Geodesic
Group analysis of the one-dimensional Boltzmann equation: III. Condition for the moment quantities to be physically meaningful
Teoretičeskaâ i matematičeskaâ fizika, Tome 195 (2018) no. 3, pp. 451-482 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We present the group classification of the one-dimensional Boltzmann equation with respect to the function $\mathcal F=\mathcal{F}(t,x,c)$ characterizing an external force field under the assumption that the physically meaningful constraints $dx=c\,dt$, $dc=\mathcal{F}\,dt$, $dt=0$, and $dx=0$ are imposed on the variables. We show that for all functions $\mathcal{F}$, the algebra is finite-dimensional, and its maximum dimension is eight, which corresponds to the equation with a zero $\mathcal{F}$.
Keywords: Boltzmann equation, symmetry group, gas dynamics equation
Mots-clés : equivalence group. 
@article{TMF_2018_195_3_a6,
     author = {K. S. Platonova and A. V. Borovskikh},
     title = {Group analysis of the~one-dimensional {Boltzmann} equation: {III.~Condition} for the~moment quantities to be physically meaningful},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {451--482},
     year = {2018},
     volume = {195},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2018_195_3_a6/}
}
TY  - JOUR
AU  - K. S. Platonova
AU  - A. V. Borovskikh
TI  - Group analysis of the one-dimensional Boltzmann equation: III. Condition for the moment quantities to be physically meaningful
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2018
SP  - 451
EP  - 482
VL  - 195
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2018_195_3_a6/
LA  - ru
ID  - TMF_2018_195_3_a6
ER  - 
%0 Journal Article
%A K. S. Platonova
%A A. V. Borovskikh
%T Group analysis of the one-dimensional Boltzmann equation: III. Condition for the moment quantities to be physically meaningful
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2018
%P 451-482
%V 195
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2018_195_3_a6/
%G ru
%F TMF_2018_195_3_a6
K. S. Platonova; A. V. Borovskikh. Group analysis of the one-dimensional Boltzmann equation: III. Condition for the moment quantities to be physically meaningful. Teoretičeskaâ i matematičeskaâ fizika, Tome 195 (2018) no. 3, pp. 451-482. http://geodesic.mathdoc.fr/item/TMF_2018_195_3_a6/

[1] I. Müller, T. Ruggeri, Extended Thermodynamics, Springer Tracts in Natural Philosophy, 37, Springer, New York, 1998 | DOI | MR

[2] L. V. Ovsyannikov, “Gruppovye svoistva uravneniya Chaplygina”, Prikladnaya mekhanika i tekhnicheskaya fizika, 1:3 (1960), 126–145

[3] K. S. Platonova, “Gruppovoi analiz uravneniya Boltsmana. I. Gruppy simmetrii”, Differents. uravneniya, 53:4 (2017), 538–546 | DOI | DOI

[4] Yu. N. Grigorev, S. V. Meleshko, “Polnaya gruppa Li i invariantnye resheniya sistemy uravnenii Boltsmana mnogokomponentnoi smesi gazov”, Sib. matem. zhurn., 38:3 (1997), 510–525 | DOI | MR | Zbl

[5] K. S. Platonova, “Gruppovoi analiz uravneniya Boltsmana. II. Gruppy ekvivalentnosti i gruppy simmetrii v spetsialnom sluchae”, Differents. uravneniya, 53:6 (2017), 801–813 | DOI | DOI

[6] S. Li, Simmetrii differentsialnykh uravnenii, v. 2, Lektsii o nepreryvnykh gruppakh s geometricheskimi i drugimi prilozheniyami, RKhD, M.–Izhevsk, 2011 | Zbl

[7] A. González-López, N. Kamran, P. J. Olver, “Lie algebras of vector fields in the real plane”, Proc. London Math. Soc., s3-64:2 (1992), 339–368 | DOI | MR