A note on statistical solutions of the three-dimensional Navier–Stokes equations: The stationary case

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

doi:10.1016/j.crma.2009.12.018

Mathematical Problems in Mechanics

A note on statistical solutions of the three-dimensional Navier–Stokes equations: The stationary case
[Note sur les solutions statistiques des équations de Navier–Stokes incompressibles en dimension trois d'espace : le cas stationnaire]

Présenté par : Philippe G. Ciarlet

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

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, et Department of Mathematics, Indiana University, Bloomington, IN 47405, USA

Comptes Rendus. Mathématique, Volume 348 (2010) no. 5-6, pp. 347-353.

Résumé
Abstract

Dans cette Note nous considérons les solutions statistiques stationnaires des équations de Navier–Stokes pour les fluides incompressibles. Les solutions statistiques stationnaires sont la formulation mathématique du concept de moyenne statistique pour des écoulements turbulents qui sont en équilibre statistique dans le temps. Ce sont aussi des généralisations de la notion de mesure invariante pour les équations de Navier–Stokes tridimensionnelles. En effet l'existence globale en temps d'une solution régulière de ces équations n'étant pas connue, on ne peut leur associer un semi-groupe d'opérateurs continus avec la définition qui en résulterait d'une solution mesure invariante. Deux définitions classiques des solutions statistiques stationnaires ont eté proposées ; elles sont ici rappelées et comparées ; les premières sont un cas particulier des secondes, et elles possèdent plusieurs propriétés utiles. De plus nous considérons les solutions statistiques stationnaires qui sont obtenues comme moyennes en temps de solutions faibles évolutives sur des intervalles de temps de plus en plus grands et nous montrons que ces moyennes temporelles appartiennent à la plus petite classe de solutions statistiques stationnaires. En outre une propriété de type récurrence est obtenue pour les solutions statistiques stationnaires qui satisfont une propriété d'accrétivité définie dans le texte. Finalement nous nous intéressons à l'attracteur global faible associé aux équations de Navier–Stokes tridimensionnelles, et nous montrons en particulier qu'il existe un sous-ensemble « topologiquement grand » de cet attracteur dont la mesure est totale par rapport à cette classe particulière de solutions statistiques stationnaires et qui présente un certain caractère de régularité.

Stationary statistical solutions of the three-dimensional Navier–Stokes equations for incompressible fluids are considered. They are a mathematical formalization of the notion of ensemble average for turbulent flows in statistical equilibrium in time. They are also a generalization of the notion of invariant measure to the case of the three-dimensional Navier–Stokes equations, for which a global uniqueness result is not known to exist and a semigroup may not be well-defined in the classical sense. The two classical definitions of stationary statistical solutions are considered and compared, one of them being a particular case of the other and possessing a number of useful properties. Furthermore, the so-called time-average stationary statistical solutions, obtained as generalized limits of time averages of weak solutions as the averaging time goes to infinity are shown to belong to this more restrictive class. A recurrent type result is also obtained for statistical solutions satisfying an accretion condition. Finally, the weak global attractor of the three-dimensional Navier–Stokes equations is considered, and in particular it is shown that there exists a topologically large subset of the weak global attractor which is of full measure with respect to that particular class of stationary statistical solutions and which has a certain regularity property.

Reçu le : 2009-12-07
Accepté le : 2009-12-29
Publié le : 2010-02-24

DOI : 10.1016/j.crma.2009.12.018

Affiliations des auteurs :

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

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, et 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 stationary case. Comptes Rendus. Mathématique, Volume 348 (2010) no. 5-6, pp. 347-353. doi : 10.1016/j.crma.2009.12.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.12.018/

Bibliographie
Cité par

[1] V.V. Chepyzhov; M.I. Vishik Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, vol. 49, American Mathematical Society, Providence, RI, 2002

[2] C. Foias; C. Foias Statistical study of Navier–Stokes equation II, Rend. Semin. Mat. Univ. Padova, Volume 48 (1972), pp. 219-348

[3] C. Foias; O.P. Manley; R. Rosa; R. Temam A note on statistical solutions of the three-dimensional Navier–Stokes equations: The time-dependent case, Comptes Rendus Acad. Sci. Paris Ser. I, Volume 348 (2010) no. 3–4, pp. 235-240

[4] C. Foias, R. Rosa, R. Temam, Properties of time-dependent statistical solutions of the three-dimensional Navier–Stokes equations, in preparation

[5] C. Foias, R. Rosa, R. Temam, Properties of stationary statistical solutions of the three-dimensional Navier–Stokes equations, in preparation

[6] C. Foias, R. Rosa, R. Temam, Topological properties of the weak-global attractor of the three-dimensional Navier–Stokes equations, in preparation

[7] C. Foias; O.P. Manley; R. Rosa; R. Temam Navier–Stokes Equations and Turbulence, Encyclopedia of Mathematics and its Applications, vol. 83, Cambridge University Press, 2001

[8] C. Foias; G. Prodi Sur les solutions statistiques des équations de Navier–Stokes, Ann. Mat. Pura Appl., Volume 111 (1976) no. 4, pp. 307-330

[9] C. Foias; R. Temam On the Stationary Solutions of the Navier–Stokes Equations and Turbulence, Publications Mathematiques d'Orsay, vol. 120-75-28, 1975 (pp. 38–77)

[10] C. Foias; R. Temam The connection between the Navier–Stokes equations, dynamical systems, and turbulence theory, Directions in Partial Differential Equations (Madison, WI, 1985), Publ. Math. Res. Center Univ. Wisconsin, vol. 54, Academic Press, Boston, MA, 1987, pp. 55-73

[11] U. Frisch Turbulence. The Legacy of A.N. Kolmogorov, Cambridge University Press, Cambridge, 1995 (xiv+296 pp)

[12] A.N. Kolmogorov The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, C. R. (Doklady) Acad. Sci. USSR (N.S.), Volume 30 (1941), pp. 301-305

[13] J. Leray Etude de diverses équations intégrales non linéaires et de quelques problèmes que pose l'hydrodynamique, J. Math. Pures Appl., Volume 12 (1933), pp. 1-82

[14] M. Lesieur Turbulence in Fluids, Fluid Mechanics and its Applications, vol. 40, Kluwer Academic Publishers Group, Dordrecht, 1997 (xxxii+515 pp)

[15] G. Prodi On probability measures related to the Navier–Stokes equations in the 3-dimensional case, Air Force Res. Div. Contr. A.P., Volume 61 (1961) no. (052)-414 (Technical Note no. 2, Trieste, 1961)

[16] O. Reynolds On the dynamical theory of incompressible viscous fluids and the determination of the criterion, Philos. Trans. R. Soc. London A, Volume 186 (1895), pp. 123-164

[17] G.R. Sell; Y. You Dynamics of Evolutionary Equations, Applied Mathematical Sciences, vol. 143, Springer-Verlag, New York, 2002

[18] G.I. Taylor Statistical theory of turbulence, Proc. R. Soc. London Ser. A, Volume 151 (1935), pp. 421-478

[19] R. Temam Navier–Stokes Equations and Nonlinear Functional Analysis, SIAM, Philadelphia, 1995

[20] M.I. Vishik; A.V. Fursikov L'équation de Hopf, les solutions statistiques, les moments correspondant aux systèmes des équations paraboliques quasilinéaires, J. Math. Pures Appl., Volume 59 (1977) no. 9, pp. 85-122

[21] M.I. Vishik; A.V. Fursikov Mathematical Problems of Statistical Hydrodynamics, Kluwer, Dordrecht, 1988

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Ciprian Foias; Ricardo M. S. Rosa; Roger M. Temam Convergence of Time Averages of Weak Solutions of the Three-Dimensional Navier–Stokes Equations, Journal of Statistical Physics, Volume 160 (2015) no. 3, p. 519 | DOI:10.1007/s10955-015-1248-3
Anne Bronzi; Ricardo Rosa On the convergence of statistical solutions of the 3D Navier-Stokes- $α$ model as $α$ vanishes, Discrete Continuous Dynamical Systems - A, Volume 34 (2014) no. 1, p. 19 | DOI:10.3934/dcds.2014.34.19
Chia Ying Lee; Boris Rozovskii On the Stochastic Navier–Stokes Equation Driven by Stationary White Noise, Malliavin Calculus and Stochastic Analysis, Volume 34 (2013), p. 219 | DOI:10.1007/978-1-4614-5906-4_10
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