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A note on statistical solutions of the three-dimensional Navier-Stokes equations: The stationary case. (Note sur les solutions statistiques de équations de Navier-Stokes incompressibles en dimension trois d’espace: le cas stationnaire.) (English) Zbl 1186.35142
Summary: 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.

MSC:
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
76D06 Statistical solutions of Navier-Stokes and related equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35B41 Attractors
35B65 Smoothness and regularity of solutions to PDEs
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[1] Chepyzhov, V.V.; Vishik, M.I., Attractors for equations of mathematical physics, American mathematical society colloquium publications, vol. 49, (2002), American Mathematical Society Providence, RI · Zbl 0986.35001
[2] Foias, C.; Foias, C., Statistical study of navier – stokes equation II, Rend. semin. mat. univ. Padova, Rend. semin. mat. univ. Padova, 49, 9-123, (1973) · Zbl 0283.76018
[3] Foias, C.; Manley, O.P.; Rosa, R.; Temam, R., A note on statistical solutions of the three-dimensional navier – stokes equations: the time-dependent case, Comptes rendus acad. sci. Paris ser. I, 348, 3-4, 235-240, (2010) · Zbl 1186.35141
[4] C. Foias, R. Rosa, R. Temam, Properties of time-dependent statistical solutions of the three-dimensional Navier-Stokes equations, in preparation · Zbl 1304.35486
[5] C. Foias, R. Rosa, R. Temam, Properties of stationary statistical solutions of the three-dimensional Navier-Stokes equations, in preparation · Zbl 1191.35204
[6] C. Foias, R. Rosa, R. Temam, Topological properties of the weak-global attractor of the three-dimensional Navier-Stokes equations, in preparation · Zbl 1191.35204
[7] Foias, C.; Manley, O.P.; Rosa, R.; Temam, R., Navier – stokes equations and turbulence, Encyclopedia of mathematics and its applications, vol. 83, (2001), Cambridge University Press · Zbl 0994.35002
[8] Foias, C.; Prodi, G., Sur LES solutions statistiques des équations de navier – stokes, Ann. mat. pura appl., 111, 4, 307-330, (1976) · Zbl 0344.76015
[9] Foias, C.; Temam, R., On the stationary solutions of the navier – stokes equations and turbulence, Publications mathematiques d’orsay, vol. 120-75-28, (1975), pp. 38-77
[10] Foias, C.; Temam, R., The connection between the navier – stokes equations, dynamical systems, and turbulence theory, (), 55-73
[11] Frisch, U., Turbulence. the legacy of A.N. Kolmogorov, (1995), Cambridge University Press Cambridge, xiv+296 pp
[12] Kolmogorov, A.N., The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, C. R. (doklady) acad. sci. USSR (N.S.), 30, 301-305, (1941) · JFM 67.0850.06
[13] Leray, J., Etude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’hydrodynamique, J. math. pures appl., 12, 1-82, (1933) · Zbl 0006.16702
[14] Lesieur, M., Turbulence in fluids, Fluid mechanics and its applications, vol. 40, (1997), Kluwer Academic Publishers Group Dordrecht, xxxii+515 pp · Zbl 0876.76002
[15] Prodi, G., On probability measures related to the navier – stokes equations in the 3-dimensional case, Air force res. div. contr. A.P., 61, (052)-414, (1961), Technical Note no. 2, Trieste, 1961
[16] Reynolds, O., On the dynamical theory of incompressible viscous fluids and the determination of the criterion, Philos. trans. R. soc. London A, 186, 123-164, (1895) · JFM 26.0872.02
[17] Sell, G.R.; You, Y., Dynamics of evolutionary equations, Applied mathematical sciences, vol. 143, (2002), Springer-Verlag New York · Zbl 1254.37002
[18] Taylor, G.I., Statistical theory of turbulence, Proc. R. soc. London ser. A, 151, 421-478, (1935) · JFM 61.0926.02
[19] Temam, R., Navier – stokes equations and nonlinear functional analysis, (1995), SIAM Philadelphia · Zbl 0833.35110
[20] Vishik, M.I.; Fursikov, A.V., L’équation de Hopf, LES solutions statistiques, LES moments correspondant aux systèmes des équations paraboliques quasilinéaires, J. math. pures appl., 59, 9, 85-122, (1977) · Zbl 0352.35072
[21] Vishik, M.I.; Fursikov, A.V., Mathematical problems of statistical hydrodynamics, (1988), Kluwer Dordrecht · Zbl 0688.35077
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