Statistically Stationary Solutions to the 3D Navier-Stokes Equations do not show Singularities

Abstract

If $\mu$ is a probability measure on the set of suitable weak solutions of the 3D Navier-Stokes equations, invariant for the time-shift, with finite mean dissipation rate, then at every time $t$ the set of singular points is empty $\mu$-a.s. The existence of a measure $\mu$ with the previous properties is also proved; it may describe a turbulent asymptotic regime.