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Journal Article

# Linearized Stability of Extreme Shock Profiles in Systems of Conservation Laws with Viscosity

Robert L. Pego
Transactions of the American Mathematical Society
Vol. 280, No. 2 (Dec., 1983), pp. 431-461
DOI: 10.2307/1999627
Stable URL: http://www.jstor.org/stable/1999627
Page Count: 31

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## Abstract

For a genuinely nonlinear hyperbolic system of conservation laws with added artificial viscosity, $u_t + f(u)_x = \varepsilon u_{xx}$, 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 $e^{cx}$ is used in components transverse to the profile, where, for an extreme shock, the linearized equation is dominated by unidirectional convection.

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