[Note sur les solutions statistiques des équations de Navier–Stokes incompressibles en dimension trois d'espace : le cas dépendant du temps]
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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@article{CRMATH_2010__348_3-4_235_0,
author = {Foias, Ciprian and Rosa, Ricardo M.S. and Temam, Roger},
title = {A note on statistical solutions of the three-dimensional {Navier{\textendash}Stokes} equations: {The} time-dependent case},
journal = {Comptes Rendus. Math\'ematique},
pages = {235--240},
publisher = {Elsevier},
volume = {348},
number = {3-4},
year = {2010},
doi = {10.1016/j.crma.2009.12.017},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2009.12.017/}
}
TY - JOUR AU - Foias, Ciprian AU - Rosa, Ricardo M.S. AU - Temam, Roger TI - A note on statistical solutions of the three-dimensional Navier–Stokes equations: The time-dependent case JO - Comptes Rendus. Mathématique PY - 2010 SP - 235 EP - 240 VL - 348 IS - 3-4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.12.017/ DO - 10.1016/j.crma.2009.12.017 LA - en ID - CRMATH_2010__348_3-4_235_0 ER -
%0 Journal Article %A Foias, Ciprian %A Rosa, Ricardo M.S. %A Temam, Roger %T A note on statistical solutions of the three-dimensional Navier–Stokes equations: The time-dependent case %J Comptes Rendus. Mathématique %D 2010 %P 235-240 %V 348 %N 3-4 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2009.12.017/ %R 10.1016/j.crma.2009.12.017 %G en %F CRMATH_2010__348_3-4_235_0
Foias, Ciprian; Rosa, Ricardo M.S.; Temam, Roger. A note on statistical solutions of the three-dimensional Navier–Stokes equations: The time-dependent case. Comptes Rendus. Mathématique, Tome 348 (2010) no. 3-4, pp. 235-240. doi : 10.1016/j.crma.2009.12.017. http://www.numdam.org/articles/10.1016/j.crma.2009.12.017/
[1] The Theory of Homogeneous Turbulence, Cambridge University Press, Cambridge, 1953
[2] Statistical study of Navier–Stokes equations I, Rend. Sem. Mat. Univ. Padova, Volume 48 (1972), pp. 219-348
[3] Sur les solutions statistiques des équations de Navier–Stokes, Ann. Mat. Pura Appl., Volume 111 (1976) no. 4, pp. 307-330
[4] Navier–Stokes Equations and Turbulence, Encyclopedia of Mathematics and Its Applications, vol. 83, Cambridge University Press, 2001
[5] A note on statistical solutions of the three-dimensional Navier–Stokes equations: the stationary case, C. R. Acad. Sci. Paris, Ser. I, Volume 348 (2010) | DOI
[6] C. Foias, R. Rosa, R. Temam, Properties of time-dependent statistical solutions of the three-dimensional Navier–Stokes equations, in preparation
[7] C. Foias, R. Rosa, R. Temam, Properties of stationary statistical solutions of the three-dimensional Navier–Stokes equations, in preparation
[8] Turbulence, The Legacy of A.N. Kolmogorov, Cambridge University Press, Cambridge, 1995 (xiv+296 pp)
[9] Turbulence, McGraw–Hill, New York, 1975
[10] Statistical hydromechanics and functional calculus, J. Ration. Mech. Anal., Volume 1 (1952), pp. 87-123
[11] The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, C. R. (Dokl.) Acad. Sci. USSR (N.S.), Volume 30 (1941), pp. 301-305
[12] On degeneration of isotropic turbulence in an incompressible viscous liquid, C. R. (Dokl.) Acad. Sci. USSR (N.S.), Volume 31 (1941), pp. 538-540
[13] 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] Turbulence in Fluids, Fluid Mechanics and Its Applications, vol. 40, Kluwer Academic Publishers Group, Dordrecht, 1997 (xxxii+515 pp)
[15] Statistical Fluid Mechanics: Mechanics of Turbulence, MIT Press, Cambridge, MA, 1975
[16] On the energy distribution in the spectrum of turbulent flow, C. R. (Dokl.) Acad. Sci. USSR, Volume 32 (1941), pp. 19-21
[17] On the dynamical theory of incompressible viscous fluids and the determination of the criterion, Phil. Trans. Roy. Soc. London A, Volume 186 (1895), pp. 123-164
[18] Statistical theory of turbulence, Proc. Roy. Soc. London Ser. A, Volume 151 (1935), pp. 421-478
[19] The spectrum of turbulence, Proc. Roy. Soc. London Ser. A, Volume 164 (1938), pp. 476-490
[20] Navier–Stokes Equations and Nonlinear Functional Analysis, SIAM, Philadelphia, 1995
[21] L'équation de Hopf, les solutions statistiques, les moments correspondant aux systémes des équations paraboliques quasilinéaires, J. Math. Pures Appl., Volume 59 (1977) no. 9, pp. 85-122
[22] Mathematical Problems of Statistical Hydrodynamics, Kluwer, Dordrecht, 1988 (+ Additional references)
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