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: For a genuinely nonlinear hyperbolic system of conservation laws with added artificial viscosity, we prove that traveling wave profiles for small amplitude extreme shocks (the slowest and fastest) are linearly stable to perturbations initial data chosen from certain spaces with weighted norm; i.e., we show that the spectrum of the linearized equation lies strictly in the left half plane, except for a simple eigenvalue at the origin (due to phase translations of the profile). The weight is used in components transverse to the profile, where, for an extreme shock, the linearized equation is dominated by unidirectional convection.

Linearized stability of extreme shock profiles in systems of conservation laws with viscosity