Group theoretic approach and analytical solutions of the compressible Navier–Stokes equations

Dina Razafindralandy; Aziz Hamdouni

doi:10.3934/dcdss.2024019

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2024, Volume 17, Issue 7: 2450-2477. Doi: 10.3934/dcdss.2024019

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Group theoretic approach and analytical solutions of the compressible Navier–Stokes equations

Dina Razafindralandy^1, , and
Aziz Hamdouni^1,

1.
LaSIE, UMR CNRS 7356 - La Rochelle Université, France

^*Corresponding author: Dina Razafindralandy
^*Corresponding author: Dina Razafindralandy

Received: December 2023

Revised: February 2024

Early access: February 27, 2024

Published online: February 27, 2024

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Abstract

A group theoretic analysis of the compressible Navier-Stokes equations of an ideal gas is carried out. The 12-dimensional Lie symmetry group is computed. The commutation table and the Levi decomposition of its Lie algebra are presented. The equations are reduced and self-similar one-, two- and three-dimensional solutions are computed.

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Mathematics Subject Classification: Primary: 76N06, 22E70, 58D19.

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Table 1. Commutation table of $ -athfrak{g} $

	$X_1$	$X_2$	$X_3$	$X_4$	$X_5$	$X_6$	$X_7$	$X_8$	$X_9$	$X_{10}$	$X_{11}$	$X_{12}$
$X_1$	0	0	0	0	$X_2$	$X_3$	$X_4$	0	0	0	$2X_1$	0
$X_2$	0	0	0	0	0	0	0	0	$-X_4$	$X_3$	$X_2$	$X_2$
$X_3$	0	0	0	0	0	0	0	$X_4$	0	$-X_2$	$X_3$	$X_3$
$X_4$	0	0	0	0	0	0	0	$-X_3$	$X_2$	0	$X_4$	$X_4$
$X_5$	$-X_2$	0	0	0	0	0	0	0	$-X_7$	$X_6$	$-X_5$	$X_5$
$X_6$	$-X_3$	0	0	0	0	0	0	$X_7$	0	$-X_5$	$-X_6$	$X_6$
$X_7$	$-X_4$	0	0	0	0	0	0	$-X_6$	$X_5$	0	$-X_7$	$X_7$
$X_8$	0	0	$-X_4$	$X_3$	0	$-X_7$	$X_6$	0	$-X_{10}$	$X_9$	0	0
$X_9$	0	$X_4$	0	$-X_2$	$X_7$	0	$-X_5$	$X_{10}$	0	$-X_8$	0	0
$X_{10}$	0	$-X_3$	$X_2$	0	$-X_6$	$X_5$	0	$-X_9$	$X_8$	0	0	0
$X_{11}$	$-2 X_1$	$-X_2$	$-X_3$	$-X_4$	$X_5$	$X_6$	$X_7$	0	0	0	0	0
$X_{12}$	0	$-X_2$	$-X_3$	$-X_4$	$-X_5$	$-X_6$	$-X_7$	0	0	0	0	0

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