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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.

MSC:

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

Citations:

Zbl 0512.76067
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References:

[1] Joseph G. Conlon, A theorem in ordinary differential equations with an application to hyperbolic conservation laws, Adv. in Math. 35 (1980), no. 1, 1 – 18. · Zbl 0426.35068
[2] I. M. Gel\(^{\prime}\)fand, Some problems in the theory of quasi-linear equations, Uspehi Mat. Nauk 14 (1959), no. 2 (86), 87 – 158 (Russian).
[3] David Gilbarg, The existence and limit behavior of the one-dimensional shock layer, Amer. J. Math. 73 (1951), 256 – 274. · Zbl 0044.21504
[4] Tai Ping Liu, The entropy condition and the admissibility of shocks, J. Math. Anal. Appl. 53 (1976), no. 1, 78 – 88. · Zbl 0332.76051
[5] Andrew Majda and Stanley Osher, A systematic approach for correcting nonlinear instabilities. The Lax-Wendroff scheme for scalar conservation laws, Numer. Math. 30 (1978), no. 4, 429 – 452. · Zbl 0368.65048
[6] Andrew Majda and Robert L. Pego, Stable viscosity matrices for systems of conservation laws, J. Differential Equations 56 (1985), no. 2, 229 – 262. · Zbl 0512.76067
[7] Akitaka Matsumura and Takaaki Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 9, 337 – 342. · Zbl 0447.76053
[8] Robert L. Pego, Nonexistence of a shock layer in gas dynamics with a nonconvex equation of state, Arch. Rational Mech. Anal. 94 (1986), no. 2, 165 – 178. · Zbl 0652.76047
[9] R. Shapiro, Shock waves as limits of progressive wave solutions of higher order equations, Ph. D. thesis, Univ. of Mchigan, 1973.
[10] J. A. Smoller and R. Shapiro, Dispersion and shock-wave structure, J. Differential Equations 44 (1982), no. 2, 281 – 305. Special issue dedicated to J. P. LaSalle. · Zbl 0486.35052
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