Stable viscosities and shock profiles for systems of conservation laws

Pego, Robert L.

doi:10.1090/S0002-9947-1984-0732117-1

Stable viscosities and shock profiles for systems of conservation laws
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by Robert L. Pego PDF

Trans. Amer. Math. Soc. 282 (1984), 749-763 Request permission

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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Shock waves as limits of progressive wave solutions of higher order equations

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