Abstract
In this paper, we consider the Navier-Stokes equations and temperature equation arising from the evolution process of the atmosphere. Under certain assumptions imposed on the initial data, we show the -stability of weak solutions for the atmospheric equations. Some ideas and delicate estimates are introduced to prove these results.
MSC:35Q35, 76D03.
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1 Introduction and main results
In this paper, we consider the atmospheric motion model under the constant external force and without the effects of topography, and the aerosphere is regarded as a spherical shell encompassing the earth. We introduce a moving frame running with the earth, , where is the longitude, is the colatitude, and is the atmospheric pressure, which can be used instead of the geocentric distance r because it is strictly monotonically decreasing function for r, where is the atmospheric pressure on the surface of the earth, and is the atmospheric pressure at a certain isobaric surface. In the coordinate system consisting of the moving frame and time, the atmospheric state functions are defined by the atmospheric horizontal velocity , the rate of pressure , the temperature T, and the geopotential Φ. All of them satisfy the following system:
with the initial data
where ω is the angular velocity of the earth; , , and R are the thermodynamics parameters; and , , are the diffusion coefficients; satisfying , namely the diffusion is related to the atmospheric pressure; Ψ is the diabatic heating of the atmosphere, which is a function of and stands for the effect of the constant external force on the atmospheric system. We have in , which denotes the Coriolis force on the atmosphere. The differential operators and on the spherical surface have the following form:
where a is the radius of the earth. The vertical scale of the atmosphere is very much smaller compared with the radius of the earth, so the geocentric distance r is replaced by the radius of the earth in the differential operators. The above equations are studied on , where .
The boundary conditions without the relief are: All the functions are 2π periodical w.r.t. λ, π periodical w.r.t. θ, and
There are many important results achieved on the atmospheric problem. Zeng [1], Li and Chou [2] have made important progress on the formulation and the analysis of the models. For different research purposes, different atmospheric models have been investigated by Pedlosky [3], Washington and Parkinson [4], Lions et al. [5–8] and references therein. Recently, Chepzhov and Vishik [9] introduced the atmospheric equations considered in this paper. Huang and Guo [10] proved the existence of the weak solutions to the atmospheric equations by the basic differential equation theory and the existence of the corresponding trajectory attractors, from which the existence of the atmospheric global attractors follows. Furthermore, Huang and Guo [11] studied the model of the climate for weather forecasts in which the pressing force of topography on atmosphere and the divergent effect of airflow are included, and they proved the existence and the asymptotic behaviors of the weak solution.
The rest of the paper is as follows. In Section 2, the main results about the -stability of weak solutions to the Navier-Stokes equations and temperature equation are stated. In Section 3, we will give several important a priori estimates. Then we will justify the stability of the weak solutions in Section 4. Finally, in Section 5, the conclusion will be given.
2 Main results
The -stability theory of weak solutions to (1.1) will be considered, and there is a simple version of system (1.1). From (1.1)3,4 and the boundary conditions (1.4), we have
and
which implies
Substitute (2.1) and (2.2) into (1.1)1,2 and define the unknown function , then we have the simplification of system (1.1):
Denote
and
then we show the definition of weak solutions of system (2.4).
Definition 2.1 (Definition of weak solution)
For any , U is said to be a weak solution of system (2.4) on , if U has the following regularities:
and satisfies the equations in the sense of distributions
Namely, we have for all and ,
where is the inner product of ,
and
where
Then we can state the main results of the present paper as follows.
Theorem 2.1 (Stability of weak solutions)
For any , let be a sequence of weak solution of system (2.4) subject to the initial data
and be such that
where and satisfies the following upper bound uniformly with respect to :
where denotes a constant, and we assume that .
Then, up to a subsequence, still denoted by the same symbol, we have
where is a weak solution of system (2.4) with the initial data .
Remark 2.1 Note that from (2.17), we can find that if
then
which is the -stability of weak solutions for the system.
Remark 2.2 Furthermore if
and
from the Egorov theorem, we have for the following. Let , then there exists a domain , such that , and for all , , for , we have
Then for we have
where we have
and
and from (2.24) and (2.25), we can find
thus, we have from Remark 2.1
which implies that
which means the weak solutions are stable almost everywhere.
3 The a priori estimates
Next, we will give the a priori estimates for the weak solution to system (2.4). Firstly, from a direct calculation, we can establish the following lemma; we omit the proof.
Lemma 3.1
where ϕ and ψ can be vector-valued functions, or the scalar functions,
where ϕ is a vector-valued function, and ψ is a scalar function.
Then we have the usual energy inequality as follows.
Lemma 3.2 Let . Under the assumptions of Theorem 2.1, we have for the weak solution to system (2.4)
where denotes a constant dependent on the initial data and time M and independent of n.
Proof Take the inner product of (2.4) with , integrating on Ω, we have
and using the boundary conditions, we have
which implies
where is a small constant such that we have
after the integration with respect to , we have
where denotes a constant dependent of the initial data and time M and independent of n. □
4 Proof of main results
With the help of the a priori estimates in (3.3), we have the following estimates for the sequence of weak solutions :
then we will prove the main results, in order to address the convergence of sequence of the weak solution; a lemma of the compactness result will be given first.
Lemma 4.1 (Lion’s compactness result)
Suppose , E, are Banach spaces, , which means is compactly embedded in E, E is embedded in , and . Denote
as the Banach space with the norm
then
Then we will give the proof of the stability of the weak solutions.
Lemma 4.2 Let be the weak solution sequence of system (2.4). Then, up to a subsequence, we have
and
for all and .
Proof From (4.1), we have the following estimates for the test function :
which implies
and we have
where denotes a constant independent of n, namely,
Next, we can find that
and we have
and
from (4.12)-(4.15), we have
and
where denotes a constant independent of n, namely,
Finally, we have
and
Then we have from (4.10), (4.18), and (4.20)
which together with Lemma 4.1 and gives
and using (4.22), we can prove (4.6) holds. □
Lemma 4.3 Let be the weak solution sequence of system (2.4). Then, up to a subsequence, we have
for all and .
Proof As , and , thus, we have
which means that the sequences converge weakly; then we have
□
Lemma 4.4 Let be the weak solution sequence of system (2.4). Then, up to a subsequence, we have
for all and .
Proof We have
and we have from (4.5) and (4.24)
where we use the fact
Using , we have
which implies
and we have
where we use the fact
applying (4.31), we also have
Finally, by means of (4.5), we can prove that
Summing (4.28), (4.32), (4.34), and (4.35), we complete the proof of (4.26). □
5 Conclusion
In this paper, the stability of weak solutions for the atmospheric equations is investigated with the constant external force and without the effects of topography; from Theorem 2.1 and Remark 2.1 and Remark 2.2, we show that if , then ; if a.e., then a.e., which means that if the difference of the initial data of two different weak solutions is small almost everywhere, then the difference of this two weak solutions is small almost everywhere as time increases. Furthermore, in the future we will consider the stability of weak solutions to the atmospheric models with the effects of topography, a non-constant external force, radiation heating, and the moist phase transformation, etc.
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Acknowledgements
The research is supported by NNSFC No. 11101145, China Postdoctoral Science Foundation No. 2012M520360.
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RXL organized and wrote this paper. QCZ examined all the steps of the proofs in this research and gave some advices. All authors read and approved the final manuscript.
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Lian, R., Zeng, Qc. Stability of weak solutions for the large-scale atmospheric equations. J Inequal Appl 2014, 455 (2014). https://doi.org/10.1186/1029-242X-2014-455
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DOI: https://doi.org/10.1186/1029-242X-2014-455