An infinite Lie group of symmetry of one-dimensional gas flow, for a class of entropy distributions
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 of symmetry of infinite order, from which an infinite number of conservation laws can be deduced by application of Noether's theorem. The Lie group has a finite sub- algebra of order eight, which has SU3 structure; the list of associated conservation laws includes each of the six ones that are derivable from general physical principles, namely: the energy, momentum and the center- of-mass integrals, two integrals expressing scale invariance, and one associated with the virial theorem; the remaining two integrals of the octet are of a new type. Such a situation reminds us of the case of the Korteweg-de Vries equation in the soliton problem, where the symmetries and infinite number of conservation laws arise as a result of the possibility to linearize through the inverse-scattering method. Thus the question is raised of whether the inverse-scattering method also applies to gas-dynamical equations (with the above equation of state), or else whether another method of linearization may be found. One should point out that somewhat similar equations of gas dynamics, but not obeying author's state laws, were approached via Cartan's exterior forms by the Novosibirsk research group.
scientific article; zbMATH DE number 3938932