Asymptotically Autonomous Robustness of Invariant Measures for the Velocity–Vorticity-Voigt Model of the 3D Navier–Stokes Equations with Delays

This paper investigates the backward asymptotic autonomy of invariant measures for the velocity–vorticity-Voigt model of the three-dimensional Navier–Stokes equations with delays. First, by overcoming the difficulties arising from delays in the higher-order regularity estimates of solutions and the asymptotic compactness of solution operators, we prove the existence of pullback attractors. Second, we construct a family of invariant Borel probability measures, which are supported on the pullback attractors. We then prove the asymptotically autonomous robustness for invariant measures, that is, any limit of the sequence of invariant measures for such a non-autonomous system must be an invariant measure of the corresponding autonomous system as time tends to negative infinity. Moreover, the asymptotically autonomous limiting set of a pullback attractor can also be characterized by the global attractor of the corresponding autonomous system with the autonomous external force terms that satisfy the convergence relationship.