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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Author: Robert L. Pego
Journal: Trans. Amer. Math. Soc. 280 (1983), 431-461
MSC: Primary 35L65; Secondary 35K55
MathSciNet review: 716831
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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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Keywords: Conservation laws, viscosity, nonlinear parabolic systems, shock profiles, traveling waves, stability, Burgers’ equation, weighted norms
Article copyright: © Copyright 1983 American Mathematical Society