# American Institute of Mathematical Sciences

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 & 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.  Google Scholar [2] M. Bisi, M. 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.  Google Scholar [3] A. V. Bobylev, Generalized Burnett hydrodynamics, J. Stat. Phys., 132 (2008), 569-580.  doi: 10.1007/s10955-008-9556-5.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] A. V. Bobylev, On the Chapman-Enskog and Grad methods for solving the Boltzmann equation, Dokl. Akad. Nauk SSSR, 262 (1982), 71-75.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [12] V. Dorodnitsyn, Applications of Lie Groups to Difference Equations, Differential and Integral Equations and Their Applications, 8, CRC Press, Boca Raton, FL, 2011.  Google Scholar [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.  Google Scholar [14] A. C. Hearn, REDUCE Users Manual, Ver. 3.3, The Rand Corporation CP 78, Santa Monica, 1987. Google Scholar [15] N. H. Ibragimov (ed.), CRC Handbook of Lie Group Analysis of Differential Equations, vol. 3, CRC Press, Boca Raton, 1996. Google Scholar [16] M. N. Kogan, Rarefied Gas Dynamics, Springer, Boston, MA, 1969. doi: 10.1007/978-1-4899-6381-9.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [2] M. Bisi, M. 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.  Google Scholar [3] A. V. Bobylev, Generalized Burnett hydrodynamics, J. Stat. Phys., 132 (2008), 569-580.  doi: 10.1007/s10955-008-9556-5.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] A. V. Bobylev, On the Chapman-Enskog and Grad methods for solving the Boltzmann equation, Dokl. Akad. Nauk SSSR, 262 (1982), 71-75.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [12] V. Dorodnitsyn, Applications of Lie Groups to Difference Equations, Differential and Integral Equations and Their Applications, 8, CRC Press, Boca Raton, FL, 2011.  Google Scholar [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.  Google Scholar [14] A. C. Hearn, REDUCE Users Manual, Ver. 3.3, The Rand Corporation CP 78, Santa Monica, 1987. Google Scholar [15] N. H. Ibragimov (ed.), CRC Handbook of Lie Group Analysis of Differential Equations, vol. 3, CRC Press, Boca Raton, 1996. Google Scholar [16] M. N. Kogan, Rarefied Gas Dynamics, Springer, Boston, MA, 1969. doi: 10.1007/978-1-4899-6381-9.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
 [1] Yi Zhou, Zhen Lei. Logarithmically improved criteria for Euler and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2715-2719. doi: 10.3934/cpaa.2013.12.2715 [2] Michele Coti Zelati. Remarks on the approximation of the Navier-Stokes equations via the implicit Euler scheme. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2829-2838. doi: 10.3934/cpaa.2013.12.2829 [3] Carlo Morosi, Livio Pizzocchero. On the constants in a Kato inequality for the Euler and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 557-586. doi: 10.3934/cpaa.2012.11.557 [4] Jishan Fan, Yasuhide Fukumoto, Yong Zhou. Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations. Kinetic & Related Models, 2013, 6 (3) : 545-556. doi: 10.3934/krm.2013.6.545 [5] Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602 [6] Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149 [7] Hamid Bellout, Jiří Neustupa, Patrick Penel. On a $\nu$-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-Type boundary condition. Discrete & Continuous Dynamical Systems, 2010, 27 (4) : 1353-1373. doi: 10.3934/dcds.2010.27.1353 [8] Tobias Breiten, Karl Kunisch. Feedback stabilization of the three-dimensional Navier-Stokes equations using generalized Lyapunov equations. Discrete & Continuous Dynamical Systems, 2020, 40 (7) : 4197-4229. doi: 10.3934/dcds.2020178 [9] Dongho Chae, Kyungkeun Kang, Jihoon Lee. Notes on the asymptotically self-similar singularities in the Euler and the Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1181-1193. doi: 10.3934/dcds.2009.25.1181 [10] Jian Su, Yinnian He. The almost unconditional convergence of the Euler implicit/explicit scheme for the three dimensional nonstationary Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3421-3438. doi: 10.3934/dcdsb.2017173 [11] Joachim Escher, Rossen Ivanov, Boris Kolev. Euler equations on a semi-direct product of the diffeomorphisms group by itself. Journal of Geometric Mechanics, 2011, 3 (3) : 313-322. doi: 10.3934/jgm.2011.3.313 [12] Hermenegildo Borges de Oliveira. Anisotropically diffused and damped Navier-Stokes equations. Conference Publications, 2015, 2015 (special) : 349-358. doi: 10.3934/proc.2015.0349 [13] Hyukjin Kwean. Kwak transformation and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 433-446. doi: 10.3934/cpaa.2004.3.433 [14] Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747 [15] C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403 [16] Igor Kukavica. On regularity for the Navier-Stokes equations in Morrey spaces. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1319-1328. doi: 10.3934/dcds.2010.26.1319 [17] Igor Kukavica. On partial regularity for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 717-728. doi: 10.3934/dcds.2008.21.717 [18] Susan Friedlander, Nataša Pavlović. Remarks concerning modified Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 269-288. doi: 10.3934/dcds.2004.10.269 [19] Marzia Bisi, Maria Paola Cassinari, Maria Groppi. Qualitative analysis of the generalized Burnett equations and applications to half--space problems. Kinetic & Related Models, 2008, 1 (2) : 295-312. doi: 10.3934/krm.2008.1.295 [20] I. Moise, Roger Temam. Renormalization group method: Application to Navier-Stokes equation. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 191-210. doi: 10.3934/dcds.2000.6.191

2020 Impact Factor: 1.432