Fast-slow dynamics for parabolic perturbations of conservation laws

We study the 'large time' behaviour of parabolic perturbations of hyperbolic systems having one linearly degenerate field, the others being genuinely non linear fields. The initial data is periodic. For 'moderated time', the perturbation can often be ignored. For 'large time', the oscillations in non linear 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 a model problem, arising in elasticity theory: the Keyfitz-Kranzer's system.