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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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