ON THE LINEARIZED STABILITY OF VISCOUS SHOCK PROFILES FOR SYSTEMS OF CONSERVATION-LAWS

This paper is concerned with the linearized stability of traveling wave solutions for systems of viscous hyperbolic conservation laws. The main purpose is to show that for a given traveling wave with shock profile from any characteristic family, there exists an appropriate weighted norm space such that the traveling wave is exponentially stable in this space. As a consequence, if the initial disturbance has average zero and decays exponentially fast as |x| → ∞, then the corresponding solution of the linearized equation decays to zero exponentially fast in t on any compact interval in x. The proof is given by applying an elementary weighted characteristic energy method to the integrated linearized system, based on the underlying wave structure. © 1992.