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

This article is cited in 44 scientific papers (total in 44 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.


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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
\by V.~V.~Kozlov
\paper The generalized Vlasov kinetic equation
\jour Uspekhi Mat. Nauk
\yr 2008
\vol 63
\issue 4(382)
\pages 93--130
\jour Russian Math. Surveys
\yr 2008
\vol 63
\issue 4
\pages 691--726

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