Regular and Chaotic Dynamics

, Volume 16, Issue 6, pp 602–622 | Cite as

The Vlasov kinetic equation, dynamics of continuum and turbulence

  • Valery V. Kozlov


We consider a continuum of interacting particles whose evolution is governed by the Vlasov kinetic equation. An infinite sequence of equations of motion for this medium (in the Eulerian description) is derived and its general properties are explored. An important example is a collisionless gas, which exhibits irreversible behavior. Though individual particles interact via a potential, the dynamics of the continuum bears dissipative features. Applicability of the Vlasov equations to the modeling of small-scale turbulence is discussed.


kinetic Vlasov’s equation Euler’s equation continuum turbulence 

MSC2010 numbers

37A60 82B30 82C05 


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  1. 1.
    Bogoliubov, N.N., Problems of Dynamic Theory in Statistical Physics, Oak Ridge, TN: Technical Information Service, 1960.Google Scholar
  2. 2.
    Uhlenbeck, G. E. and Ford, G.W., Lectures in Statistical Mechanics, Providence, RI: AMS, 1963.zbMATHGoogle Scholar
  3. 3.
    Chetverushkin, B.N., Kinetic Schemes and Quasi-Gas Dynamic System of Equations, Moscow: Maks Press, 2004 [Barcelona: CIMNE, 2008].zbMATHGoogle Scholar
  4. 4.
    Kac, M., Some Stochastic Problems in Physics and Mathematics, Colloquium lectures in pure and applied science, vol. 2, Magnolia Petroleum Co., 1956.Google Scholar
  5. 5.
    Vlasov, A.A., Statistical Distribution Functions, Moscow: Nauka, 1966 (Russian).Google Scholar
  6. 6.
    Vedenyapin, V.V., Boltzmann and Vlasov Kinetic Equations, Moscow: Fizmatlit, 2001 (Russian).Google Scholar
  7. 7.
    Maslov V.P., Equations of the Self-Consistent Field, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat., vol. 11, Moscow: VINITI, 1978, pp. 153–234] [J. Soviet Math., 1979, vol. 11, pp. 123–195].Google Scholar
  8. 8.
    Dobrushin, R. L., Vlasov Equations, Funktsional. Anal. i Prilozhen., 1979, vol. 13, no. 2, pp. 48–58 [Funct. Anal. Appl., 1979, vol. 13, no. 2, pp. 115–123].CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Arsen’ev, A.A., Some Estimates of the Solution of Vlasov’s Equation, Zh. Vychisl. Mat. Mat. Fiz., 1985, vol. 25, no. 1, pp. 80–87 [USSR Comput. Math. Math. Phys., 1985, vol. 25, no. 1, pp. 52–57].MathSciNetGoogle Scholar
  10. 10.
    Kozlov, V.V., The Generalized Vlasov Kinetic Equation // Uspekhi Mat. Nauk, 2008, vol. 63, no. 4(382), pp. 93–130 [Russian Math. Surveys, 2008, vol. 63, no. 4, pp. 691–726].Google Scholar
  11. 11.
    Benney, D. J., Some Properties of Long Nonlinear Waves, Stud. Appl. Math., 1973, vol. 52, no. 1, pp. 45–50.zbMATHGoogle Scholar
  12. 12.
    Zakharov, V.E., Benney Equations and Quasiclassical Approximation in the Method of the Inverse Problem, Funktsional. Anal. i Prilozhen., 1980, vol. 14, no. 2, pp. 15–24 [Funct. Anal. Appl., 1980, vol. 14, no. 2, pp. 89–98].CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Gibbons, J., Collisionless Boltzmann Equations and Integrable Moment Equations, Phys. D, 1981, vol. 3, pp. 503–511.CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Lebedev, D.R. and Manin, Yu. I., Conservation Laws and Lax Representation for Benney’s Long Wave Equation, Phys. Lett. A, 1979, vol. 74, pp. 154–156.CrossRefMathSciNetGoogle Scholar
  15. 15.
    Gibbons, J. and Tsarev, S.P., Reductions of the Benney Equations, Phys. Lett. A, 1996, vol. 211, pp. 19–24.CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Akhiezer, N. I., The Classical Moment Problem and Some Related Questions in Analysis, Edinburgh-London: Oliver & Boyd, 1965.zbMATHGoogle Scholar
  17. 17.
    Poincaré, H., Figures d’équilibre d’une masse fluide, Paris: Gauthier-Villars, 1902.Google Scholar
  18. 18.
    Poincaré, H., Réfflexions sur la théorie cinétique des gaz, J. Phys. théoret. et appl., sér. 4, 1906, vol. 5, pp. 369–403.Google Scholar
  19. 19.
    Kozlov, V.V., Kinetics of Collisionless Continuous Medium, Regul. Chaotic Dyn., 2001, vol. 6, no. 3, pp. 235–251.CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Kozlov, V.V., Notes on Diffusion in Collisionless Medium, Regul. Chaotic Dyn., 2004, vol. 9, no. 1, pp. 29–34.CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Pokhozhaev, S. I., On Stationary Solutions of the Vlasov-Poisson Equations, Differ. Uravn., 2010, vol. 46, no. 4, pp. 527–534 [Differ. Equ., 2010, vol. 46, no. 4, pp. 530–537].MathSciNetGoogle Scholar
  22. 22.
    Batt, J., Faltenbacher, W., and Horst, E., Stationary Spherically Symmetric Models in Stellar Dynamics, Arch. Ration. Mech. Anal., 1986, vol. 93, no. 2, pp. 159–183.CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Batt, J., Berestycki, H., Decond, P., and Pertname, B., Some Families of Solutions of the Vlasov-Poisson System, Arch. Ration. Mech. Anal., 1988, vol. 104, no. 1, pp. 79–103.CrossRefzbMATHGoogle Scholar
  24. 24.
    Vedenyapin, V.V., On the Classification of Steady-State Solutions of Vlasov’s Equation on the Torus, and a Boundary Value Problem Dokl. Ross. Akad. Nauk, 1992, vol. 323, no. 6, pp. 1004–1006 [Russ. Acad. Sci., Dokl. Math., 1992, vol. 45, no. 2, pp. 459–462].MathSciNetGoogle Scholar
  25. 25.
    Carleman, T., Problèmes mathématiques dans la théorie cinétique des gaz, Upsala: Almqvist & Wiksells, 1957.zbMATHGoogle Scholar
  26. 26.
    Poincaré, H., Les méthodes nouvelles de la Mécanique céleste: T. 1, Paris: Gauthier-Villars, 1892.Google Scholar
  27. 27.
    Kozlov, V.V., Symmetries, Topology, and Resonances in Hamiltonian Mechanics, Berlin: Springer, 1995.Google Scholar
  28. 28.
    Belotserkovskii, O. M., Fimin, N.N., and Chechetkin, V. M., Application of the Kac Equation to Turbulence Simulation, Zh. Vychisl. Mat. Mat. Fiz., 2010, vol. 50, no. 3, pp. 575–584 [Comput. Math. Math. Phys., 2010, vol. 50, no. 3, pp. 549–557.MathSciNetzbMATHGoogle Scholar
  29. 29.
    Pfaffelmoser, K., Global Classical Solutions of the Vlasov-Poisson System in Three Dimensions for General Initial Data, J. Differential Equations, 1992, vol. 95, no. 2, pp. 281–303.CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    Lions, P. L. and Perthame, B., Propagation of Moments and Regularity for the 3-Dimensional Vlasov-Poisson System, Invent. Math., 1991, vol. 105, no. 2, pp. 415–430.CrossRefMathSciNetzbMATHGoogle Scholar
  31. 31.
    Fridman, A. M., Prediction and Discovery of New Structures in Spiral Galaxies, Uspekhi Fiz. Nauk, 2007, vol. 177, no. 2, pp. 121–148 [Physics-Uspekhi, 2007, vol. 50, no. 2, pp. 115–139].CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  1. 1.V.A. Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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