An infinite Lie group of symmetry of one-dimensional gas flow, for a class of entropy distributions
type: Item
instance of: scholarly article
title: An infinite Lie group of symmetry of one-dimensional gas flow, for a class of entropy distributions
zbMATH Open document ID: 0585.76100
DOI: 10.1016/0167-2789(84)90012-5
author: B. Gaffet
published in: Physica D
publication date: 1984-01-01T00:00:00Z
review text: The author refers to his previous papers [J. Fluid Mechanics 134, 179-194 (1983; Zbl 0537.76043) and J. Math. Phys. 25, 245 (1984)], concerning 3- symmetries in one-dimensional adiabatic gas flow. In the present paper he considers a specific entropy distribution law: \[ \sigma (M)\equiv P/\rho \gamma =(a_ 0M^ 2+a_ 1M+a_ 2)^{-4}\quad (with\quad \gamma =3), \] where \(M=\int \rho dr\) is the Lagrangian mass coordinate, and \(\gamma\), P, \(\rho\), r represent the adiabatic index, pressure, density and position coordinate as usual, the three (discrete) symmetries generate an infinite multiplicative group (G)-i.e. one that contains an infinite number of elements. Further, by combination with the continuous group of space-translations, (\({\mathcal C}_ r)\), a Lie group of symmetry of infinite order, \((\hat G)\), is generated. The velocity \(\nu\) and density \(\rho\) are then the following partial derivatives of the position coordinate \(r: \nu =\partial r/\partial t;\) \(w\equiv 1/\rho =\partial r/\partial M\). The pressure P and sound velocity c are related to the density by \(P=C\rho^ 3/M^ 4;\) \(c=\sqrt{3P/\rho}\), where the value of the numerical constant C need not be specified here. Such flows are governed by the following partial differential equation: \(\partial \nu /\partial t+\partial P/\partial M=0\), expressing, as is well known, momentum conservation. We recall that in the one-dimensional case any equation of the form \(\partial X/\partial t+\partial Y/\partial M=0\) constitutes a conservation law. The author introduced the three Bäcklund transformations a) Transformation \((T'):\) \(M'=P;\) \(t'=\nu;\) \(r'=\nu t-r-3M/\rho\); b) Transformation \((T^*):\) \(M^*=-M;\) \(t^*=1/t;\) \(r^*=-r/t\); c) Transformation \((\bar T):\) \(\bar M=1/M;\) \(\bar t=-t;\) \(\bar P=P/M\) in his earlier work. Now he proceeds in the following manner, making use of Noether's theorem. Introducing the remaining symmetry of space- translation enlarges the group into a Lie group …
Mathematics Subject Classification ID: 76N15
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 22E70
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 58D05
zbMATH DE Number: 3938932
zbMATH Keywords: one-dimensional adiabatic gas flow
zbMATH Keywords: entropy distribution law
zbMATH Keywords: continuous group of space-translations
zbMATH Keywords: Lie group of symmetry of infinite order
zbMATH Keywords: Noether's theorem
zbMATH Keywords: infinite number of conservation laws
zbMATH Keywords: finite sub- algebra
zbMATH Keywords: scale invariance
zbMATH Keywords: soliton problem
zbMATH Keywords: inverse-scattering method
zbMATH Keywords: method of linearization
Wikidata QID: Q115363850
reviewed by: Vadim Komkov
MaRDI profile type: Publication
full work available at URL: 0167-2789(84)90012-5
OpenAlex ID: W1988862277
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description: scientific article; zbMATH DE number 3938932