Abstract
The chapter deals with applications of the group analysis method to the full Boltzmann kinetic equation and some similar equations. These equations form the foundation of the kinetic theory of rarefied gas and coagulation. They typically include special integral operators with quadratic nonlinearity and multiple kernels which are called collision integrals.
Calculations of the 11-parameter Lie group G 11 admitted by the full Boltzmann equation with arbitrary intermolecular potential and its extensions for power potentials are presented. The found isomorphism of these Lie groups with the Lie groups admitted by the ideal gas dynamics equations allowed one to obtain an optimal system of admitted subalgebras and to classify all invariant solutions of the full Boltzmann equation. For equations similar to the full Boltzmann equation complete admitted Lie groups are derived by solving determining equations. The corresponding optimal systems of admitted subalgebras are constructed and representations of all invariant solutions are obtained.
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Notes
- 1.
See the review [15].
- 2.
The text follows [17], see also Chap. 2.
- 3.
- 4.
See also Chap. 1.
- 5.
See Appendix A.
- 6.
The part of the optimal system of six and seven dimensional subalgebras of [36] is presented in Appendix A.
- 7.
The operator X is considered as the equivalent canonical Lie–Bäcklund operator.
- 8.
The previous results [29] obtained in this direction were far from being complete.
- 9.
The spectral properties of the linearized system corresponding to (3.4.18) are substantially used in the proof.
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Grigoriev, Y.N., Ibragimov, N.H., Kovalev, V.F., Meleshko, S.V. (2010). The Boltzmann Kinetic Equation and Various Models. In: Symmetries of Integro-Differential Equations. Lecture Notes in Physics, vol 806. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3797-8_3
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