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Izv. RAN. Ser. Mat., 2017, Volume 81, Issue 3, Pages 45–82 (Mi izv8444)  

This article is cited in 7 scientific papers (total in 7 papers)

Vlasov-type and Liouville-type equations, their microscopic, energetic and hydrodynamical consequences

V. V. Vedenyapinab, M. A. Negmatovc, N. N. Fimina

a Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow
b Peoples Friendship University of Russia, Moscow
c The Central Research Institute of Machinery

Abstract: We give a derivation of the Vlasov–Maxwell and Vlasov–Poisson–Poisson equations from the Lagrangians of classical electrodynamics. The equations of electromagnetic hydrodynamics (EMHD) and electrostatics with gravitation are derived from them by means of a ‘hydrodynamical’ substitution. We obtain and compare the Lagrange identities for various types of Vlasov equations and EMHD equations. We discuss the advantages of writing the EMHD equations in Godunov's double divergence form. We analyze stationary solutions of the Vlasov–Poisson–Poisson equation, which give rise to non-linear elliptic equations with various properties and various kinds of behaviour of the trajectories of particles as the mass passes through a critical value. We show that the classical equations can be derived from the Liouville equation by the Hamilton–Jacobi method and give an analogue of this procedure for the Vlasov equation as well as in the non-Hamiltonian case.

Keywords: Liouville equation, Hamilton–Jacobi method, hydrodynamical substitution, Vlasov–Maxwell equation, Vlasov–Poisson–Poisson equation, Lagrange identity.

Funding Agency Grant Number
Russian Foundation for Basic Research 16-02-00656
Russian Academy of Sciences - Federal Agency for Scientific Organizations 7
1.3.1
Ministry of Education and Science of the Russian Federation 5-100
This paper was written with the support of RFBR grant no. 16-02-00656 and RAS Presidium Programme no. 7 (N. N. Fimin) and with the financial support of the Ministry of Education and Science of the Russian Federation under the programme ‘5-100’ of raising the competitive ability of PFUR among leading scientific and educational centres in 2016–2020, as well as with support of the RAS DMS programme 1.3.1 for problems of computational mathematical physics (V. V. Vedenyapin).

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/im8444

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English version:
Izvestiya: Mathematics, 2017, 81:3, 505–541

Bibliographic databases:

UDC: 517.9
PACS: 02.30.Jr
MSC: 35Q83
Received: 17.09.2015

Citation: V. V. Vedenyapin, M. A. Negmatov, N. N. Fimin, “Vlasov-type and Liouville-type equations, their microscopic, energetic and hydrodynamical consequences”, Izv. RAN. Ser. Mat., 81:3 (2017), 45–82; Izv. Math., 81:3 (2017), 505–541

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. V. Vedenyapin, T. S. Kazakova, Ya. K. V., B. N. Chetverushkin, “Schrödinger equation as a self-consistent field”, Dokl. Math., 97:3 (2018), 240–242  mathnet  crossref  crossref  zmath  isi  elib  scopus
    2. V. V. Vedenyapin, A. A. Andreeva, V. V. Vorobyeva, “Euler and Navier–Stokes equations as self-consistent fields”, Dokl. Math., 97:3 (2018), 283–285  mathnet  crossref  crossref  zmath  isi  elib  scopus
    3. V. V. Vedenyapin, S. Z. Adzhiev, V. V. Kazantseva, “Entropiya po Boltsmanu i Puankare, ekstremali Boltsmana i metod Gamiltona–Yakobi v negamiltonovoi situatsii”, Differentsialnye i funktsionalno-differentsialnye uravneniya, SMFN, 64, no. 1, Rossiiskii universitet druzhby narodov, M., 2018, 37–59  mathnet  crossref
    4. V. V. Vedenyapin, “Uravnenie Vlasova–Maksvella–Einshteina”, Preprinty IPM im. M. V. Keldysha, 2018, 188, 20 pp.  mathnet  crossref
    5. V. V. Vedenyapin, N. N. Fimin, V. M. Chechetkin, “Ob uravnenii Vlasova–Maksvella–Einshteina i ego nerelyativistskikh i slaborelyativistskikh analogakh”, Preprinty IPM im. M. V. Keldysha, 2018, 265, 30 pp.  mathnet  crossref
    6. V. V. Vedenyapin, N. I. Karavaeva, O. A. Kostyuk, B. N. Chetverushkin, “Uravnenie Shredingera kak sledstvie novykh uravnenii tipa Vlasova”, Preprinty IPM im. M. V. Keldysha, 2019, 026, 11 pp.  mathnet  crossref
    7. Vedenyapin V.V., Salnikova T.V., Stepanov S.Ya., “Vlasov-Poisson-Poisson Equations, Critical Mass, and Kordylewski Clouds”, Dokl. Math., 99:2 (2019), 221–224  crossref  isi
  • Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya Izvestiya: Mathematics
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