Abstract and Applied Analysis

Volume 2012 (2012), Article ID 172956, 15 pages

doi:10.1155/2012/172956

Research Article

Global Attractor of Atmospheric Circulation Equations with Humidity Effect

Hong Luo

College of Mathematics and Software Science, Sichuan Normal University, Sichuan, Chengdu 610066, China

Received 1 June 2012; Accepted 15 July 2012

Academic Editor: Jinhu Lü

Copyright © 2012 Hong Luo. 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.

Global attractor of atmospheric circulation equations is considered in this paper. Firstly, it is proved that this system possesses a unique global weak solution in . Secondly, by using C-condition, it is obtained that atmospheric circulation equations have a global attractor in .

1. Introduction

This paper is concerned with global attractor of the following initial-boundary problem of atmospheric circulation equations involving unknown functions at ( is a period of field ):

where , , , and are constants, , , , and denote velocity field, temperature, humidity, and pressure, respectively; , are known functions, and is constant matrix:

The problems (1.1)–(1.4) are supplemented with the following Dirichlet boundary condition at and periodic condition for :

and initial value conditions

The partial differential equations (1.1)–(1.7) were firstly presented in atmospheric circulation with humidity effect [1]. Atmospheric circulation is one of the main factors affecting the global climate, so it is very necessary to understand and master its mysteries and laws. Atmospheric circulation is an important mechanism to complete the transports and balance of atmospheric heat and moisture and the conversion between various energies. On the contrary, it is also the important result of these physical transports, balance and conversion. Thus, it is of necessity to study the characteristics, formation, preservation, change and effects of the atmospheric circulation and master its evolution law, which is not only the essential part of human’s understanding of nature, but also the helpful method of changing and improving the accuracy of weather forecasts, exploring global climate change, and making effective use of climate resources.

The atmosphere and ocean around the earth are rotating geophysical fluids, which are also two important components of the climate system. The phenomena of the atmosphere and ocean are extremely rich in their organization and complexity, and a lot of them cannot be produced by laboratory experiments. The atmosphere or the ocean or the couple atmosphere and ocean can be viewed as an initial and boundary value problem [2–5], or an infinite dimensional dynamical system [6–8]. We deduce the atmospheric circulation model (1.1)–(1.7) which is able to show features of atmospheric circulation and is easy to be studied from the very complex atmospheric circulation model based on the actual background and meteorological data, and we present global solutions of atmospheric circulation equations with the use of the -weakly continuous operator [1]. In fact, there are numerous papers on this topic [9–13]. Compared with some similar papers, we add humidity function in this paper. We propose firstly the atmospheric circulation equation with humidity function which does not appear in the previous literature.

As far as the theory of infinite-dimensional dynamical system is concerned, we refer to [9–11, 14–18]. In the study of infinite dimensional dynamical system, the long-time behavior of the solution to equations is an important issue. The long-time behavior of the solution to equations can be shown by the global attractor with the finite-dimensional characteristics. Some authors have already studied the existence of the global attractor for some evolution equations [2, 3, 13, 19–21]. The global attractor strictly defined as -limit set of ball, which under additional assumptions is nonempty, compact, and invariant [13, 17]. Attractor theory has been intensively investigated within the science, mathematics, and engineering communities. Lü et al. [22–25] apply the current theoretical results or approaches to investigate the global attractor of complex multiscroll chaotic systems. We obtain existence of global attractor for the atmospheric circulation equations from the mathematical perspective in this paper.

The paper is organized as follows. In Section 2, we recall preliminary results. In Section 3, we present uniqueness of the solution to the atmospheric circulation equations. In Section 4, we obtain global attractor of the equations.

denote norm of the space ; and are variable constants. Let satisfy (1.4), (1.6)}, and satisfy (1.4), (1.6)}.

2. Preliminaries

Let and be two Banach spaces, a compact and dense inclusion. Consider the abstract nonlinear evolution equation defined on , given by

where is an unknown function, a linear operator, and a nonlinear operator.

A family of operators () is called a semigroup generated by (2.1) if it satisfies the following properties:

(1) is a continuous map for any ;(2) is the identity;(3), for all . Then, the solution of (2.1) can be expressed as

Next, we introduce the concepts and definitions of invariant sets, global attractors, and -limit sets for the semigroup .

Definition 2.1. Let be a semigroup defined on . A set is called an invariant set of if , for all . An invariant set is an attractor of if is compact, and there exists a neighborhood of such that for any ,

In this case, we say that attracts . Particularly, if attracts any bounded set of , is called a global attractor of in .

For a set , we define the -limit set of as follows:

where the closure is taken in the -norm. Lemma 2.2 is the classical existence theorem of global attractor by Temam [13].

Lemma 2.2. Let be the semigroup generated by (2.1). Assume that the following conditions hold:

(1) has a bounded absorbing set , that is, for any bounded set there exists a time such that ,  for  all   and ;(2) is uniformly compact, that is, for any bounded set and some sufficiently large, the set is compact in .

Then the -limit set of is a global attractor of (2.1), and is connected providing is connected.

Definition 2.3 (see [19]). We say that satisfies -condition, if for any bounded set and , there exist and a finite dimensional subspace such that is bounded, and

where is a projection.

Lemma 2.4 (see [19]). Let () be a dynamical systems. If the following conditions are satisfied:

(1)there exists a bounded absorbing set ;(2) satisfies -condition,

then has a global attractor in .

From Linear elliptic equation theory, one has the following.

Lemma 2.5. The eigenvalue equation:

has eigenvalue , and

3. Uniqueness of Global Solution

Theorem 3.1. If , and is the first eigenvalue of elliptic equation (2.6), then the weak solution to (1.1)–(1.7) is unique.

Proof. From [1], , is the weak solution to (1.1)–(1.7). Then for all , , we have

Set and are two weak solutions to (1.1)–(1.7), which satisfy (3.1). Let . Then,

Let . We obtain from (3.2) the following:

Then,

By using the Gronwall inequality, it follows that

which imply . Thus, the weak solution to (1.1)–(1.7) is unique.

4. Existence of Global Attractor

Theorem 4.1. If , and is the first eigenvalue of elliptic equation (2.6), then (1.1)–(1.7) have a global attractor in .

Proof. According to Lemma 2.4, we prove Theorem 4.1 in the following two steps.

Step  1. Equations (1.1)–(1.7) have an absorbing set in .

Multiply (1.1) by and integrate the product in :

Then,

Multiply (1.2) by and integrate the product in :

Then,

Multiply (1.3) by and integrate the product in :

Then,

We deduce from (4.2)–(4.6) the following:

Let be appropriate small such that

Then,

Applying the Gronwall inequality, it follows that

Then, when , for any , here is a bounded in , there exists such that

where is a ball in , at of radius . Thus, (1.1)–(1.7) have an absorbing in .

Step  2. -condition is satisfied.

The eigenvalue equation:

has eigenvalues and eigenvector , and . If , then . constitutes an orthogonal base of .

For all , we have

When , . Let be small positive constant, and . There exists positive integer such that

Introduce subspace . Let be an orthogonal subspace of in .

For all , we find that

Let be the orthogonal projection. Thanks to Definition 2.3, we will prove that for any bounded set and , there exists such that

From Step  1, has an absorbing set . Then for any bounded set , there exists such that , for all , which imply (4.16).

Multiply (1.1) by and integrate over ). We obtain

Then,

where is a constant which needs to be determined.

From (4.14), we find that

where .

Thanks to and , it follows that

We deduce from (4.11) the following:

Using (4.19)–(4.22), we find that

Let satisfy and . Then,

By the Gronwall inequality, we find that

Then, there exists satisfying

Since , for it follows that

Multiply (1.2) by and integrate over ). We obtain

Then,

where is a constant which needs to be determined.

From (4.14), we find that

where .

Since and , it follows that

Using (4.22) and (4.29)–(4.31), we find that

Let satisfy . Then

By the Gronwall inequality, we find that

Then, there exists satisfying

Since , for , it follows that

Multiply (1.3) by and integrate over ). We obtain

Then,

where is a constant which needs to be determined.

From (4.14), we find that

where .

Since and , we see that

Using (4.22) and (4.38)–(4.40), we obtain

Let satisfy . Then,

By the Gronwall inequality, we find that

Then, there exists satisfying

Since , for , it follows that

From (4.27); (4.36) and (4.45) for all there exists such that when , it follows that

which imply (4.17). From Lemma 2.4, (1.1)–(1.7) have a global attractor in .

Acknowledgments

This work was supported by national natural science foundation of China (no. 11271271) and the NSF of Sichuan Education Department of China (no. 11ZA102).

References

H. Luo, “Global solution of atmospheric circulation equations with humidity effect,” submitted

T. Ma and S. H. Wang, Phase Transition Dynamics in Nonlinear Sciences, Springer, New York, NY, USA, 2012.

T. Ma, Theories and Methods in Partial Differential Equations, Academic Press, Beijing, China, 2011 (Chinese).

N. A. Phillips, “The general circulation of the atmosphere: A numerical experiment,” Quarterly Journal of the Royal Meteorological Society, vol. 82, no. 352, pp. 123–164, 1956.

C. G. Rossby, “On the solution of problems of atmospheric motion by means of model experiment,” Monthly Weather Review, vol. 54, pp. 237–240, 1926.

J.-L. Lions, R. Temam, and S. H. Wang, “New formulations of the primitive equations of atmosphere and applications,” Nonlinearity, vol. 5, no. 2, pp. 237–288, 1992.

J.-L. Lions, R. Temam, and S. H. Wang, “On the equations of the large-scale ocean,” Nonlinearity, vol. 5, no. 5, pp. 1007–1053, 1992.

J.-L. Lions, R. Temam, and S. H Wang, “Models for the coupled atmosphere and ocean. (CAO I),” Computational Mechanics Advances, vol. 1, no. 1, pp. 5–54, 1993.

C. Foias, O. Manley, and R. Temam, “Attractors for the Bénard problem: existence and physical bounds on their fractal dimension,” Nonlinear Analysis: Theory, Methods & Applications, vol. 11, no. 8, pp. 939–967, 1987.

B. L. Guo, “Spectral method for solving two-dimensional Newton-Boussinesq equations,” Acta Mathematicae Applicatae Sinica. English Series, vol. 5, no. 3, pp. 208–218, 1989.

B. Guo and B. Wang, “Approximate inertial manifolds to the Newton-Boussinesq equations,” Journal of Partial Differential Equations, vol. 9, no. 3, pp. 237–250, 1996.

T. Ma and S. Wang, “El Niño southern oscillation as sporadic oscillations between metastable states,” Advances in Atmospheric Sciences, vol. 28, no. 3, pp. 612–622, 2011.

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2nd edition, 1997.

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, vol. 25 of Studies in Mathematics and Its Applications, North-Holland Publishing, Amsterdam, The Netherlands, 1992.

J.-M. Ghidaglia, “Finite-dimensional behavior for weakly damped driven Schrödinger equations,” Annales de l'Institut Henri Poincaré, vol. 5, no. 4, pp. 365–405, 1988.

J.-M. Ghidaglia, “A note on the strong convergence towards attractors of damped forced KdV equations,” Journal of Differential Equations, vol. 110, no. 2, pp. 356–359, 1994.

J. K. Hale, Asymptotic Behavior of Dissipative Systems, vol. 25 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, USA, 1988.

O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Lincei Lectures, Cambridge University Press, Cambridge, UK, 1991.

Q. Ma, S. Wang, and C. Zhong, “Necessary and sufficient conditions for the existence of global attractors for semigroups and applications,” Indiana University Mathematics Journal, vol. 51, no. 6, pp. 1541–1559, 2002.

T. Ma and S. H. Wang, Stability and Bifurcation of Nonlinear Evolution Equations, Science Press, Beijing, China, 2007 (Chinese).

T. Ma and S. Wang, Bifurcation Theory and Applications, vol. 53 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific, Hackensack, NJ, USA, 2005.

J. Lü and G. Chen, “Generating multiscroll chaotic attractors: theories, methods and applications,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 16, no. 4, pp. 775–858, 2006.

J. Lü, F. Han, X. Yu, and G. Chen, “Generating 3-D multi-scroll chaotic attractors: a hysteresis series switching method,” Automatica, vol. 40, no. 10, pp. 1677–1687, 2004.

J. Lü, G. Chen, X. Yu, and H. Leung, “Design and analysis of multiscroll chaotic attractors from saturated function series,” IEEE Transactions on Circuits and Systems. I, vol. 51, no. 12, pp. 2476–2490, 2004.

J. Lü, S. Yu, H. Leung, and G. Chen, “Experimental verification of multidirectional multiscroll chaotic attractors,” IEEE Transactions on Circuits and Systems I, vol. 53, no. 1, pp. 149–165, 2006.