Convergence of Time Averages of Weak Solutions of the Three-Dimensional Navier–Stokes Equations

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• 作者：Ciprian Foias ; Ricardo M. S. Rosa ; Roger M. Temam
• 关键词：Navier–Stokes equations ; Stationary statistical solutions ; Time averages ; Recurrence ; Sojourn time ; 35Q30 ; 76D05 ; 76D06 ; 37A05 ; 37L40e2
• 刊名：Journal of Statistical Physics
• 出版年：2015
• 出版时间：August 2015
• 年：2015
• 卷：160
• 期：3
• 页码：519-531
• 全文大小：465 KB
• 参考文献：1.Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd edn. Springer, Berlin (2006)
  2.Batchelor, G.K.: The Theory of Homogeneous Turbulence. Cambridge University Press, Cambridge (1953)MATH
  3.Bercovici, H., Constantin, P., Foias, C., Manley, O.P.: Exponential decay of the power spectrum of turbulence. J. Stat. Phys. 80(3-), 579-02 (1995)MATH MathSciNet ADS View Article
  4.Chekroun, M.D., Glatt-Holtz, N.E.: Invariant measures for dissipative dynamical systems: abstract results and applications. Commun. Math. Phys. 316(3), 723-61 (2012)MATH MathSciNet ADS View Article
  5.Constantin, P., Foias, C.: Navier–Stokes Equations. University of Chicago Press, Chicago (1989)
  6.Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems. Cambridge University Press, Cambridge (1996)MATH View Article
  7.Dunford, N., Schwartz, J.T.: Linear Operators, I. General Theory. Pure and Applied Mathematics. Interscience, New York (1958)MATH
  8.Foias, C.: Statistical study of Navier–Stokes equations I. Rend. Sem. Mat. Univ. Padova 48, 219-48 (1972)MathSciNet
  9.Foias, C.: Statistical study of Navier–Stokes equations II. Rend. Sem. Mat. Univ. Padova 49, 9-23 (1973)MATH
  10.Foias, C., Temam, R.: On the stationary statistical solutions of the Navier–Stokes equations. Publications Mathématiques d’Orsay 120-5-28 (1975)
  11.Foias, C., Prodi, G.: Sur les solutions statistiques des équations de Navier–Stokes. Ann. Mater. Pur. Appl. 111(4), 307-30 (1976)MATH MathSciNet View Article
  12.Foias, C., Manley, O.P., Rosa, R., Temam, R.: Navier–Stokes Equations and Turbulence. Encyclopedia of Mathematics and its Applications, vol. 83. Cambridge University Press, Cambridge (2001)MATH View Article
  13.Foias, C., Jolly, M., Manley, O.P., Rosa, R.: Statistical estimates for the Navier–Stokes equations and the Kraichnan theory of 2-D fully developed turbulence. J. Stat. Phys. 108(3-), 591-46 (2002)MATH MathSciNet View Article
  14.Foias, C., Rosa, R., Temam, R.: A note on statistical solutions of the three-dimensional Navier–Stokes equations: the stationary case. C. R. Math. 348, 235-40 (2010)MATH MathSciNet View Article
  15.Foias, C., Rosa, R., Temam, R.: A note on statistical solutions of the three-dimensional Navier–Stokes equations: the time-dependent case. C. R. Math. 348, 347-53 (2010)MATH MathSciNet View Article
  16.Foias, C., Rosa, R., Temam, R.: Properties of time-dependent statistical solutions of the three-dimensional Navier–Stokes equations. Ann. Inst. Fourier 63(6), 2515-573 (2013)MATH MathSciNet View Article
  17.Frisch, U.: Turbulence. The legacy of A. N. Kolmogorov. Cambridge University Press, Cambridge (1995). xiv+296 ppMATH
  18.Hinze, J.O.: Turbulence. McGraw-Hill, New York (1975)
  19.Krengel, U.: Ergodic Theorems. With a supplement by Antoine Brunel. De Gruyter Studies in Mathematics, vol. 6. Walter de Gruyter & Co., Berlin (1985)MATH View Article
  20.Krylov, N., Bogoliubov, N.N.: La théorie générale de la mesure dans son application à l’étude des systèmes dynamiques de la mécanique non linéaire. Ann. Math. 38, 65-13 (1937)MATH View Article
  21.Ladyzhenskaya, O.: 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)
  22.Lesieur, M.: Turbulence in fluids. Fluid Mechanics and its Applications, vol. 40, 3rd edn. Kluwer Academic Publishers Group, Dordrecht (1997). xxxii+515 pp
  23.Lukaszewicz, G., Real, J., Robinson, J.C.: Invariant measures for dissipative systems and generalised Banach limits. J. Dyn. Diff. Equ. 23(2), 225-50 (2011)MATH MathSciNet View Article
  24.Monin, A.S., Yaglom, A.M.: Statistical Fluid Mechanics: Mechanics of Turbulence. MIT Press, Cambridge, MA (1975)
  25.Schwartz, L.: Radon measures on arbitrary topological spaces and cylindrical measures, Tata Institute of Fundamental Research Studies in Mathematics, No. 6. Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London (1973)
  26.Sell, G.R.: Global attractors for the three-dimensional Navier–Stokes equations. J. Dyn. Differ. Equ. 8(1), 1-3 (1996)MATH MathSciNet View Article
  27.Taylor, G.I.: Statistical theory of turbulence. Proc. R. Soc. Lond. Ser. A 151, 421-78 (1935)MATH ADS View Article
  28.Taylor, G.I.: The spectrum of turbulence. Proc. R. Soc. Lond. Ser. A 164, 476-90 (1938)ADS View Article
  29.Temam, R.: Navier–Stokes Equations. Theory and Numerical Analysis. Studies in Mathematics and its Applications, vol. 2, 3rd edn. North-Holland Publishing Co., Amsterdam (1984). Reedition in the AMS Chelsea series, AMS, Providence (2001)MATH
  30.Vishik, M.I., Fursikov, A.V.: Translationally homogeneous statistical solutions and individual solutions with infinite energy of a system of Navier–Stokes equations. Sib. Math. J. 19(5), 710-29 (1978). (Translated
• 作者单位：Ciprian Foias (1)
  Ricardo M. S. Rosa (2)
  Roger M. 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, Ilha do Fund?o, Caixa Postal 68530, Rio de Janeiro, RJ, 21945-970, Brazil
  3. Department of Mathematics, Indiana University, Bloomington, IN, 47405, USA
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Statistical Physics
  Mathematical and Computational Physics
  Physical Chemistry
  Quantum Physics
  
• 出版者：Springer Netherlands
• ISSN：1572-9613

文摘

Using the concept of stationary statistical solution, which generalizes the notion of invariant measure, it is proved that, in a suitable sense, time averages of almost every Leray–Hopf weak solution of the three-dimensional incompressible Navier–Stokes equations converge as the averaging time goes to infinity. This system of equations is not known to be globally well-posed, and the above result answers a long-standing problem, extending to this system a classical result from ergodic theory. It is also shown that, from a measure-theoretic point of view, the stationary statistical solution obtained from a generalized limit of time averages is independent of the choice of the generalized limit. Finally, any Borel subset of the phase space with positive measure with respect to a stationary statistical solution is such that for almost all initial conditions in that Borel set and for at least one Leray–Hopf weak solution starting with that initial condition, the corresponding orbit is recurrent to that Borel subset and its mean sojourn time within that Borel subset is strictly positive.