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Some problems concerning the solvability of the nonlinear stationary Boltzmann equation in the framework of the BGK model


Authors: Agavard Kh. Khachatryan and Khachatur A. Khachatryan
Translated by: V. E. Nazaikinskii
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 77 (2016), vypusk 1.
Journal: Trans. Moscow Math. Soc. 2016, 87-106
MSC (2010): Primary 47H30; Secondary 34K30, 35Q20
DOI: https://doi.org/10.1090/mosc/255
Published electronically: November 28, 2016
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Abstract: In the framework of the BGK (Bhatnagar-Gross-Krook) model, we derive a system of nonlinear integral equations for the macroscopic variables both in a finite plane channel $ \Pi _{r}$ of thickness $ r$ $ (r<+\infty )$ and in the subspace $ \Pi _\infty $ $ (r=+\infty )$ from the nonlinear integro-differential Boltzmann equation. Solvability problems are discussed and solution methods are suggested for these systems of nonlinear integral equations. Theorems on the existence of bounded positive solutions are proved and two-sided estimates of these solutions are obtained for the resulting nonlinear integral equations of the Urysohn type describing the temperature (Theorems 1 and 3). A theorem on the existence of a unique solution in the space $ L_1[0,r]$ is proved for the linear integral equations describing the velocity and density. Integral estimates for the solutions are obtained (see Theorem 2 and the Corollary).

The nonlinear system of integral equations in the subspace obtained for the macroscopic variables in the framework of the nonlinear BGK model of the Boltzmann equation is shown to have no bounded solutions with finite limit at infinity other than a constant solution.

The solution of the linear problem obtained by linearizing the corresponding nonlinear system is proved to be $ O(x)$ as $ x\rightarrow +\infty $ (Theorem 3).


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Additional Information

Agavard Kh. Khachatryan
Affiliation: Armenian National Agrarian University, Yerevan, Armenia
Email: aghavard@hotbox.ru

Khachatur A. Khachatryan
Affiliation: Institute of Mathematics, National Academy of Sciences of Republic Armenia, Yerevan, Armenia
Email: khach82@rambler.ru

DOI: https://doi.org/10.1090/mosc/255
Keywords: Nonlinearity, monotonicity, iteration, symbol of an operator, model Boltzmann equation, Urysohn equation.
Published electronically: November 28, 2016
Additional Notes: This research was supported by the State Committee of Science, Ministry of Education and Science of Armenia, under projects no. SCS 13-1A068 and no. SCS 15T-1A033.
Article copyright: © Copyright 2016 American Mathematical Society