The asymptotic behavior of primitive equations with multiplicative noise

This paper is concerned with the existence of invariant measure for 3D stochastic primitive equations driven by linear multiplicative noise under non-periodic boundary conditions. The common method is to apply Sobolev imbedding theorem to proving the tightness of the distribution of the solution. However, this method fails because of the non-linearity and non-periodic boundary conditions of the stochastic primitive equations. To overcome the difficulties, we show the existence of random attractor by proving the compact property and the regularity of the solution operator. Then we show the existence of invariant measure.