  The Navier-Stokes-Voigt model of viscoelastic incompressible fluid has been
recently proposed as a regularization of the three-dimensional Navier-Stokes
equations for the purpose of direct numerical simulations. Besides the
kinematic viscosity parameter, $\nu>0$, this model possesses a regularizing
parameter, $\alpha> 0$, a given length scale parameter, so that
$\frac{\alpha^2}{\nu}$ is the relaxation time of the viscoelastic fluid. In
this work, we derive several statistical properties of the invariant measures
associated with the solutions of the three-dimensional Navier-Stokes-Voigt
equations. Moreover, we prove that, for fixed viscosity, $\nu>0$, as the
regularizing parameter $\alpha$ tends to zero, there exists a subsequence of
probability invariant measures converging, in a suitable sense, to a strong
stationary statistical solution of the three-dimensional Navier-Stokes
equations, which is a regularized version of the notion of stationary
statistical solutions - a generalization of the concept of invariant measure
introduced and investigated by Foias. This fact supports earlier numerical
observations, and provides an additional evidence that, for small values of the
regularization parameter $\alpha$, the Navier-Stokes-Voigt model can indeed be
considered as a model to study the statistical properties of the
three-dimensional Navier-Stokes equations and turbulent flows via direct
numerical simulations.
