Comptes Rendus
Mathematical Problems in Mechanics
A note on statistical solutions of the three-dimensional Navier–Stokes equations: The time-dependent case
[Note sur les solutions statistiques des équations de Navier–Stokes incompressibles en dimension trois d'espace : le cas dépendant du temps]
Comptes Rendus. Mathématique, Volume 348 (2010) no. 3-4, pp. 235-240.

Dans cette Note nous considérons les solutions statistiques des équations de Navier-Stokes incompressibles en dimension trois d'espace. Elles constituent une formalisation mathématique de la notion de moyenne statistique dans la théorie de la turbulence et forment l'un des fondements de la théorie mathématique de la turbulence. Deux notions différentes de solutions statistiques ont été introduites ; nous les rappelons et donnons une formulation nouvelle de l'une d'elles. Nous établissons en outre un théorème d'existence de solutions pour cette nouvelle notion, et donnons un certain nombre de propriétés utiles des solutions statistiques.

Time-dependent statistical solutions of the three-dimensional Navier–Stokes equations for incompressible fluids are considered. They are a mathematical formalization of the notion of ensemble averages in turbulence theory and form the backbone for a mathematical foundation of the theory of turbulence. The two main notions of statistical solutions, previously introduced, are revisited and a new formulation of one of them is given. An existence proof for this new formulation is given, along with a number of useful properties.

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DOI : 10.1016/j.crma.2009.12.017

Ciprian Foias 1 ; Ricardo M.S. Rosa 2 ; Roger Temam 3, 4

1 Department of Mathematics, Texas A&M University, College Station, TX 77843, USA
2 Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530 Ilha do Fundão, Rio de Janeiro, RJ 21945-970, Brazil
3 Académie des Sciences, Paris
4 Department of Mathematics, Indiana University, Bloomington, IN 47405, USA
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Ciprian Foias; Ricardo M.S. Rosa; Roger Temam. A note on statistical solutions of the three-dimensional Navier–Stokes equations: The time-dependent case. Comptes Rendus. Mathématique, Volume 348 (2010) no. 3-4, pp. 235-240. doi : 10.1016/j.crma.2009.12.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.12.017/

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