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

Foias, Ciprian; Rosa, Ricardo M. S.; Temam, Roger

Annales de l'Institut Fourier, Tome 63 (2013) no. 6, pp. 2515-2573.

Ce travail est dédié au concept de solutions statistiques des équations de Navier-Stokes qui a été proposé comme un objet mathématique rigoureux permettant de décrire et étudier le concept fondamental de moyennes statistiques (ensemble averages en Anglais) dans la théorie conventionnelle de la turbulence développée. Deux concepts de solutions statistiques ont été proposés dans les années 1970 par Foias et Prodi d’une part et par Vishik et Fursikov d’autre part. Dans cet article nous introduisons et étudions un nouveau concept intermédiaire de solutions statistiques. Les solutions que nous considérons sont des solutions statistiques au sens de Foias et Prodi d’un type particulier et elles sont construites par une procédure proche de celle de Vishik et Fursikov, si bien qu’elles possèdent un certain nombre de propriétés analytiques utiles.

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 is a particular type of a statistical solution in the sense of Foias and Prodi which is constructed in a way akin to the definition given by Vishik and Fursikov, in such a way that it possesses a number of useful analytical properties.

Classification : 35Q30, 76D06, 37A60, 28C20
Keywords: Navier-Stokes equations, statistical solutions, turbulence, measure theory, functional analysis
Mot clés : équations de Navier-Stokes, solutions statistiques, turbulence, théorie de la mesure, analyse fonctionnelle

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