On the convergence of statistical solutions of the 3D Navier-Stokes-$\alpha$ model as $\alpha$ vanishes

TL;DR

This paper proves that statistical solutions of the 3D Navier-Stokes-$$ model converge to those of the exact 3D Navier-Stokes equations as the parameter $$ approaches zero, under natural conditions.

Contribution

It establishes the convergence of statistical solutions from the Navier-Stokes-$$ model to the true Navier-Stokes solutions as $$ vanishes, providing a rigorous link between the models.

Findings

Statistical solutions of the Navier-Stokes-$$ model converge to those of the Navier-Stokes equations as $ o 0$.

Convergence is proved under certain natural conditions.

The solutions are characterized as families of time-projections of measures in trajectory spaces.

Abstract

In this paper statistical solutions of the 3D Navier-Stokes-$\alpha$ model with periodic boundary condition are considered. It is proved that under certain natural conditions statistical solutions of the 3D Navier-Stokes-$\alpha$ model converge to statistical solutions of the exact 3D Navier-Stokes equations as $\alpha$ goes to zero. The statistical solutions considered here arise as families of time-projections of measures in suitable trajectory spaces.