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 Tr. Mat. Inst. Steklova, 2014, Volume 285, Pages 264–287 (Mi tm3549)

Microscopic solutions of kinetic equations and the irreversibility problem

A. S. Trushechkinab

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
b National Engineering Physics Institute "MEPhI", Moscow, Russia

Abstract: As established by N. N. Bogolyubov, the Boltzmann–Enskog kinetic equation admits the so-called microscopic solutions. These solutions are generalized functions (have the form of sums of delta functions); they correspond to the trajectories of a system of a finite number of balls. However, the existence of these solutions has been established at the “physical” level of rigor. In the present paper, these solutions are assigned a rigorous meaning. It is shown that some other kinetic equations (the Enskog and Vlasov–Enskog equations) also have microscopic solutions. In this sense, one can speak of consistency of these solutions with microscopic dynamics. In addition, new kinetic equations for a gas of elastic balls are obtained through the analysis of a special limit case of the Vlasov equation.

 Funding Agency Grant Number Russian Foundation for Basic Research 12-01-37273-mol_a Ministry of Education and Science of the Russian Federation NSh-864.2014.18215 This work was supported in part by the Russian Foundation for Basic Research (project no. 12-01-37273-mol_a), by a grant of the President of the Russian Federation (project no. NSh-864.2014.1), and by the Ministry of Education and Science of the Russian Federation (project no. 8215).

DOI: https://doi.org/10.1134/S0371968514020186

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English version:
Proceedings of the Steklov Institute of Mathematics, 2014, 285, 251–274

Bibliographic databases:

Document Type: Article
UDC: 517.958+517.968.7

Citation: A. S. Trushechkin, “Microscopic solutions of kinetic equations and the irreversibility problem”, Selected topics of mathematical physics and analysis, Collected papers. In commemoration of the 90th anniversary of Academician Vasilii Sergeevich Vladimirov's birth, Tr. Mat. Inst. Steklova, 285, MAIK Nauka/Interperiodica, Moscow, 2014, 264–287; Proc. Steklov Inst. Math., 285 (2014), 251–274

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3549
• https://doi.org/10.1134/S0371968514020186
• http://mi.mathnet.ru/eng/tm/v285/p264

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This publication is cited in the following articles:
1. Trushechkin A., “Microscopic and Soliton-Like Solutions of the Boltzmann Enskog and Generalized Enskog Equations For Elastic and Inelastic Hard Spheres”, Kinet. Relat. Mod., 7:4 (2014), 755–778
2. M. Pulvirenti, S. Simonella, A. Trushechkin, “Microscopic solutions of the Boltzmann-Enskog equation in the series representation”, Kinet. Relat. Mod., 11:4, SI (2018), 911–931
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