Fast-slow dynamics for parabolic perturbations of conservation laws

Abstract

We study the {open_quote}large time{close_quote} behaviour of parabolic perturbations of hyperbolic systems having on linearly degenerate field, the others being genuinely non linear fields. The initial data is periodic. For {open_quote}moderated time{close_quote}, the perturbation can often be ignored. For {open_quote}large time{close_quote}, the oscillations in non liner modes should be damped whereas the linear one behaves as a travelling wave. Because of the coupling, all the modes are modulated by the slow time. The purpose of the article is three-fold. First we give a mathematical description of this problem by means of an asymptotic expansion. We then formally describe the slow evolution for the Navier-Stokes equations of a compressible viscous heat conductive fluid. Finally, we justify the asymptotic development for model problem, arising in elasticity theory: the Keyfitz-Kranzer`s system. 8 refs.