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 TMF, 1984, Volume 60, Number 2, Pages 280–310 (Mi tmf5284)

Exact solutions of the nonlinear Boltzmann equation and the theory of relaxation of a Maxwellian gas

A. V. Bobylev

Abstract: Results obtained in recent years in the theory of the nonlinear Boltzmann equation for Maxwellian molecules are reviewed. The general theory of spatially homogeneous relaxation based on Fourier transformation with respect to the velocity is presented. The behavior of the distribution function $f({\mathbf v},t)$ is studied in the limit $|{\mathbf v}|\rightarrow\infty$ (the formation of the MaxwelIian tails) and $t\rightarrow\infty$ (relaxation rate). An analytic transformation relating the nonlinear and linearized equations is constructed. It is shown that the nonlinear equation has a countable set of invariants, families of particular solutions of special form are constructed, and an analogy with equations of Korteweg–de Vries type is noted.

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English version:
Theoretical and Mathematical Physics, 1984, 60:2, 820–841

Bibliographic databases:

Citation: A. V. Bobylev, “Exact solutions of the nonlinear Boltzmann equation and the theory of relaxation of a Maxwellian gas”, TMF, 60:2 (1984), 280–310; Theoret. and Math. Phys., 60:2 (1984), 820–841

Citation in format AMSBIB
\Bibitem{Bob84} \by A.~V.~Bobylev \paper Exact solutions of the nonlinear Boltzmann equation and the theory of relaxation of a Maxwellian gas \jour TMF \yr 1984 \vol 60 \issue 2 \pages 280--310 \mathnet{http://mi.mathnet.ru/tmf5284} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=762269} \zmath{https://zbmath.org/?q=an:0565.76074} \transl \jour Theoret. and Math. Phys. \yr 1984 \vol 60 \issue 2 \pages 820--841 \crossref{https://doi.org/10.1007/BF01018983} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1984ACL9200011} 

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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. V. V. Vedenyapin, “Differential forms in spaces without a norm. A theorem on the uniqueness of Boltzmann's $H$-function”, Russian Math. Surveys, 43:1 (1988), 193–219
2. A. V. Mishchenko, D. Ya. Petrina, “Linearization and exact solutions of a class of Boltzmann equations”, Theoret. and Math. Phys., 77:1 (1988), 1096–1109
3. A. A. Raines, “Numerical testing of Bird's VHS model”, Comput. Math. Math. Phys., 35:8 (1995), 1025–1031
4. M. S. Ivanov, M. A. Korotchenko, G. A. Mikhailov, S. V. Rogazinskii, “Global weighted Monte Carlo method for the nonlinear Boltzmann equation”, Comput. Math. Math. Phys., 45:10 (2005), 1792–1801
5. M. A. Korotchenko, G. A. Mikhailov, S. V. Rogazinskii, “Modifications of weighted Monte Carlo algorithms for nonlinear kinetic equations”, Comput. Math. Math. Phys., 47:12 (2007), 2023–2033
6. Malkov E.A., Ivanov M.S., “Determinirovannyi metod chastits-v-yacheikakh dlya resheniya zadach dinamiki razrezhennogo gaza. chast i”, Vychislitelnye metody i programmirovanie: novye vychislitelnye tekhnologii, 12:1 (2011), 368–374
7. Evsevleeva L.G., Sverdlova O.L., Kirik M.S., Gozbenko V.E., “Analiticheskaya model vzaimodeistviya atomov kisloroda s poverkhnostyu adsorbenta”, Sovremennye tekhnologii. sistemnyi analiz. modelirovanie, 2012, no. 3, 137–140
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