Invariant measures for the $3$DNavier-Stokes-Voigt equations and their Navier-Stokes limit

The Navier-Stokes-Voigt model ofviscoelastic incompressible fluid has been recently proposed as aregularization of the three-dimensional Navier-Stokes equations forthe purpose of direct numerical simulations. Besides the kinematicviscosity parameter, ν>0, this model possesses a regularizingparameter, α> 0, a given length scale parameter, so thatα²/ν is the relaxation time of the viscoelasticfluid. In this work, we derive several statistical properties ofthe invariant measures associated with the solutions of thethree-dimensional Navier-Stokes-Voigt equations. Moreover, we provethat, for fixed viscosity , ν>0, as the regularizing parameterα tends to zero, there exists a subsequence of probabilityinvariant measures converging, in a suitable sense, to a strongstationary statistical solution of the three-dimensionalNavier-Stokes equations, which is a regularized version of thenotion of stationary statistical solutions - a generalization of theconcept of invariant measure introduced and investigated by Foias.This fact supports earlier numerical observations, and provides anadditional evidence that, for small values of the regularizationparameter α, the Navier-Stokes-Voigt model can indeed beconsidered as a model to study the statistical properties of thethree-dimensional Navier-Stokes equations and turbulent flows viadirect numerical simulations.