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.

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 ):

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 :

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 [2-5], or an infinite dimensional dynamical system [6-8]. 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.

We consider the divergence form of the linear elliptic equation

where , , , is uniformly elliptic, i.e., there exist constants such that

The problem (2.1) is supplemented with the following Dirichlet boundary condition:

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

Letbe afield, , , , . Ifis a solution of Eqs. (2.1), (2.2), then

wheredepends onn, p, λ, Ω and-norm or-norm of the coefficient functions.

We consider the Stokes equation

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

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

wheredepends onμ, 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

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

Definition 2.5 A mapping is called T-weakly continuous if for , and uniformly weakly converges to , we have

Lemma 2.6[3]

AssumeisT-weakly continuous and satisfies:

(A1) there exists, (), such that

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

(A2) there existsfor anyand,

wheredepends only onT, , , 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

Lemma 2.7In addition to the assumptions about the existence of a global solution in Lemma 2.6, ifis Frechét differentiable and satisfies (2.6), (2.7), then Eq. (2.4) has a unique global solution

for all.

Lemma 2.8[13]

LetLbe a generator of a strongly continuous semigroup. Ifis a weak solution to the equation

and, then the solutioncan 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[13-15] (Imbedding theorem of factional order spaces)

Letbe a Lipschitz field, be a sectorial operator, , and. Then for, the fractional order spacessatisfy the following relations:

and the inequalities

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

Lemma 2.10Letbe a sectorial operator which generates an analytic semigroup. If all eigenvaluesλofLsatisfyfor some real number, then for, we have

(1) is bounded for alland,

(2) , ,

(3) for each, is bounded and

where some, is a constant only depending onα,

(4) the-norm can be defined by

(5) if ℒ is symmetric, for any, we have

We introduce the spaces

Let

Then Eqs. (1.1)-(1.7) can be rewritten as an abstract equation

Theorem 3.1If, , then the global solutionϕof Eqs. (1.1)-(1.7) can be read as

whereis an analytic semigroup generated byL, andis 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,

From the Hölder inequality and the Sobolev imbedding theorem, we see

Then . Thus,

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 generated by L can be read as

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

Theorem 4.1If, , 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

By the imbedding theorem of factional order spaces, we have

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

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

We consider the Stokes equation

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

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

We consider the elliptic equation

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

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

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, , then Eqs. (1.1)-(1.7) have a higher regular solutionand

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 .

The author declares that he has no competing interests.

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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