arXiv:math/0408227v1 [math.AP] 17 Aug 2004 L p Asymptotic Behavior of Perturbed Viscous Shock Profiles Mohammadreza Raoofi * June 7, 2018 Abstract We investigate the L p asymptotic behavior (1 ≤ p ≤∞) of a per- turbation of a Lax or overcompressive type shock wave solution to a system of conservation law in one dimension. The system of the equa- tions can be strictly parabolic, or have real viscosity matrix (partially parabolic, e.g., compressible Navier–Stokes equations or equations of Magnetohydrodynamics). We use known pointwise Green function bounds for the linearized equation around the shock to show that the perturbation of such a solution can be decomposed into a part corre- sponding to shift in shock position or shape, a part which is the sum of diffusion waves, i.e., the solutions to a viscous Burger’s equation, conserving the initial mass and convecting away from the shock profile in outgoing modes, and another part which is more rapidly decaying in L p . 1 Introduction Consider the system of conservation laws with viscosity: (1.1) u t + f (u) x = ν (B(u)u x ) x with u ∈ R n is the conserved quantity, and ν is a constant measuring trans- port effects (e.g. viscosity or heat conduction). As we are not considering the vanishing-viscosity limit ν → 0, we can assume ν =1. An important * Indiana University, Bloomington, IN 47405; mraoofi@indiana.edu. This work was carried out as part of the author’s doctoral thesis at Indiana University, Bloomington, under the direction of Kevin Zumbrun. The author would like to thank professor Zumbrun for his advice, encouragement and support. This project was supported in part by National Science Foundation under Grant DMS-0070765 1