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Criterium for the gradient catastrophe for the non‐isentropic gas dynamics equations in the one dimensional case
Author(s) -
Rozanova Olga
Publication year - 2007
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200700695
Subject(s) - polytropic process , ode , isentropic process , mathematics , nonlinear system , mathematical analysis , impulse (physics) , space (punctuation) , gas dynamics , momentum (technical analysis) , ordinary differential equation , kinetic energy , classical mechanics , physics , differential equation , mechanics , computer science , quantum mechanics , finance , economics , operating system
A method to describe the motion of non‐isentropic polytropic gas by means of integral functionals is proposed. They are as a matter of fact the integrals of the solution components over a material volume. Included among the functionals are the kinetic and potential energies, impulse, momentum of mass, and a number of specific integrals. These integral functionals are functions of time and satisfy to a system of ODE, which is closed provided we consider a linear velocity field. In 1D case the system can be reduced to a certain nonlinear second order ODE for the first derivative in space of velocity. The problem about the data yielding the gradient catastrophe (the unboundedness of the solution gradients) can be reduced to the problem of blow‐up of the solutions to the latter equation. This problem can be completely solved, so, the criterium of the gradient catastrophe can be obtained. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)