Properties of time-dependent statistical solutions of the three-dimensional Navier-Stokes equations
Annales de l'Institut Fourier, Volume 63 (2013) no. 6, pp. 2515-2573.

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.

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.

DOI: 10.5802/aif.2836
Classification: 35Q30,  76D06,  37A60,  28C20
Keywords: Navier-Stokes equations, statistical solutions, turbulence, measure theory, functional analysis
Foias, Ciprian 1; Rosa, Ricardo M. S. 2; Temam, Roger 3

1 Texas A&M University Department of Mathematics College Station, TX 77843 (USA)
2 Universidade Federal do Rio de Janeiro Instituto de Matemática Caixa Postal 68530 Ilha do Fundão Rio de Janeiro, RJ 21945-970 (Brazil)
3 Indiana University Department of Mathematics Bloomington, IN 47405 (USA)
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Foias, Ciprian; Rosa, Ricardo M. S.; Temam, Roger. Properties of time-dependent statistical solutions of the three-dimensional Navier-Stokes equations. Annales de l'Institut Fourier, Volume 63 (2013) no. 6, pp. 2515-2573. doi : 10.5802/aif.2836. https://aif.centre-mersenne.org/articles/10.5802/aif.2836/

[1] Aliprantis, Charalambos D.; Border, Kim C. Infinite dimensional analysis, Springer, Berlin, 2006, pp. xxii+703 (A hitchhiker’s guide) | MR | Zbl

[2] Ball, J. M. Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci., Volume 7 (1997) no. 5, pp. 475-502 | DOI | MR | Zbl

[3] Basson, Arnaud Homogeneous statistical solutions and local energy inequality for 3D Navier-Stokes equations, Comm. Math. Phys., Volume 266 (2006) no. 1, pp. 17-35 | DOI | MR | Zbl

[4] Batchelor, G. K. The theory of homogeneous turbulence, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge at the University Press, 1953, pp. x+197 | MR | Zbl

[5] Bercovici, H.; Constantin, P.; Foias, C.; Manley, O. P. Exponential decay of the power spectrum of turbulence, J. Statist. Phys., Volume 80 (1995) no. 3-4, pp. 579-602 | DOI | MR | Zbl

[6] Bourbaki, N. Éléments de mathématique. Fasc. XXXV. Livre VI: Intégration. Chapitre IX: Intégration sur les espaces topologiques séparés, Actualités Scientifiques et Industrielles, No. 1343, Hermann, Paris, 1969, pp. 133 | MR | Zbl

[7] Brown, Arlen; Pearcy, Carl Introduction to operator theory. I, Springer-Verlag, New York, 1977, pp. xiv+474 (Elements of functional analysis, Graduate Texts in Mathematics, No. 55) | MR | Zbl

[8] Constantin, P.; Foias, C.; Temam, R. Attractors representing turbulent flows, Mem. Amer. Math. Soc., Volume 53 (1985) no. 314, pp. vii+67 | MR | Zbl

[9] Constantin, Peter; Doering, Charles R. Variational bounds on energy dissipation in incompressible flows: shear flow, Phys. Rev. E (3), Volume 49 (1994) no. 5, part A, pp. 4087-4099 | DOI | MR

[10] Constantin, Peter; Doering, Charles R. Variational bounds on energy dissipation in incompressible flows. II. Channel flow, Phys. Rev. E (3), Volume 51 (1995) no. 4, part A, pp. 3192-3198 | DOI | MR

[11] Constantin, Peter; Foias, Ciprian Navier-Stokes equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988, pp. x+190 | MR | Zbl

[12] Dascaliuc, R. A generalization of the energy inequality for the Leray-Hopf solutions of the 3D periodic Navier-Stokes equations (arXiv:1010.5535v1)

[13] Doering, Charles R.; Titi, Edriss S. Exponential decay rate of the power spectrum for solutions of the Navier-Stokes equations, Phys. Fluids, Volume 7 (1995) no. 6, pp. 1384-1390 | DOI | MR | Zbl

[14] Dunford, Nelson; Schwartz, Jacob T. Linear Operators. I. General Theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York, 1958, pp. xiv+858 | MR | Zbl

[15] Foias, Ciprian Statistical study of Navier-Stokes equations. I, Rend. Sem. Mat. Univ. Padova, Volume 48 (1972), pp. 219-348 | Numdam | MR | Zbl

[16] Foias, Ciprian A functional approach to turbulence, Russian Math. Survey, Volume 29 (1974) no. 2, pp. 293-336 | DOI | Zbl

[17] Foias, Ciprian; Jolly, M. S.; Manley, O. P.; Rosa, R.; Temam, R. Kolmogorov theory via finite-time averages, Phys. D, Volume 212 (2005) no. 3-4, pp. 245-270 | DOI | MR | Zbl

[18] Foias, Ciprian; Manley, O.; Rosa, R.; Temam, R. Navier-Stokes equations and turbulence, Encyclopedia of Mathematics and its Applications, 83, Cambridge University Press, Cambridge, 2001, pp. xiv+347 | MR | Zbl

[19] Foias, Ciprian; Manley, Oscar P.; Rosa, Ricardo M. S.; Temam, Roger Cascade of energy in turbulent flows, C. R. Acad. Sci. Paris Sér. I Math., Volume 332 (2001) no. 6, pp. 509-514 | DOI | MR | Zbl

[20] Foias, Ciprian; Manley, Oscar P.; Rosa, Ricardo M. S.; Temam, Roger Estimates for the energy cascade in three-dimensional turbulent flows, C. R. Acad. Sci. Paris Sér. I Math., Volume 333 (2001) no. 5, pp. 499-504 | DOI | MR | Zbl

[21] Foias, Ciprian; Prodi, G. Sur les solutions statistiques des équations de Navier-Stokes, Ann. Mat. Pura Appl. (4), Volume 111 (1976), pp. 307-330 | DOI | MR | Zbl

[22] Foias, Ciprian; Rosa, Ricardo M. S.; Temam, Roger Properties of stationary statistical solutions of the three-dimensional Navier-Stokes equations (in preparation)

[23] Foias, Ciprian; Rosa, Ricardo M. S.; Temam, Roger A note on statistical solutions of the three-dimensional Navier-Stokes equations: the stationary case, C. R. Math. Acad. Sci. Paris, Volume 348 (2010) no. 5-6, pp. 347-353 | DOI | MR | Zbl

[24] Foias, Ciprian; Rosa, Ricardo M. S.; Temam, Roger A note on statistical solutions of the three-dimensional Navier-Stokes equations: the time-dependent case, C. R. Math. Acad. Sci. Paris, Volume 348 (2010) no. 3-4, pp. 235-240 | DOI | MR | Zbl

[25] Foias, Ciprian; Rosa, Ricardo M. S.; Temam, Roger Topological properties of the weak global attractor of the three-dimensional Navier-Stokes equations, Discrete Contin. Dyn. Syst., Volume 27 (2010) no. 4, pp. 1611-1631 | DOI | MR | Zbl

[26] Frisch, U. Turbulence, The legacy of A. N. Kolmogorov, Cambridge University Press (1995), pp. xiv+296 | MR | Zbl

[27] Hinze, J. O. Turbulence, McGraw-Hill Book Co., New York, 1975

[28] Hopf, Eberhard Statistical hydromechanics and functional calculus, J. Rational Mech. Anal., Volume 1 (1952), pp. 87-123 | MR | Zbl

[29] Howard, L.N. Bounds on flow quantities, Annu. Rev. Fluid Mech., Volume 4 (1972), pp. 473-494 | DOI | Zbl

[30] 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.), Volume 30 (1941), pp. 301-305 | MR

[31] Kolmogorov, A. N. On degeneration of isotropic turbulence in an incompressible viscous liquid, C. R. (Doklady) Acad. Sci. USSR (N. S.), Volume 31 (1941), pp. 538-540 | MR | Zbl

[32] Kuratowski, K. Topology. Vol. I, New edition, revised and augmented. Translated from the French by J. Jaworowski, Academic Press, New York, 1966, pp. xx+560 | MR | Zbl

[33] Ladyzhenskaya, O. A. The mathematical theory of viscous incompressible flow, Revised English edition. Translated from the Russian by Richard A. Silverman, Gordon and Breach Science Publishers, New York, 1963, pp. xiv+184 | MR | Zbl

[34] Leray, Jean Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math., Volume 63 (1934) no. 1, pp. 193-248 | DOI | MR

[35] Lesieur, Marcel Turbulence in fluids, Fluid Mechanics and its Applications, 40, Kluwer Academic Publishers Group, Dordrecht, 1997, pp. xxxii+515 | MR | Zbl

[36] Monin, A. S.; Yaglom, A. M. Statistical fluid mechanics: mechanics of turbulence, Vol. I and II, Dover Publications Inc., Mineola, NY, 2007, pp. xii+769 and xii+874 (Translated from the 1965 Russian original, Edited and with a preface by John L. Lumley, English edition updated, augmented and revised by the authors, Reprinted from the 1975 edition) | MR | Zbl

[37] Moschovakis, Yiannis N. Descriptive set theory, Studies in Logic and the Foundations of Mathematics, 100, North-Holland Publishing Co., Amsterdam, 1980, pp. xii+637 | MR | Zbl

[38] Ramos, F.; Rosa, R.; Temam, R. Statistical estimates for channel flows driven by a pressure gradient, Phys. D, Volume 237 (2008) no. 10-12, pp. 1368-1387 | DOI | MR | Zbl

[39] Reynolds, O. On the dynamical theory of incompressible viscous fluids and the determination of the criterion, Proc. Roy. Soc. London Ser. A, Volume 451 (1995) no. 1941, pp. 5-47 | DOI | MR

[40] Rudin, Walter Real and complex analysis, McGraw-Hill Book Co., New York, 1987, pp. xiv+416 | MR | Zbl

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

[42] Taylor, G.I. The spectrum of turbulence, Proc. Roy. Soc. London Ser. A, Volume 164 (1938), pp. 476-490 | DOI

[43] Temam, Roger Navier-Stokes equations, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam, 1984, pp. xii+526 (Reedition in 2001 in the AMS Chelsea Publishing, Providence, RI) | MR | Zbl

[44] Temam, Roger Navier-Stokes equations and nonlinear functional analysis, CBMS-NSF Regional Conference Series in Applied Mathematics, 66, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995, pp. xiv+141 | MR | Zbl

[45] Višik, M. I.; Fursikov, A. V. Translationally homogeneous statistical solutions and individual solutions with infinite energy of a system of Navier-Stokes equations, Siberian Mathematical Journal, Volume 19 (1978) no. 5, pp. 710-729 (Translated from Sibirskii Matematicheskii Sbornik, Vol. 19, no. 5, 1005–1031, September-October 1978) | DOI | MR | Zbl

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

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