Abstract
We consider a system of kinetic equations with one-dimensional velocity space. The system is a simple mathematical model that describes the evolution of a two-component gas mixture at the molecular level. We study some qualitative properties of its solutions, in particular, the conservation laws and spectrum of the linearized problem. In the spatially homogeneous case we present the widest Lie algebra of admissible operators and construct some exact solutions in closed form. We indicate some methods for constructing numerical schemes conservative with respect to fulfillment of the discrete conservation laws of energy and the concentrations of the components.
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References
Ferziger J. H. and Kaper H. G., Mathematical Theory of Transport Processes in Gases [Russian translation], Mir, Moscow (1976).
Osipov A. I. and Uvarov A. V., “Kinetic and gasdynamic processes in nonequilibrium molecular physics, ” Uspekhi Fiz. Nauk, 162, No.11, 1–42 (1992).
Grigor'ev Yu. N. and Mikhalitsyn A. N., “The spectral method for numerical solution of the kinetic Boltzmann equation,” Zh. Vychisl. Mat. i Mat. Fiz., 23, No.6, 1454–1463 (1983).
Kac M., “Foundation of kinetic theory,” Proc. 3rd Berkeley Sympos. Math. Stat. and Prob., Berkeley: Univ. Calif. Press, 1956, 8, pp. 171–197.
Grigor'ev Yu. N., “A class of exact solutions of a nonlinear kinetic equation,” Dinamika Sploshn. Sredy, No. 26, 30–43 (1976).
Cornille H., Gervois A., and Protopopesku V., “Closed similarity solutions for a class of stationary nonlinear Boltzmann-like equations,” J. Phys. A., 16, L343–L350 (1984).
Cornille H., “Closed solution for the spatially homogeneous Kac's model of the nonlinear Boltzmann equation,” J. Phys. A, 17, L235–L242 (1984).
Cornille H., “Solution for the spatially inhomogeneous nonlinear Kac model of the Boltzmann equation, ” J. Math. Phys., 26, No.6, 1203–1214 (1985).
Cornille H., “Oscillating Maxwellians,” J. Phys. A., 18, L839–L844 (1985).
Ernst M. H., “Nonlinear model-Boltzmann equations and exact solutions,” Phys. Rep., 78, No.1, 1–171 (1981).
Roshal' A. S., “The fast Fourier transform in computational physics,” Izv. Vyssh. Uchebn. Zaved. Radiofiz., 19, No.10, 1425–1454 (1976).
Korn G. and Korn T., Handbook of Mathematics [Russian translation], Nauka, Moscow (1973).
Grigor'ev Yu. N. and Meleshko S. V., Study of Invariant Solutions of the Nonlinear Kinetic Boltzmann Equation and Their Models [Preprint, No. 18–86] [in Russian], Inst. Teoret. i Prikl. Mat., Novosibirsk (1986).
Grigor'ev Yu. N. and Meleshko S. V., “The full Lie group and invariant solutions of the system of Boltzmann equations of a multicomponent gas mixture,” Siberian Math. J., 38, No.3, 434–448 (1997).
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Original Russian Text Copyright © 2005 Grigor'ev Yu. N. and Omel'yanchuk M. I.
The authors were supported by the Russian Foundation for Basic Research (Grant 05-01-00359).
In memory of Tadei Ivanovich Zelenyak.
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Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 5, pp. 1021–1035, September– October, 2005.
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Grigor'ev, Y.N., Omel'yanchuk, M.I. Qualitative Properties of a Certain Kinetic Model of a Binary Gas. Sib Math J 46, 813–825 (2005). https://doi.org/10.1007/s11202-005-0080-4
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DOI: https://doi.org/10.1007/s11202-005-0080-4