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

Abstract

For a genuinely nonlinear hyperbolic system of conservation laws with added artificial viscosity, u t + f ( u ) x = ε u x x {u_t} + f{(u)_x} = \varepsilon {u_{xx}} , we prove that traveling wave profiles for small amplitude extreme shocks (the slowest and fastest) are linearly stable to perturbations in 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 e c x {e^{cx}} is used in components transverse to the profile, where, for an extreme shock, the linearized equation is dominated by unidirectional convection.