Properties of time-dependent statistical solutions of the three-dimensional Navier-Stokes equations

Details Properties of time-dependent statistical solutions of the three-dimensional Navier-Stokes equations Properties of time-dependent statistical solutions of the three-dimensional Navier-Stokes equations

2011-11-27

arxiv.org/abs/1111.6257v1

Mathematics

Analysis of PDEs

Scientific paper

The concept of statistical solution of the three-dimensional Navier-Stokes equations is addressed, and a new type of statistical solution is given. The concept of statistical solution have been introduced as a rigorous mathematical object to formalize the notion of ensemble average in the conventional statistical theory of turbulence; a better understanding of this concept is of fundamental importance for a rigorous mathematical approach to the theory of turbulence. The concept of statistical solution, as introduced by Foias and Prodi in the early 1970's, is that of a time-dependent family of Borel probability measures in the phase space of the system representing the probability distribution of the velocity field of the flow at each given time. The new type of statistical solution is obtained as a projection of a measure in the trajectory space, akin to the notion introduced by Vishik and Fursikov in the late 1970's. This new type of statistical solution has a number of useful analytical properties, which we study here and which will be further investigated in subsequent works. The current work is devoted only to the time-dependent case. The particular case of stationary statistical solutions will be addressed elsewhere.

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