Abstract and Applied Analysis

VolumeΒ 2012Β (2012), Article IDΒ 172956, 15 pages

http://dx.doi.org/10.1155/2012/172956

## Global Attractor of Atmospheric Circulation Equations with Humidity Effect

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

Received 1 June 2012; Accepted 15 July 2012

Academic Editor: JinhuΒ LΓΌ

Copyright Β© 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)β(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)β(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 humanβs 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 [2β5], or an infinite dimensional dynamical system [6β8]. We deduce the atmospheric circulation model (1.1)β(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 [9β13]. 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 [9β11, 14β18]. 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, 19β21]. 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. LΓΌ et al. [22β25] 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 ,ββforββallββ 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)β(1.7) is unique.*

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

Set and are two weak solutions to (1.1)β(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)β(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)β(1.7) have a global attractor in .*

* Proof. *According to Lemma 2.4, we prove Theorem 4.1 in the following two steps.*Stepββ1.* Equations (1.1)β(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)β(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)β(1.7) have an absorbing in .*Stepββ2. *-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 Stepββ1, 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)β(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)β(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)β(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)β(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).

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