Stochastic 3D Globally Modified Navier-Stokes Equations: Weak Attractors, Invariant Measures and Large Deviations

This paper is mainly concerned with the asymptotic dynamics of nonautonomous stochastic 3D globally modified Navier-Stokes equations driven by nonlinear noise. Based on the well-posedness of such equations, we first show the existence and uniqueness of weak pullback mean random attractors. Then we investigate the existence of (periodic) invariant measures, the zero-noise limit of periodic invariant measures and their limit as the modification parameter N → N0 ∈ (0, +∞). Furthermore, under weaker conditions, we obtain the existence of invariant measures as well as their limiting behaviors when the external term is independent of time. Finally, by using weak convergence method, we establish the large deviation principle for the solution processes.