8, Article ID 083508, 23 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Hu, R. Temam, and M. Ziane, “The primitive equations on the large scale ocean under the small depth hypothesis,”
*Discrete and Continuous Dynamical Systems A*, vol. 9, no. 1, pp. 97–131, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Hu, “Asymptotic analysis of the primitive equations under the small depth assumption,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 61, no. 3, pp. 425–460, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Petcu, R. Temam, and D. Wirosoetisno, “Existence and regularity results for the primitive equations in two space dimensions,”
*Communications on Pure and Applied Analysis*, vol. 3, no. 1, pp. 115–131, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Tachim Medjo, “On the uniqueness of $z$-weak solutions of the three-dimensional primitive equations of the ocean,”
*Nonlinear Analysis: Real World Applications*, vol. 11, no. 3, pp. 1413–1421, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - F. Guillén-González, N. Masmoudi, and M. A. Rodríguez-Bellido, “Anisotropic estimates and strong solutions of the primitive equations,”
*Differential and Integral Equations*, vol. 14, no. 11, pp. 1381–1408, 2001. View at Zentralblatt MATH · View at MathSciNet - R. Temam and M. Ziane, “Some mathematical problems in geophysical fluid dynamics,” in
*Handbook of Mathematical Fluid Dynamics*, vol. 3, pp. 535–658, 2005. View at Zentralblatt MATH · View at MathSciNet - B. You and S. Ma, “Global attractors for three dimensional viscous primitive equations of large-scale atmosphere in log-pressure coordinate,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 758730, 16 pages, 2013. View at Publisher · View at Google Scholar - N. Ju, “The global attractor for the solutions to the 3D viscous primitive equations,”
*Discrete and Continuous Dynamical Systems A*, vol. 17, no. 1, pp. 159–179, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Temam,
*Infinite-Dimensional Dynamical Systems in Mechanics and Physics*, Springer, New York, NY, USA, 1997. View at MathSciNet - Y. Li and C. K. Zhong, “Pullback attractors for the norm-to-weak continuous process and application to the non-autonomous reaction-diffusion equations,”
*Applied Mathematics and Computation*, vol. 190, no. 2, pp. 1020–1029, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Wang and C. Zhong, “Pullback $D$-attractors for nonautonomous Sine-Gordon equations,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 67, no. 7, pp. 2137–2148, 2007. View at Publisher · View at Google Scholar · View at MathSciNet