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Abstract
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.
Mathematics Subject Classification: 76D06, 76D05, 76F20, 76F55, 76A10.
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