Abstract
We consider the one-dimensional Boltzmann equation \(f_t+cf_x+(\mathcal{F}f)_c=0\) with a function \(\mathcal{F}\) depending on (t,x,c,f) and obtain the complete group classification of such equations in the class of point changes of whole set of variables (t,x,c,f). for this, we impose additional conditions on the transformations for the invariance of (a) the relations dx = c dt and \(dc=\mathcal{F}dt\), (b) the lines dt = dx = 0, and (c) the form f dx dc, which fix the physical meaning of the used variables and the relations between them.
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References
K. S. Platonova, “Group analysis of the one-dimensional Boltzmann equation I: Symmetry groups,” Differ. Equ., 53, 530–538 (2017).
K. S. Platonova, “Group analysis of the one-dimensional Boltzmann equation: II. Equivalence groups and symmetry groups in the special case,” Differ. Equ., 53, 796–808 (2017).
K. S. Platonova and A. V. Borovskikh, “Group analysis of the one-dimensional Boltzmann equation: III. Condition for the moment quantities to be physically meaningful,” Theor. Math. Phys., 195, 886–915 (2018).
J. Maxwell, “On the dynamical theory of gases,” in: The Scientific Papers of James Clerk Maxwell (W. Niven, ed.), Cambridge Univ. Press, Cambridge (2011), pp. 26–78.
I. Müller and T. Ruggeri, Extended Thermodynamics (Springer Tracts Nat. Philos., Vol. 37), Springer, New York (1998).
S. Chapmen and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge Univ. Press, Cambridge (1970).
V. V. Vedenyapin, Boltzmann and Vlasov Kinetic Equations [in Russian], Fizmatlit, Moscow (2001).
C. Cercignani, The Boltzmann Equation and Its Applications (Appl. Math. Sci., Vol. 67), Springer, New York (1988).
L. V. Ovsiannikov, Group Analysis of Differential Equations [in Russian], Nauka, Moscow (1978); English transl., Acad. Press, New York (1982).
P. J. Olver, Applications of Lie groups to Differential Equations, Springer, New York (1986).
N. H. Ibragimov, ed., CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 1, Symmetries, Exact Solutions, and Conservation Laws, CRC, Boca Raton, Fla. (1994).
A. Bobylev and V. Dorodnitsyn, “Symmetries of evolution equations with non-local operators and applications to the Boltzmann equation,” Discrete Contin. Dyn. Syst., 24, 35–57 (2009).
Yu. N. Grigoriev, N. H. Ibragimov, V. F. Kovalev, and S. V. Meleshko, Symmetries of Integro-Differential Equations: With Applications in Mechanics and Plasma Physics (Lect. Notes Phys., Vol. 806), Springer, Dordrecht (2010).
S. Lie, Symmetries of Differential Equations [in Russian], Vol. 2, Lectures on Continuous Groups with Geometric and Other Applications, RKhD, Moscow (2011).
A. Gonzalez-Lopez, N. Kamran, and P. J. Olver, “Lie algebras of vector fields in the real plane,” Proc. London Math. Soc., 64, 339–368 (1992).
A. V. Borovskikh and K. S. Platonova, “Group analysis of the one-dimensional Boltzmann equation,” arXiv: 1905.08873v1 [math.AP] (2019).
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 201, No. 2, pp. 232–265, November, 2019.
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Borovskikh, A.V., Platonova, K.S. Group Analysis of the One-Dimensional Boltzmann Equation: IV. Complete Group Classification in the General Case. Theor Math Phys 201, 1614–1643 (2019). https://doi.org/10.1134/S0040577919110072
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DOI: https://doi.org/10.1134/S0040577919110072