Journal of Applied Mathematics

Volume 2012 (2012), Article ID 867310, 18 pages

doi:10.1155/2012/867310

Research Article

Steady State Solution to Atmospheric Circulation Equations with Humidity Effect

Hong Luo

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

Received 26 April 2012; Accepted 17 September 2012

Academic Editor: B. V. Rathish Kumar

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.

The steady state solution to atmospheric circulation equations with humidity effect is studied. A sufficient condition of existence of steady state solution to atmospheric circulation equations is obtained, and regularity of steady state solution is verified.

1. Introduction

This paper is concerned with steady state solution of the following initial-boundary problem of atmospheric circulation equations involving unknown functions at ( is a period of field ),

where are constants, denote velocity field, temperature, humidity, and pressure, respectively, are known functions, and is a 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 presented in atmospheric circulation with humidity effect. 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 [1–4] or an infinite dimensional dynamical system [5–7]. We deduce atmospheric circulation models which are able to show features of atmospheric circulation and are 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 [8].

We investigate steady state solution of the atmospheric circulation equations in this paper. The steady state solution is a special state of evolution equations and the time-independent solution, which plays a very important role on understanding the dynamical behavior of the evolution equations and is the main directions and important content in studying evolution equations. Steady state solutions of some systems are studied [9–12]. The purpose to consider with the steady state solution of atmospheric circulation equations is to seek the conditions under which atmospheric circulation is stable and to understand structure of the circulation cell.

We discuss the existence and regularity of steady state solution to atmospheric circulation equations (1.1)–(1.4) with the boundary condition (1.6). In other words, we discuss the following equations:

The paper is organized as follows. In Section 2 we present preliminary results. In Section 3, we prove that the systems (1.8)–(1.13) possess steady state solutions in , by using space sequence method. In Section 4, by using Sard-Smale Theorem and energy method, we obtain regularity of the solutions to the models (1.8)–(1.13).

Let , and denote norm of the space .

2. Preliminaries

We introduce theory of linear elliptic equation and ADN theory of Stokes equation.

We consider with divergence form of linear elliptic equation:

where , , , is uniformly elliptic, that is, there exist constants such that

The problem (1.1) is supplemented with the following Dirichlet boundary condition

We define three classes of solutions of (2.1) and (2.3).

(1)Classical solution: if there is a function satisfying (2.1), (2.3), we say is a classical solution to (2.1) and (2.3). (2)Strong solution: if there is a function satisfying (2.1), (2.3) almost everywhere for some , we say is a strong solution to (2.1) and (2.3). (3)Weak solution: if there is a function satisfying

and (2.3), we say is a weak solution to (2.1) and (2.3).

Lemma 2.1 2.1 (see [13] (Schauder Theorem)). Let be a field, , . If is a solution to (2.1) and (2.3), then

where depends on and -norm of the coefficient functions , , .

Lemma 2.2 2.2 (see [13] ( Theorem)). Let be a field, , , , . If is a solution of (2.1) and (2.3), then

where depends on and -norm or -norm of the coefficient functions.

One considers with Stokes equation

Lemma 2.3 (see [14, 15] (ADN theory of Stokes equation)). (1) Let , , . If is a solution of (2.7), then the solution , and

where depends on .

(2) Let , , . If is a solution of (2.7), then the solution , and

where depends on .

Lemma 2.4. The eigenvalue equation:

has eigenvalue , and

Let be a linear space, , two Banach space, separable, and reflexive. Let . There exists a linear mapping

Definition 2.5. A mapping is called weakly continuous provided

for all , in .

Lemma 2.6 (see [2]). If is weakly continuous, is bounded open set, , and

then the equation has a solution in .

One introduces the Sard-Smale Theorem of infinite dimensional operator. Let , be two separable Banach Spaces, be a mapping. is called a Fredholm operator provided the derivative operator is a Fredholm operator for all .

Lemma 2.7 (see [16, 17] (Sard-Smale Theorem)). Let be a Fredholm operator with zero index. Then regular value of is dense in . If is critical value of , then is discrete set.

3. Existence of Steady State Solution

Theorem 3.1. If , and is the first eigenvalue of the elliptic equation (2.10), then for all , (1.8)–(1.13) have a solution , .

Proof. Let (1.11)–(1.13)}, and (1.11)–(1.13)}.

Define , for all ,

Firstly, we prove the coercivity of .

Let be appropriate small. Then

From , it follows that

Then there exists an appropriate large constant such that

Furthermore, we verify that is weakly continuous.

Let in , we have from the Sobolev imbedding Theorem

By , in , it follows that

Combining the general Hölder inequality and (3.6), we deduce

Then,

Thus,

As , in , we find

Combining the general Hölder inequality and (3.6), we deduce

Then,

By , in , we have

Combining the general Hölder inequality and (3.6), we deduce

Then,

Thus,

Combining (3.10)–(3.17), we have

which imply that is weakly continuous. According to Lemma 2.6, (1.8)–(1.13) have a solution .

Lastly we prove that , .

From the Hölder inequality, we see

Then, .

For the Stokes equation:

since , according to ADN theory, (3.20) has a solution:

By the Hölder inequality, we have

thus, .

For the elliptic equation:

as , according to theory of linear elliptic equation, (3.23) has a solution

From the Hölder inequality, we see

thus, .

For the elliptic equation:

as , according to theory of linear elliptic equation, (3.26) has a solution

By the Sobolev imbedding Theorem, we see

Then , and

Thus, . Consequently . According to ADN theory, (3.20) has a solution

Similarly, we deduce

By doing the same procedures as above, (1.8)–(1.13) have a solution , .

4. Regularity of Steady State Solution

Theorem 4.1. If , and is the first eigenvalue of elliptic equation (2.10), then there exists a dense open set , the solution to (1.8)–(1.13) is finite for all .

Proof. There are the following estimates for (1.8)–(1.13):

As is a solution to (1.8)–(1.13), we have . Then

Thus,

Consequently,

Choosing an appropriate constant , we see

According to the Sobolev imbedding Theorem and (4.5), we deduce

Using the Gagliardo-Nironberg inequality and Young inequality, we have

Combining the Hölder inequality and (4.6)–(4.9), we see

Since are solutions to (3.20), (3.23), and (3.26), according to ADN theory and theory of linear elliptic equation, we have

From (4.6) and (4.10), it follows that

Let . Then

which imply (4.1).

We introduce the mappings:

Let

Then, (1.8)–(1.13) can be rewrite as the following mapping

Clearly, is a completely continuous field. Thus is a Fredholm operator with zero index. According to the Sard-Smale Theorem, the regular value of is dense in , and is discrete in for all . in is finite for from (4.1). Consequently, is interior point and is an open set.

Theorem 4.2. If , and is the first eigenvalue of elliptic equation (2.10), then

(1)the Equations (1.8)–(1.13) have a classical solution for all , (2)there exists a tense open set , such that the solution to (1.8)–(1.13) is finite for all , (3)the solution to (1.8)–(1.13) is in if .

Proof. We prove the assertion (1). As , for all , for all . From Theorem 4.1, (1.8)–(1.13) have a strong solution , .

When , , from the Sobolev imbedding theorem. Then

Thus,

For Stokes equation:

as , according to ADN theory, (4.19) has a solution:

For the elliptic equation:

as , according to theory of linear elliptic equation, (4.21) has a solution:

For the elliptic equation:

as , according to theory of linear elliptic equation, (4.23) has a solution:

Thus, (1.8)–(1.13) have a solution .

Secondly, we prove the assertion (2). Combining ADN theory, theory of linear elliptic equation, and (4.19)–(4.23), we have

We introduce the mappings:

Let

then, (1.8)–(1.13) can be rewritten as

Clearly, is complete continuous field. Then is a Fredholm operator with zero index. The regular value is dense from Sard-Smale theorem, and is discrete in for all . From (4.25), we find that is finite in . Thus, is an interior point and is an open set.

Finally we prove the assertion (3). Since , it is true that ( is arbitrary integer). According to Theorem 4.1, we conclude that ( is arbitrary integer). From the Sobolev imbedding theorem, ( is arbitrary integer). Then .

5. Remark

is a sufficient condition, not a necessary condition. In fact, if the condition does not hold, (1.8)–(1.13) may have not solution for some .

Returning to the problem of atmospheric circulation, as the temperature source and the moisture source are changed, the state of the atmospheric circulation changes, but there is still a corresponding steady state.

Funding

The National Natural Science Foundation of China (no. 11271271) and the NSF of Sichuan Education Department of China (no. 11ZA102).

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