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Uspekhi Mat. Nauk, 2008, Volume 63, Issue 4(382), Pages 93–130 (Mi umn9216)  

This article is cited in 39 scientific papers (total in 39 papers)

The generalized Vlasov kinetic equation

V. V. Kozlov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: This paper is concerned with the investigation of a generalized kinetic equation describing the evolution of the density of a probability measure. In the general case this is a non-linear integro-differential equation. On the one hand, this equation includes as a special case the simpler linear Liouville equation (which underlies classical statistical mechanics) and the equation of a self-consistent field (the Vlasov kinetic equation). On the other hand, some other well-known equations also reduce to this equation, for instance, the vorticity equation for plane flows of an ideal incompressible fluid. The main aim of the paper is to study the problem of the weak limits, as the time tends to infinity, of solutions of the generalized kinetic equation. This problem plays a significant role in the transition from a micro- to a macrodescription, when the behaviour of the averages (most probable values) of dynamical quantities is considered. The theory of weak limits of solutions of the Liouville equation is closely connected with ideas and methods of ergodic theory. The case under consideration presents greater difficulties, which stem from the non-trivial problem of the existence of invariant countably-additive measures for dynamical systems in infinite-dimensional spaces. General results are applied to the analysis of continua of interacting particles and to the investigation of statistical properties of plane flows of an ideal fluid.

DOI: https://doi.org/10.4213/rm9216

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English version:
Russian Mathematical Surveys, 2008, 63:4, 691–726

Bibliographic databases:

UDC: 517
MSC: Primary 37A60, 82B30, 82C05; Secondary 82C35
Received: 21.05.2008

Citation: V. V. Kozlov, “The generalized Vlasov kinetic equation”, Uspekhi Mat. Nauk, 63:4(382) (2008), 93–130; Russian Math. Surveys, 63:4 (2008), 691–726

Citation in format AMSBIB
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    4. Finkelshtein D., Kondratiev Yu., Kutoviy O., “Vlasov scaling for the Glauber dynamics in continuum”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 14:4 (2011), 537–569  crossref  zmath  isi  elib  scopus
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    6. V. V. Vedenyapin, M. A. Negmatov, “Derivation and classification of Vlasov-type and magnetohydrodynamics equations: Lagrange identity and Godunov's form”, Theoret. and Math. Phys., 170:3 (2012), 394–405  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    7. Skubachevskii A.L., “On the unique solvability of initial boundary value problems for the Vlasov–Poisson system of equations in a half-space”, Dokl. Math., 85:2 (2012), 255–258  crossref  mathscinet  zmath  isi  elib  elib  scopus
    8. Manita O.A. Shaposhnikov S.V., “Nonlinear parabolic equations for measures”, Dokl. Math., 86:3 (2012), 857–860  crossref  mathscinet  zmath  isi  elib  scopus
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    11. B. I. Sadovnikov, N. G. Inozemtseva, E. E. Perepelkin, “Generalized phase space and conservative systems”, Dokl. Math., 88:1 (2013), 457  crossref  crossref  mathscinet  zmath  isi  elib  elib  scopus
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    13. V. V. Vedenyapin, M. A. Negmatov, “On derivation and classification of Vlasov type equations and equations of magnetohydrodynamics. The Lagrange identity, the Godunov form, and critical mass”, Journal of Mathematical Sciences, 202:5 (2014), 769–782  mathnet  crossref
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    15. A. S. Trushechkin, “Microscopic solutions of kinetic equations and the irreversibility problem”, Proc. Steklov Inst. Math., 285 (2014), 251–274  mathnet  crossref  crossref  isi  elib  elib
    16. B. I. Sadovnikov, E. E. Perepelkin, N. G. Inozemtseva, “Coordinate uncertainty principle in a generalized phase space”, Dokl. Math, 90:2 (2014), 628  crossref  mathscinet  zmath  isi  scopus
    17. B. I. Sadovnikov, E. E. Perepelkin, N. G. Inozemtseva, “New class of special functions based on the exact solution of a nonlinear partial differential equation of divergence type”, Dokl. Math, 91:1 (2015), 105  crossref  mathscinet  zmath  isi  scopus
    18. O. A. Manita, M. S. Romanov, S. V. Shaposhnikov, “Uniqueness of probability solutions to nonlinear Fokker-Planck-Kolmogorov equation”, Dokl. Math, 91:2 (2015), 142  crossref  mathscinet  zmath  isi  scopus
    19. A. L. Skubachevskii, “Nonlocal Problems for the Vlasov–Poisson Equations in an Infinite Cylinder”, Funct. Anal. Appl., 49:3 (2015), 234–238  mathnet  crossref  crossref  isi  elib
    20. Kozlov V.V. Smolyanov O.G., “Invariant and Quasi-Invariant Measures on Infinite-Dimensional Spaces”, 92, no. 3, 2015, 743–746  crossref  mathscinet  zmath  isi  scopus
    21. Berns Ch., Kondratiev Yu., Kutoviy O., “Markov Jump Dynamics With Additive Intensities in Continuum: State Evolution and Mesoscopic Scaling”, 161, no. 4, 2015, 876–901  crossref  mathscinet  zmath  isi  scopus
    22. Manita O.A., Romanov M.S., Shaposhnikov S.V., “on Uniqueness of Solutions To Nonlinear Fokker-Planek-Kolmogorov Equations”, 128, 2015, 199–226  crossref  mathscinet  zmath  isi  scopus
    23. Kozlov V.V., “Coarsening in Ergodic Theory”, 22, no. 2, 2015, 184–187  crossref  mathscinet  zmath  isi  scopus
    24. Yu. O. Belyaeva, “Statsionarnye resheniya uravnenii Vlasova dlya vysokotemperaturnoi dvukomponentnoi plazmy”, Trudy seminara po differentsialnym i funktsionalno-differentsialnym uravneniyam v RUDN pod rukovodstvom A. L. Skubachevskogo, SMFN, 62, RUDN, M., 2016, 19–31  mathnet
    25. Skubachevskii A.L., “Nonlocal elliptic problems in infinite cylinder and applications”, Discret. Contin. Dyn. Syst.-Ser. S, 9:3 (2016), 847–868  crossref  mathscinet  zmath  isi  elib  scopus
    26. Lukashev E.A., Yakovlev N.N., Radkevich E.V., Vasil'yeva O.A., “On problems of the laminar–turbulent transition”, Dokl. Math., 94:3 (2016), 649–653  crossref  mathscinet  zmath  isi  scopus
    27. Skubachevskii A.L., Tsuzuki Y., “Vlasov–Poisson equations for a two-component plasma in a half-space”, Dokl. Math., 94:3 (2016), 681–683  crossref  mathscinet  zmath  isi  scopus
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    29. V. I. Bogachev, A. I. Kirillov, S. V. Shaposhnikov, “Distances between stationary distributions of diffusions and solvability of nonlinear Fokker–Planck–Kolmogorov equations”, Theory Probab. Appl., 62:1 (2018), 12–34  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
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    31. V. V. Vedenyapin, M. A. Negmatov, N. N. Fimin, “Vlasov-type and Liouville-type equations, their microscopic, energetic and hydrodynamical consequences”, Izv. Math., 81:3 (2017), 505–541  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    32. E. A. Lukashev, E. V. Radkevich, N. N. Yakovlev, O. A. Vasileva, “Vvedenie v obobschennuyu teoriyu neravnovesnykh fazovykh perekhodov Kana—Khillarda (termodinamicheskii analiz zadach mekhaniki sploshnoi sredy)”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 21:3 (2017), 437–472  mathnet  crossref  zmath  elib
    33. V. V. Vedenyapin, “Uravnenie Vlasova–Maksvella–Einshteina”, Preprinty IPM im. M. V. Keldysha, 2018, 188, 20 pp.  mathnet  crossref  elib
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    39. Ha S.-Y., Kim J., Kuchling P., Kutoviy L., “Infinite Particle Systems With Collective Behaviour and Related Mesoscopic Equations”, J. Math. Phys., 60:12 (2019), 122704  crossref  isi
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