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Properties of time-dependent statistical solutions of the three-dimensional Navier-Stokes equations. (Propriétés des solutions statistiques des équations de Navier-Stokes tridimensionnelles dans le cas évolutif.) (English. French summary) Zbl 1304.35486
In [the first author and G. Prodi, Ann. Mat. Pura Appl., IV. Ser. 111 (1975) 111, 307–330 (1976; Zbl 0344.76015)] was introduced the concept of statistical solutions, as a family of measures on the phase space of Navier-Stokes equations verifying a particular Liouville-type equation. A different type of statistical solutions was introduced by M. I. Vishik and A. V. Fursikov [Sib. Math. J. 19, 710–729 (1979; Zbl 0412.35078)], given by measures on a suitable trajectory spaces. The present paper is introducing a new kind of statistical solutions, related with a Borel probability measure on a suitable space, carried by trajectory space of Leray-Hopf weak solutions of Navier-Stokes equations. This new definition is slightly different from those of Visik-Fursikov and has some useful additional analytical properties, being in fact a particular case of the Foias-Prodi concept. The main parts of the paper are concerning the properties of the weak solutions of Navier-Stokes equations, some elements of measure theory, time dependent function spaces, trajectory spaces, mean energy inequalities. The presented theory is very useful for the study of fully developed turbulence phenomenon.

MSC:
35Q30 Navier-Stokes equations
76D06 Statistical solutions of Navier-Stokes and related equations
37A60 Dynamical aspects of statistical mechanics
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
35D30 Weak solutions to PDEs
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