## Statistically stationary solutions to the 3-D Navier-Stokes equation do not show singularities.(English)Zbl 0973.35148

Summary: 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.e. The existence of a measure $$\mu$$ with the previous properties is also proved; it may describe a turbulent asymptotic regime.

### MSC:

 35Q30 Navier-Stokes equations 76D06 Statistical solutions of Navier-Stokes and related equations 35R60 PDEs with randomness, stochastic partial differential equations
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