Stability of weak solutions for the large-scale atmospheric equations
© Lian and Zeng; licensee Springer. 2014
Received: 15 July 2014
Accepted: 23 October 2014
Published: 11 November 2014
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.
1 Introduction and main results
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 .
There are many important results achieved on the atmospheric problem. Zeng , Li and Chou  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 , Washington and Parkinson , Lions et al. [5–8] and references therein. Recently, Chepzhov and Vishik  introduced the atmospheric equations considered in this paper. Huang and Guo  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  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
then we show the definition of weak solutions of system (2.4).
Definition 2.1 (Definition of weak solution)
Then we can state the main results of the present paper as follows.
Theorem 2.1 (Stability of weak solutions)
where denotes a constant, and we assume that .
where is a weak solution of system (2.4) with the initial data .
which is the -stability of weak solutions for the system.
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.
where ϕ is a vector-valued function, and ψ is a scalar function.
Then we have the usual energy inequality as follows.
where denotes a constant dependent on the initial data and time M and independent of n.
where denotes a constant dependent of the initial data and time M and independent of n. □
4 Proof of main results
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)
Then we will give the proof of the stability of the weak solutions.
for all and .
and using (4.22), we can prove (4.6) holds. □
for all and .
for all and .
Summing (4.28), (4.32), (4.34), and (4.35), we complete the proof of (4.26). □
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.
The research is supported by NNSFC No. 11101145, China Postdoctoral Science Foundation No. 2012M520360.
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