June  2021, 14(3): 469-482. doi: 10.3934/krm.2021012

On group symmetries of the hydrodynamic equations for rarefied gas

1. 

Keldysh Institute of Applied Mathematics, Russian Academy of Science, Miusskaya Pl. 4, Moscow, 125047, Russia

2. 

School of Mathematics, Institute of Science, Suranaree University of Technology, Nakhon Ratchasima, 30000, Thailand

* Corresponding author: Alexander V. Bobylev

Received  October 2020 Revised  January 2021 Published  June 2021 Early access  February 2021

The invariant group transformations of three-dimensional hydrodynamic equations derived from the Boltzmann equation are studied. Three levels (with respect to the Knudsen number) of hydrodynamic description are considered and compared: (a) Euler equations, (b) Navier-Stokes equations, (c) Generalized Burnett equations (GBEs), which replace the original (ill-posed) Burnett equations. The main attention is paid to group analysis of GBEs in their most general formulation because this and related questions have not been studied before in the literature. The results of group analysis of GBEs and, for comparison, of similar results for Euler and Navier-Stokes equations are presented in two theorems and discussed in detail. It is remarkable that the use of computer code greatly simplifies the proof of the results for GBEs, which are very cumbersome equations with many undetermined parameters.

Citation: Alexander V. Bobylev, Sergey V. Meleshko. On group symmetries of the hydrodynamic equations for rarefied gas. Kinetic and Related Models, 2021, 14 (3) : 469-482. doi: 10.3934/krm.2021012
References:
[1]

V. I. Arnol'd, Geometrical Methods in the Theory of Ordinary Differential Equations, Fundamental Principles of Mathematical Science, 250, Springer-Verlag, New York-Berlin, 1983.

[2]

M. BisiM. P. Cassinari and M. Groppi, Qualitative analysis of the generalized Burnett equations and applications to half-space problems, Kinet. Relat. Models, 1 (2008), 295-312.  doi: 10.3934/krm.2008.1.295.

[3]

A. V. Bobylev, Generalized Burnett hydrodynamics, J. Stat. Phys., 132 (2008), 569-580.  doi: 10.1007/s10955-008-9556-5.

[4]

A. V. Bobylev, Instabilities in the Chapman-Enskog expansion and hyperbolic Burnett equations, J. Stat. Phys., 124 (2006), 371-399.  doi: 10.1007/s10955-005-8087-6.

[5]

A. V. Bobylev, Kinetic Equations. Boltzmann Equation, Maxwell Models, and Hydrodynamics Beyond Navier–Stokes, De Gruyter Series in Applied and Numerical Mathematics, 1, De Gruyter, Berlin/Boston, 2020. doi: 10.1515/9783110550986.

[6]

A. V. Bobylev, On the Chapman-Enskog and Grad methods for solving the Boltzmann equation, Dokl. Akad. Nauk SSSR, 262 (1982), 71-75. 

[7]

A. V. Bobylev and N. K. Ibragimov, Relationships between the symmetry properties of the equations of gas kinetics and hydrodynamics, Math. Modeling Comput. Experiment, 1 (1993), 291-300.

[8]

A. V. Bobylev and S. V. Meleshko, Group analysis of the generalized Burnett equations, J. Nonlinear Math. Phys., 27 (2020), 494-508.  doi: 10.1080/14029251.2020.1757238.

[9]

A. V. Bobylev and Å. Windfäll, Boltzmann equation and hydrodynamics at the Burnett level, Kinet. Relat. Models, 5 (2012), 237-260.  doi: 10.3934/krm.2012.5.237.

[10]

S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction, and Diffusion in Gases, Cambridge University Press, New York 1960.

[11]

A. A. Cherevko, Optimal system of subalgebras of the Lie algebra of generators admitted by the system of the gas dynamics equations of a polytropic gas with the state equation $p = f(s)\rho^{5/3}$, preprint of Lavrent'ev Institute of Hydrodynamics, Novosibirsk, 1996, 4–96.

[12]

V. Dorodnitsyn, Applications of Lie Groups to Difference Equations, Differential and Integral Equations and Their Applications, 8, CRC Press, Boca Raton, FL, 2011.

[13]

Y. N. Grigoriev, N. H. Ibragimov, V. F. Kovalev and S. V. Meleshko, Symmetries of Integro-Differential Equations, Lecture Notes in Physics, 806, Springer, Dordrecht, 2010. doi: 10.1007/978-90-481-3797-8.

[14]

A. C. Hearn, REDUCE Users Manual, Ver. 3.3, The Rand Corporation CP 78, Santa Monica, 1987.

[15]

N. H. Ibragimov (ed.), CRC Handbook of Lie Group Analysis of Differential Equations, vol. 3, CRC Press, Boca Raton, 1996.

[16]

M. N. Kogan, Rarefied Gas Dynamics, Springer, Boston, MA, 1969. doi: 10.1007/978-1-4899-6381-9.

[17]

S. V. Meleshko, Methods for Constructing Exact Solutions of Partial Differential Equations, Mathematical and Analytical Techniques with Applications to Engineering, Springer, New York, 2005. doi: 10.1007/b107051.

[18]

P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, 107, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4684-0274-2.

[19]

L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, Inc., Harcourt Brace Jovanovich, Publishers, New York-London, 1982. doi: 10.1016/C2013-0-07470-1.

[20]

L. V. Ovsiannikov, The program 'Submodels'. Gas dynamics, J. Appl. Math. Mech., 58 (1994), 601-627.  doi: 10.1016/0021-8928(94)90137-6.

show all references

References:
[1]

V. I. Arnol'd, Geometrical Methods in the Theory of Ordinary Differential Equations, Fundamental Principles of Mathematical Science, 250, Springer-Verlag, New York-Berlin, 1983.

[2]

M. BisiM. P. Cassinari and M. Groppi, Qualitative analysis of the generalized Burnett equations and applications to half-space problems, Kinet. Relat. Models, 1 (2008), 295-312.  doi: 10.3934/krm.2008.1.295.

[3]

A. V. Bobylev, Generalized Burnett hydrodynamics, J. Stat. Phys., 132 (2008), 569-580.  doi: 10.1007/s10955-008-9556-5.

[4]

A. V. Bobylev, Instabilities in the Chapman-Enskog expansion and hyperbolic Burnett equations, J. Stat. Phys., 124 (2006), 371-399.  doi: 10.1007/s10955-005-8087-6.

[5]

A. V. Bobylev, Kinetic Equations. Boltzmann Equation, Maxwell Models, and Hydrodynamics Beyond Navier–Stokes, De Gruyter Series in Applied and Numerical Mathematics, 1, De Gruyter, Berlin/Boston, 2020. doi: 10.1515/9783110550986.

[6]

A. V. Bobylev, On the Chapman-Enskog and Grad methods for solving the Boltzmann equation, Dokl. Akad. Nauk SSSR, 262 (1982), 71-75. 

[7]

A. V. Bobylev and N. K. Ibragimov, Relationships between the symmetry properties of the equations of gas kinetics and hydrodynamics, Math. Modeling Comput. Experiment, 1 (1993), 291-300.

[8]

A. V. Bobylev and S. V. Meleshko, Group analysis of the generalized Burnett equations, J. Nonlinear Math. Phys., 27 (2020), 494-508.  doi: 10.1080/14029251.2020.1757238.

[9]

A. V. Bobylev and Å. Windfäll, Boltzmann equation and hydrodynamics at the Burnett level, Kinet. Relat. Models, 5 (2012), 237-260.  doi: 10.3934/krm.2012.5.237.

[10]

S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction, and Diffusion in Gases, Cambridge University Press, New York 1960.

[11]

A. A. Cherevko, Optimal system of subalgebras of the Lie algebra of generators admitted by the system of the gas dynamics equations of a polytropic gas with the state equation $p = f(s)\rho^{5/3}$, preprint of Lavrent'ev Institute of Hydrodynamics, Novosibirsk, 1996, 4–96.

[12]

V. Dorodnitsyn, Applications of Lie Groups to Difference Equations, Differential and Integral Equations and Their Applications, 8, CRC Press, Boca Raton, FL, 2011.

[13]

Y. N. Grigoriev, N. H. Ibragimov, V. F. Kovalev and S. V. Meleshko, Symmetries of Integro-Differential Equations, Lecture Notes in Physics, 806, Springer, Dordrecht, 2010. doi: 10.1007/978-90-481-3797-8.

[14]

A. C. Hearn, REDUCE Users Manual, Ver. 3.3, The Rand Corporation CP 78, Santa Monica, 1987.

[15]

N. H. Ibragimov (ed.), CRC Handbook of Lie Group Analysis of Differential Equations, vol. 3, CRC Press, Boca Raton, 1996.

[16]

M. N. Kogan, Rarefied Gas Dynamics, Springer, Boston, MA, 1969. doi: 10.1007/978-1-4899-6381-9.

[17]

S. V. Meleshko, Methods for Constructing Exact Solutions of Partial Differential Equations, Mathematical and Analytical Techniques with Applications to Engineering, Springer, New York, 2005. doi: 10.1007/b107051.

[18]

P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, 107, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4684-0274-2.

[19]

L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, Inc., Harcourt Brace Jovanovich, Publishers, New York-London, 1982. doi: 10.1016/C2013-0-07470-1.

[20]

L. V. Ovsiannikov, The program 'Submodels'. Gas dynamics, J. Appl. Math. Mech., 58 (1994), 601-627.  doi: 10.1016/0021-8928(94)90137-6.

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