Existence and continuity of random exponential attractors for stochastic 3D globally modified non-autonomous Navier-Stokes equation

In this paper, we deal with four problems. (i) Based on criteria for a continuous non-autonomous deterministic dynamical system (NDDS) and a continuous non-autonomous random dynamical system (NRDS), we construct an exponential attractor for a continuous NDDS and a family of random exponential attractors for a family of continuous non-autonomous random dynamical systems (NRDS), respectively. (ii) We prove that this family of random exponential attractors is continuous (or stable, robust, i.e., upper and lower semi-continuous) in the sense of the symmetric Hausdorff distance as the intensity of stochastic perturbations approaches zero. (iii) We prove that for two conjugate NRDS, if one has a random exponential attractor, then the other has a random exponential attractor, and that for two families of conjugate NRDS, if a family of random exponential attractors for one family is continuous, then a corresponding family of random exponential attractors for the other family is continuous. (iv) We apply our abstract result to study the existence and continuity of random exponential attractors for 3D globally modified non-autonomous Navier-Stokes equation with additive noise.