Asymptotical behavior of the 2D stochastic partial dissipative Boussinesq system with memory

The objective of this paper is to consider the asymptotical behavior of solutions for the two-dimensional partial dissipative Boussinesq system with memory and additive noise. We first establish the existence of a random absorbing set in the phase space. However, due to the presence of the memory term, we cannot obtain some kind of compactness of the corresponding cocycle through Sobolev compactness embedding theorem or by verifying the pullback flattening property. To overcome this difficulty, we first prove the asymptotical compactness of the velocity component of weak solutions, and then we prove the asymptotical compactness of other components based on some energy estimates and the Aubin–Lions compactness lemma, which implies the asymptotical compactness of the corresponding cocycle. Thus, the existence of a random attractor is obtained. Finally, we establish an abstract result about some kind of upper semi-continuity of the random attractor, which is applied to the two-dimensional partial dissipative Boussinesq system.