Communications in Mathematical Sciences

Volume 19 (2021)

Number 1

Asymptotic behavior of 3-D stochastic primitive equations of large-scale moist atmosphere with additive noise

Pages: 1 – 38



Lidan Wang (School of Statistics and Data Science, Nankai University, Tianjin, China)

Guoli Zhou (School of Statistics and Mathematics, Chongqing University, Chongqing, China)

Boling Guo (Institute of Applied Physics and Computational Mathematics, Beijing, China)


The primitive equations (PEs) are a basic model in the study of large scale oceanic and atmospheric dynamics. Its high non-linearity and anisotropic structure attract much attention from mathematicians.

In the present article, we consider the corresponding stochastic model. As studies from climate sciences show that the complex multi-scale nature of the earth’s climate system results in many uncertainties that should be accounted for in the basic dynamical models of atmospheric and oceanic processes. It is further pointed out by [Palmer, Q.J.R. Meteorol. Soc., 127:279–304, 2001] and [Majda, Timofeyev and Vanden-Eijinden, Commun. Pure Appl. Math., 54:891–974, 2001] that stochastic modeling for climate is important for understanding the intrinsic variability of dominant low-frequency teleconnection patterns in climate, to provide cheap low-dimensional computational models for the coupled atmosphere-ocean system and to reduce model error in standard deterministic computer models for extended-range prediction through appropriate stochastic noise.

This is the first attempt to consider stochastic moist PEs defined on manifolds. Using a new and general way, we prove the existence of random attractor (strong attractor) for the three dimensional stochastic moist primitive equations defined on a manifold in 3D space improving the existence of weak attractor for the corresponding deterministic model [Guo, Huang, J. Diff. Eqs., 251:457–491, 2011]. As an application of the result, we show the existence of the invariant measure. The technique presented in this work can be applied to common classes of dissipative stochastic partial differential equations and it has some advantages over the common method of using compact Sobolev imbedding theorem, i.e., if the absorbing set in some Sobolev space does exist in view of the common method, our method would then further imply the existence of random attractor in this space.


stochastic moist primitive equations, manifold, random attractor, invariant measure

2010 Mathematics Subject Classification

35Q35, 60H15

The author Lidan Wang’s research was partially supported by NNSF of China (Grant No. 11801283), Key Laboratory for Medical Data Analysis and Statistical Research of Tianjin (KLMDASR). The author Guoli Zhou’s research was partially supported by NNSF of China (Grant No. 11971077), Natural Science Foundation Project of CQ (Grant No. cstc2020jcyjmsxmX0441), Fundamental Research Funds for the Central Universities (Grant No. 2020CDJ-LHZZ-027) and Chongqing Key Laboratory of Analytic Mathematics and Applications, Chongqing University, Chongqing, 401331, China.

Received 5 March 2019

Accepted 12 July 2020

Published 24 March 2021