On group symmetries of the hydrodynamic equations for rarefied gas

Alexander V. Bobylev; Sergey V. Meleshko

doi:10.3934/krm.2021012

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2021, Volume 14, Issue 3: 469-482. Doi: 10.3934/krm.2021012

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On group symmetries of the hydrodynamic equations for rarefied gas

Alexander V. Bobylev^1, , and
Sergey V. Meleshko^2,

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
^* Corresponding author: Alexander V. Bobylev

Received: September 30, 2020

Revised: December 31, 2020

Early access: February 2021

Published: June 2021

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Abstract

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.

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Mathematics Subject Classification: Primary: 35Q35, 76P05; Secondary: 76M60.

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