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Transactions of the American Mathematical Society

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

 

 

Stable viscosities and shock profiles for systems of conservation laws


Author: Robert L. Pego
Journal: Trans. Amer. Math. Soc. 282 (1984), 749-763
MSC: Primary 35L65; Secondary 35L67, 76L05
DOI: https://doi.org/10.1090/S0002-9947-1984-0732117-1
MathSciNet review: 732117
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Abstract: Wide classes of high order "viscosity" terms are determined, for which small amplitude shock wave solutions of a nonlinear hyperbolic system of conservation laws $ {u_t} + f{(u)_x} = 0$ are realized as limits of traveling wave solutions of a dissipative system $ {u_t} + f{(u)_x} = \nu {({D_1}{u_x})_x} + \cdots + {\nu ^n}{({D_n}{u^{(n)}})_x}$. The set of such "admissible" viscosities includes those for which the dissipative system satisfies a linearized stability condition previously investigated in the case $ n = 1$ by A. Majda and the author. When $ n = 1$ we also establish admissibility criteria for singular viscosity matrices $ {D_1}(u)$, and apply our results to the compressible Navier-Stokes equations with viscosity and heat conduction, determining minimal conditions on the equation of state which ensure the existence of the "shock layer" for weak shocks.


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DOI: https://doi.org/10.1090/S0002-9947-1984-0732117-1
Keywords: Shock profiles, viscosity, traveling waves, center manifold, compressible Navier-Stokes equations, shock layer
Article copyright: © Copyright 1984 American Mathematical Society