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TMF, 2020, Volume 203, Number 3, Pages 417–450 (Mi tmf9849)  

This article is cited in 1 scientific paper (total in 1 paper)

Group analysis of the Boltzmann and Vlasov equations

K. S. Platonova*, A. V. Borovskikh

Lomonosov Moscow State University, Moscow, Russia

Abstract: We present results of a group analysis of the multidimensional Boltzmann and Vlasov equations. For the Boltzmann equation, we obtain the equivalence group and classifying relations for the symmetry group and study these relations in the case where external forces are absent. We discover a scale paradox: we show that for any collision integral, an equation that is invariant with respect to the shift group does not admit uniform dilations, because the left- and right-hand sides of the equation scale differently. In particular, this holds for the classical Boltzmann equation. For the Vlasov equation, we also obtain the equivalence group and classifying relations for the symmetry group and classify the interparticle interactions for which the Vlasov equation admits groups containing the Galilean group in the case where external forces are absent.

Keywords: Boltzmann equation, Vlasov equation, symmetry group, equivalence group, kinetic equation.
* Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/tmf9849

Full text: PDF file (709 kB)
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English version:
Theoretical and Mathematical Physics, 2020, 203:3, 794–823

Bibliographic databases:

MSC: 35Q20, 35Q83
Received: 11.11.2019
Revised: 11.11.2019

Citation: K. S. Platonova, A. V. Borovskikh, “Group analysis of the Boltzmann and Vlasov equations”, TMF, 203:3 (2020), 417–450; Theoret. and Math. Phys., 203:3 (2020), 794–823

Citation in format AMSBIB
\Bibitem{PlaBor20}
\by K.~S.~Platonova, A.~V.~Borovskikh
\paper Group analysis of the~Boltzmann and Vlasov equations
\jour TMF
\yr 2020
\vol 203
\issue 3
\pages 417--450
\mathnet{http://mi.mathnet.ru/tmf9849}
\crossref{https://doi.org/10.4213/tmf9849}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=4104785}
\elib{https://elibrary.ru/item.asp?id=45519184}
\transl
\jour Theoret. and Math. Phys.
\yr 2020
\vol 203
\issue 3
\pages 794--823
\crossref{https://doi.org/10.1134/S0040577920060070}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85087515122}


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  • http://mi.mathnet.ru/eng/tmf9849
  • https://doi.org/10.4213/tmf9849
  • http://mi.mathnet.ru/eng/tmf/v203/i3/p417

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. K. S. Platonova, A. V. Borovskikh, “Group analysis of the one-dimensional Boltzmann equation. Invariants and the problem of moment system closure”, Theoret. and Math. Phys., 208:3 (2021), 1165–1181  mathnet  crossref  crossref  isi  elib
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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