Pullback Measure Attractors and Periodic Measures of Stochastic Non-autonomous Tamed 3D Navier–Stokes Equation

This paper devotes to studying the asymptotic behavior of solutions for stochastic non-autonomous tamed 3D Navier–Stokes equations driven by nonlinear multiplicative noise. We first show that such equation possesses a unique strong solution, which implies the existence and uniqueness of weak solutions. Based on this, we investigate the existence and uniqueness of pullback measure attractors for the inhomogeneous Markov process generated by the stochastic tamed 3D Navier–Stokes equations, which is the first instance of three-dimensional hydrodynamic system that characterizes the existence of a pullback measure attractor. In addition, we prove the existence of periodic invariant measures for the non-autonomous transition semigroup associated with the stochastic equations, along with an analysis of their limiting behavior as the intensity of noise tends to zero.