Random Exponential Attractor for the 3D Non-autonomous Stochastic Damped Navier-Stokes Equation

In this paper, we prove the existence of a random exponential attractor (a positively invariant, compact, random set with finite fractal dimension that attracts any trajectory exponentially) for the 3D non-autonomous damped Navier-Stokes equation with additive noise, which implies that the asymptotic behavior of solutions for the equation can be described by finite independent parameters. The key and difficult point of this proof lies in proving the Lipschitz continuity and the random squeezing property in mean sense for the non-autonomous random dynamical system generated by solutions of the equation.