# Regularity of global solution to atmospheric circulation equations with humidity effect

Boundary Value Problems20122012:143

DOI: 10.1186/1687-2770-2012-143

Accepted: 14 November 2012

Published: 5 December 2012

## Abstract

In this article, the regularity of the global solutions to atmospheric circulation equations with humidity effect is considered. Firstly, the formula of the global solutions is obtained by using the theory of linear operator semigroups. Secondly, the regularity of the global solutions to atmospheric circulation equations is presented by using mathematical induction and regularity estimates for the linear semigroups.

MSC:35D35, 35K20, 35Q35.

### Keywords

global solution regularity atmospheric circulation equations humidity effect

## 1 Introduction

This paper is concerned with the regularity of solutions to the following initial-boundary problem of atmospheric circulation equations involving unknown functions at ( is a period of field ):
(1.1)
(1.2)
(1.3)
(1.4)
where , , , are constants, , T, q, p denote velocity field, temperature, humidity and pressure respectively, Q, G are known functions, and σ is a constant matrix
The problems (1.1)-(1.4) are supplemented with the following Dirichlet boundary condition at and the periodic condition for :
(1.5)
(1.6)
and initial value conditions
(1.7)

Partial differential equations (1.1)-(1.7) are 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. Moreover, 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 a 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 initial and boundary value problems [25], or an infinite dimensional dynamical system [68]. We deduce atmospheric circulation models which are able to show the 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 T-weakly continuous operator [1]. In [9], the steady state solutions to atmospheric circulation equations with humidity effect are studied. A sufficient condition of the existence of the steady state solutions to atmospheric circulation equations is obtained, and the regularity of the steady state solutions is verified. In this article, we investigate the regularity of the solutions to atmospheric circulation equations (1.1)-(1.7).

The paper is organized as follows. In Section 2, we present preliminary results. In Section 3, we present the formula of the solution to the atmospheric circulation equations. In Section 4, we obtain the regularity of the solutions to equations (1.1)-(1.7).

denotes the norm of the space X, and C, are variable constants.

## 2 Preliminaries

We consider the divergence form of the linear elliptic equation
(2.1)
where , , , is uniformly elliptic, i.e., there exist constants such that
The problem (2.1) is supplemented with the following Dirichlet boundary condition:
(2.2)

Lemma 2.1 [10] (Theory of linear elliptic equations)

Let be a field, , , , . If is a solution of Eqs. (2.1), (2.2), then

where depends on n, p, λ, Ω and -norm or -norm of the coefficient functions.

We consider the Stokes equation
(2.3)

Lemma 2.2 [11, 12] (ADN theory of the Stokes equation)

Let , , . If () is a solution of Eq. (2.3), then the solution , and

where depends on μ, n, k, α, Ω.

Let X be a linear space, , be two separable reflexive Banach spaces, and H be a Hilbert space. , , and H are completion spaces of X under the respective norm. are dense embedding. is a continuous mapping. We consider the abstract equation
(2.4)

where , is unknown.

Definition 2.3 Let be a given initial value. () is called a global solution of Eq. (2.4) if u satisfies
Definition 2.4 Let . is called uniformly weak convergence in if is bounded, and
(2.5)
Definition 2.5 A mapping is called T-weakly continuous if for , and uniformly weakly converges to , we have

Lemma 2.6 [3]

Assume is T-weakly continuous and satisfies:

(A1) there exists , (), such that

where , are constants, (), , is a seminorm of , ,

(A2) there exists for any and ,

where depends only on T, , , and .

Then for any , Eq. (2.4) has a global weak solution

If is Frechét differentiable, then the regular solution can be presented under some condition.

We introduce a space sequence
where X, , H are such as in Lemma 2.6, is a Banach space, is a Hilbert space, and are compact including. There exist a constant and a nonnegative function () such that
(2.6)
(2.7)
Lemma 2.7 In addition to the assumptions about the existence of a global solution in Lemma  2.6, if is Frechét differentiable and satisfies (2.6), (2.7), then Eq. (2.4) has a unique global solution

for all .

Lemma 2.8 [13]

Let L be a generator of a strongly continuous semigroup . If is a weak solution to the equation
(2.8)
and , then the solution can be read as
Note that we used to assume that the linear operator L in (2.8) is a sectorial operator which generates an analytic semigroup . It is known that there exists a constant such that generates the fractional power operators and fractional order spaces for , where . Without loss of generality, we assume that ℒ generates the fractional power operators and fractional order spaces as follows:

where is the domain of . By the semigroup theory of linear operators (Pazy [13]), we know that is a compact inclusion for any .

Lemma 2.9 [1315] (Imbedding theorem of factional order spaces)

Let be a Lipschitz field, be a sectorial operator, , and . Then for , the fractional order spaces satisfy the following relations:
and the inequalities

For sectorial operators, we also have the following properties which can be found in [13].

Lemma 2.10 Let be a sectorial operator which generates an analytic semigroup . If all eigenvalues λ of L satisfy for some real number , then for , we have
1. (1)

is bounded for all and ,

2. (2)

, ,

3. (3)
for each , is bounded and

where some , is a constant only depending on α,
1. (4)
the -norm can be defined by
(2.9)

2. (5)
ifis symmetric, for any , we have

## 3 Formula of global solutions

We introduce the spaces
Let
Then Eqs. (1.1)-(1.7) can be rewritten as an abstract equation
Theorem 3.1 If , , then the global solution ϕ of Eqs. (1.1)-(1.7) can be read as
(3.1)

where is an analytic semigroup generated by L, and is a Leray projection.

Proof As is a weak solution to Eqs. (1.1)-(1.7) [1], from the Hölder inequality and the Sobolev imbedding theorem, it follows that
Then . Hence,
(3.2)
From the Hölder inequality and the Sobolev imbedding theorem, we see
Then . Thus,
(3.3)
Similarly, we have
(3.4)
According to the ADN theory and the theory of linear elliptic equations, we have that
is a sectorial operator and

Therefore, L generates the analytic semigroup .

It follows from (3.2), (3.3), and (3.4) that
Applying Lemma 2.8 yields

□

Remark 3.2 The analytic semigroup generated by L can be read as
Remark 3.3 The semigroup generated by Eqs. (1.1)-(1.7) can be rewritten as

## 4 Regularity of global solution

Theorem 4.1 If , , then Eqs. (1.1)-(1.7) have a unique solution , and

for all .

Proof Let and . Define as
Then

for any and . Then (2.7) holds.

We prove (2.6).
By the interpolation inequality [16], we see
(4.1)
By the imbedding theorem of factional order spaces, we have
(4.2)
Then it follows from (4.1) and 4.2) that
Since is a weak solution to Eqs. (1.1)-(1.7), , . Then (2.6) and (2.7) hold. From Lemma 2.7, we deduce that the solution ϕ is unique and
(4.3)
Multiplying (1.1) by u and integrating over Ω, we get
Using the Young inequality, we obtain
where is a real constant satisfying . Then there exists a constant such that
Thanks to (4.3), we have
(4.4)
We consider the Stokes equation
(4.5)
From (4.3), (4.4), and the Sobolev imbedding theorem, we find that , . By the ADN theorem, Eq. (4.5) has a solution
Then and . Using the ADN theorem, we obtain
(4.6)
Multiplying (1.2) by T and integrating over Ω, we get
where is a constant. Then there exists a constant such that
Using (4.3), we have
(4.7)
We consider the elliptic equation
(4.8)
It follows from (4.3), (4.7), and the Sobolev imbedding theorem that , . Using the theory of linear elliptic equations, Eq. (4.8) has a solution
Then and . Using the theory of linear elliptic equations, we have that
(4.9)
Multiplying (1.3) by q and integrating over Ω, we get
where is a constant. Then there exists a constant such that
Using (4.3), we have
(4.10)
We consider the elliptic equation
(4.11)
Using the arguments similar to those for (4.8), we get
(4.12)
It follows from (4.6), (4.9), and (4.12) that

□

Theorem 4.2 If , , then Eqs. (1.1)-(1.7) have a higher regular solution and

for .

Proof We prove the theorem using mathematical induction.

If , , , then , . Using Theorem 4.1, we find that .

Thanks to the Sobolev imbedding theorem, if . We obtain

Then and .

We have from the formula (3.1)
Then there exists α satisfying such that

Thus, . Then in Eq. (4.5). By the ADN theory, . Thus, and .

We have

which implies . Then in Eq. (4.5). Using the ADN theory, and . Thus, and . Then and from the formula (3.1).

Similarly,

Then and .

We have from the formula (3.1)
Then there exists α satisfying such that

which implies . Then in Eq. (4.8). It follows from the linear elliptic equation . Thus, and .

Then

which implies . We obtain that in Eq. (4.8). Then from the theory of linear elliptic equations. Thus, . From the formula (3.1), and .

Similarly,

Then and .

We have from the formula (3.1)
Then there exists α satisfying such that

which implies . Then in Eq. (4.11). Thus, from the theory of linear elliptic equations. Then and .

Thus,

which implies . We see in Eq. (4.11). Then from the theory of linear elliptic equations. Thus, . We have and from the formula (3.1).

It follows from Eq. (4.5) that
Clearly, the right-hand side of the above equality is continuous in . Thus,

If , , and , then and . From the hypothesis of mathematical induction, we see .

By the Sobolev imbedding theorem, we have if . Then it follows from the Sobolev imbedding theorem and the interpolation inequality that

Then and .

We have from the formula (3.1)
Then there exists α satisfying such that
Then . We see that in Eq. (4.5). Thus, from the ADN theory. Hence, and . Then
which implies . Then in Eq. (4.5). Using the ADN theory, , and , we get
From the formula (3.1), we have
Similarly,

Then and .

We have from the formula (3.1)
Then there exists α satisfying such that
which implies . Then in Eq. (4.8). It follows from the linear elliptic equation that and . We obtain
Then . We have in Eq. (4.8). Then from the theory of linear elliptic equations. Thus,
From the formula (3.1), we induce
Similarly,

Then and .

We have from the formula (3.1)
Then there exists α satisfying such that
which implies . Then in Eq. (4.11). Thus, from the theory of linear elliptic equations. Then and . Thus,
which implies . We find in Eq. (4.11). Then from the theory of linear elliptic equations. We have
From the formula (3.1), we see
It follows from Eq. (4.5) that
Clearly, the right-hand side of the above equality is continuous in . Then

The proof is completed. □

Since the differentiability of time and of space can be transformed into each other, we obtain

Remark 4.3 If , , then Eqs. (1.1)-(1.7) have a higher regular solution , and

for , where l, r, α, β are positive integers satisfying and .

## Declarations

### Acknowledgements

The author is very grateful to the anonymous referees whose careful reading of the manuscript and valuable comments enhanced presentation of the manuscript. The project is supported by the National Natural Science Foundation of China (11271271), the NSF of Sichuan Science and Technology Department of China (2010JY0057) and the NSF of Sichuan Education Department of China (11ZA102).

## Authors’ Affiliations

(1)
College of Mathematics and Software Science, Sichuan Normal University

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