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Some results on the Navier-Stokes equations in connection with the statistical theory of stationary turbulence

Ricardo M. S. Rosa — 2002

Applications of Mathematics

Some rigorous results connected with the conventional statistical theory of turbulence in both the two- and three-dimensional cases are discussed. Such results are based on the concept of stationary statistical solution, related to the notion of ensemble average for turbulence in statistical equilibrium, and concern, in particular, the mean kinetic energy and enstrophy fluxes and their corresponding cascades. Some of the results are developed here in the case of nonsmooth boundaries and a less regular...

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

Ciprian Foias;  Ricardo M. S. Rosa;  Roger Temam — 2013

Annales de l’institut Fourier

This work is devoted to the concept of statistical solution of the Navier-Stokes equations, proposed as a rigorous mathematical object to address the fundamental concept of ensemble average used in the study of the conventional theory of fully developed turbulence. Two types of statistical solutions have been proposed in the 1970’s, one by Foias and Prodi and the other one by Vishik and Fursikov. In this article, a new, intermediate type of statistical solution is introduced and studied. This solution...

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