Fast-slow dynamics for parabolic perturbations of conservation laws

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