Abstract:
We consider problems for the nonlinear Boltzmann equation in the framework of two models{:} a new nonlinear model and the Bhatnagar–Gross–Krook model. The corresponding transformations reduce these problems to nonlinear systems of integral equations. In the framework of the new nonlinear model, we prove the existence of a positive bounded solution of the nonlinear system of integral equations and present examples of functions describing the nonlinearity in this model. The obtained form of the Boltzmann equation in the framework of the Bhatnagar–Gross–Krook model allows analyzing the problem and indicates a method for solving it. We show that there is a qualitative difference between the solutions in the linear and nonlinear cases{\rm:} the temperature is a bounded function in the nonlinear case, while it increases linearly at infinity in the linear approximation. We establish that in the framework of the new nonlinear model, equations describing the distributions of temperature, concentration, and mean-mass velocity are mutually consistent, which cannot be asserted in the case of the Bhatnagar–Gross–Krook model.
Keywords:
model Boltzmann equation, nonlinearity, monotonicity, bounded solution,
temperature jump, system of integral equations.
Citation:
A. Kh. Khachatryan, Kh. A. Khachatryan, “Qualitative difference between solutions of stationary model Boltzmann equations in the linear and nonlinear cases”, TMF, 180:2 (2014), 272–288; Theoret. and Math. Phys., 180:2 (2014), 990–1004
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\paper Qualitative difference between solutions of stationary model Boltzmann equations in the~linear and nonlinear cases
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\pages 272--288
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\jour Theoret. and Math. Phys.
\yr 2014
\vol 180
\issue 2
\pages 990--1004
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Linking options:
https://www.mathnet.ru/eng/tmf8623
https://doi.org/10.4213/tmf8623
https://www.mathnet.ru/eng/tmf/v180/i2/p272
This publication is cited in the following 10 articles:
Kh. A. Khachatryan, A. R. Hakobyan, “On nontrivial solvability of one class of nonlinear integral equations with conservative kernel on the positive semi-axis”, Uch. zapiski EGU, ser. Fizika i Matematika, 56:1 (2022), 7–18
Ch.-j. Liu, S. Pang, Q. Xu, L. He, Sh.-p. Yang, Yu.-j. Qing, “The study of the Boltzmann equation of solid-gas two-phase flow with three-dimensional BGK model”, International Conference on Civil, Mechanical and Material Engineering (Iccmme 2018), AIP Conf. Proc., 1973, eds. J. Jung, D. Kim, Amer. Inst. Phys., 2018, UNSP 020004-1
Kh. A. Khachatryan, Ts. É. Terdzhyan, T. G. Sardanyan, “On the Solvability of One System of Nonlinear Hammerstein-Type Integral Equations on the Semiaxis”, Ukr Math J, 69:8 (2018), 1287
Kh. A. Khachatryan, T. G. Sardaryan, “O razreshimosti odnogo klassa nelineinykh integralnykh uravnenii tipa Urysona na vsei pryamoi”, Izv. Sarat. un-ta. Nov. ser. Ser.: Matematika. Mekhanika. Informatika, 17:1 (2017), 40–50
A. Kh. Khachatryan, Kh. A. Khachatryan, “A one-parameter family of positive solutions of the non-linear stationary Boltzmann equation (in the framework of a modified model)”, Russian Math. Surveys, 72:3 (2017), 571–573
Kh. A. Khachatryan, A. S. Petrosyan, A. A. Sisakyan, “O netrivialnoi razreshimosti odnogo klassa nelineinykh integralnykh uravnenii tipa Urysona”, Tr. IMM UrO RAN, 23, no. 2, 2017, 266–273
Kh. A. Khachatryan, “On the solvability of one class of two-dimensional Urysohn integral equations”, Siberian Adv. Math., 28:3 (2018), 166–174
H. H. Azizyan, Kh. A. Khachatryan, “One-parametric family of positive solutions for a class of nonlinear discrete Hammerstein–Volterra equations”, Ufa Math. J., 8:1 (2016), 13–19
A. Kh. Khachatryan, Kh. A. Khachatryan, “Some problems concerning the solvability of the nonlinear stationary Boltzmann equation in the framework of the BGK model”, Trans. Moscow Math. Soc., 77 (2016), 87–106
K. A. Khachatryan, T. E. Terdzhyan, “On the solvability of one class of nonlinear integral equations in $L_1(0,+\infty)$”, Siberian Adv. Math., 25:4 (2015), 268–275
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