Stable viscosities and shock profiles for systems of conservation laws. (English) Zbl 0512.76068

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 \[ u_t + f(u)_x = \nu (D_1u_x)_x + \cdots + \nu ^n(D_nu^{(n)})_x. \] 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 this author [J. Differ. Equ. 56, 229–262 (1985; Zbl 0512.76067)]. 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.


76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
76L05 Shock waves and blast waves in fluid mechanics
35L65 Hyperbolic conservation laws
35C07 Traveling wave solutions
76E99 Hydrodynamic stability
76D99 Incompressible viscous fluids


Zbl 0512.76067
Full Text: DOI


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