Invariant measures for non-autonomous Navier–Stokes–Voight equations with variable delay

This paper is devoted to the dynamical stability of retarded three-dimensional Navier-Stokes-Voight equations driven by non-autonomous forcing term. We first prove the existence and uniqueness of pullback attractors for such equations. Second, we show that the existence and uniqueness of a family of invariant Borel probability measures supported on the pullback attractors. Third, we study the upper semicontinuity of pullback attractors as the delay time approaches to zero. Finally, we investigate the convergence of invariant measures as the delay time tends to zero. In this paper, we need to overcome two difficulties. On the one hand, due to the lack of higher regularity of solutions, we prove the asymptotic compactness of solutions using the spectrum decomposition method. On the other hand, we introduce an invertible function to solve the effect of the variable delay, and then establish the convergence of solutions for the delay system to the non-delay one.