Random attractors and invariant measures for 3D stochastic globally modified Navier-Stokes equations with time-dependent delay and coefficient

In this article, we investigate the dynamics of three-dimensional, non-autonomous, and globally modified Navier-Stokes equations with time-dependent delay and additive noise. Based on the well-posedness of solutions, we demonstrate that the solution operators of such equations generate a non-autonomous random dynamical system. We further prove the existence and uniqueness of random attractors as well as the periodicity and the upper semicontinuity for this random dynamical system. We also prove the existence of invariant measures supported on the random attractors. Moreover, we rigorously find that every limit of a sequence of invariant measures must be an invariant measure of the corresponding limiting equations as the noise intensity $ \varepsilon\rightarrow0 $ or the modification parameter $ N\rightarrow N_0\in (0, +\infty) $.