A note on shock profiles in dissipative hyperbolic and parabolic models

This note presents a comparative study of shock profiles in dissi- pative systems. Main assumption is that both hyperbolic and parabolic model are reducible to the same underlying equilibrium system when dissipative ef- fects are neglected. It will be shown that the highest characteristic speed of equilibrium system determines the critical value of the shock speed for which downstream equilibrium state bifurcates. It will be also shown that it obeys the same transcritical bifurcation pattern in hyperbolic, as well as in parabolic case. 1. Introduction and preliminaries Mathematical models of dynamical processes in continuous media may have different structure and may posses different degrees of complexity. Their main ingredients are conservation laws of continuum physics adjoined with constitutive relations which describe material response. Structure of the model mainly depends on assumptions used in building up constitutive relations. Typical outcomes are hyperbolic and parabolic PDE's. Complete models of physical phenomena could be rather complicated. There- fore, analysis may be pursued in a different direction—development of so-called model equations which have simpler form, but capture all important qualitative features of the complete model. The purpose of this paper is to analyze common properties of shock profiles which appear both in parabolic and hyperbolic model equations. To motivate this study some preliminary assumptions will be established first. Models which will be dealt with will be confined to one space dimension, say x, without loss of generality. The basic system—we call it the equilibrium system in the remainder of the paper—will be the system of hyperbolic conservation laws