Communications in Mathematical Sciences

Volume 16 (2018)

Number 6

The asymptotic behavior of primitive equations with multiplicative noise

Pages: 1685 – 1711



Rangrang Zhang (Department of Mathematics and Statistics, Beijing Institute of Technology, Beijing, China)

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

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


This article is concerned with the existence of random attractor and the existence of the invariant measure for 3D stochastic primitive equations driven by linear multiplicative noise under non-periodic boundary conditions. To achieve these goals, the crucial step is to establish the uniform a priori estimates in a functional space which is more regular than the solution space. But, it is very difficult because of the high nonlinearity and non-periodic boundary conditions of the stochastic primitive equations. To overcome the difficulties, we firstly obtain the existence of the absorbing ball in the solution space. Then, we use Aubin–Lions lemma and the regularity of the solution to prove that the solution operator is compact. Finally, by operating the absorbing ball with the compact solution operator, we obtain a compact absorbing ball in the solution space, which ensures the existence of the random attractor. Since the solution is Markov, the asymptotic compactness of the solution operator implies the existence of an invariant measure.


stochastic primitive equations, random attractor, invariant measure

2010 Mathematics Subject Classification

35Q35, 60H15

Received 16 February 2018

Accepted 8 June 2018

Published 7 February 2019