The asymptotic behavior of primitive equations with multiplicative noise

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.