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 effect1 Introduction
This paper is concerned with the regularity of solutions to the following initialboundary
problem of atmospheric circulation equations involving unknown functions
where
The problems (1.1)(1.4) are supplemented with the following Dirichlet boundary condition
at
and initial value conditions
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 Tweakly 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).
2 Preliminaries
We consider the divergence form of the linear elliptic equation
where
The problem (2.1) is supplemented with the following Dirichlet boundary condition:
Lemma 2.1[10] (Theory of linear elliptic equations)
Let
where
We consider the Stokes equation
Lemma 2.2[11,12] (ADN theory of the Stokes equation)
Let
where
Let X be a linear space,
where
Definition 2.3 Let
Definition 2.4 Let
Definition 2.5 A mapping
Lemma 2.6[3]
Assume
(A1) there exists
where
(A2) there exists
where
Then for any
If
We introduce a space sequence
where X,
Lemma 2.7In addition to the assumptions about the existence of a global solution in Lemma 2.6, if
for all
Lemma 2.8[13]
LetLbe a generator of a strongly continuous semigroup
and
Note that we used to assume that the linear operator L in (2.8) is a sectorial operator which generates an analytic semigroup
where
Lemma 2.9[1315] (Imbedding theorem of factional order spaces)
Let
and the inequalities
For sectorial operators, we also have the following properties which can be found in [13].
Lemma 2.10Let
(1)
(2)
(3) for each
where some
(4) the
(5) if ℒ is symmetric, for any
3 Formula of global solutions
We introduce the spaces
Let
Then Eqs. (1.1)(1.7) can be rewritten as an abstract equation
Theorem 3.1If
where
Proof As
Then
From the Hölder inequality and the Sobolev imbedding theorem, we see
Then
Similarly, we have
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
Remark 3.3 The semigroup generated by Eqs. (1.1)(1.7) can be rewritten as
4 Regularity of global solution
Theorem 4.1If
for all
Proof Let
Then
for any
We prove (2.6).
By the interpolation inequality [16], we see
By the imbedding theorem of factional order spaces, we have
Then it follows from (4.1) and 4.2) that
Since
Multiplying (1.1) by u and integrating over Ω, we get
Using the Young inequality, we obtain
where
Thanks to (4.3), we have
We consider the Stokes equation
From (4.3), (4.4), and the Sobolev imbedding theorem, we find that
Then
Multiplying (1.2) by T and integrating over Ω, we get
where
Using (4.3), we have
We consider the elliptic equation
It follows from (4.3), (4.7), and the Sobolev imbedding theorem that
Then
Multiplying (1.3) by q and integrating over Ω, we get
where
Using (4.3), we have
We consider the elliptic equation
Using the arguments similar to those for (4.8), we get
It follows from (4.6), (4.9), and (4.12) that
□
Theorem 4.2If
for
Proof We prove the theorem using mathematical induction.
If
Thanks to the Sobolev imbedding theorem,
Then
We have from the formula (3.1)
Then there exists α satisfying
Thus,
We have
which implies
Similarly,
Then
We have from the formula (3.1)
Then there exists α satisfying
which implies
Then
which implies
Similarly,
Then
We have from the formula (3.1)
Then there exists α satisfying
which implies
Thus,
which implies
It follows from Eq. (4.5) that
Clearly, the righthand side of the above equality is continuous in
If
By the Sobolev imbedding theorem, we have
Then
We have from the formula (3.1)
Then there exists α satisfying
Then
which implies
From the formula (3.1), we have
Similarly,
Then
We have from the formula (3.1)
Then there exists α satisfying
which implies
Then
From the formula (3.1), we induce
Similarly,
Then
We have from the formula (3.1)
Then there exists α satisfying
which implies
which implies
From the formula (3.1), we see
It follows from Eq. (4.5) that
Clearly, the righthand side of the above equality is continuous in
The proof is completed. □
Since the differentiability of time and of space can be transformed into each other, we obtain
Remark 4.3 If
for
Competing interests
The author declares that he has no competing interests.
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).
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