Majda, Andrew; Pego, Robert L. Stable viscosity matrices for systems of conservation laws. (English) Zbl 0512.76067 J. Differ. Equations 56, 229-262 (1985). We study a natural class of appropriate viscosity matrices for strictly hyperbolic systems of conservation laws in one space dimension, \(u_t + f(u)_x = 0\), \(u\in\mathbb R^m\). These matrices are admissible in the sense that small-amplitude shock wave solutions of the hyperbolic system are shown to be limits of smooth traveling wave solutions of the parabolic system \(u_t + f(u)_x = \nu(Du_x)_x\) as \(\nu\to 0\) if \(D\) is in this class. The class is determined by a linearized stability requirement: The Cauchy problem for the equation \(u_t + f'(u_0)u_x = \nu(Du_x)_{xx}\) should be well posed in \(L^2\) uniformly in \(\nu\) as \(\nu\to 0\). Previous examples of inadmissible viscosity matrices are accounted for through violation of the stability criterion. Reviewer: Andrew Majda (Berkeley) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 4 ReviewsCited in 62 Documents MSC: 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 35L65 Hyperbolic conservation laws 76L05 Shock waves and blast waves in fluid mechanics 76D99 Incompressible viscous fluids 76E99 Hydrodynamic stability Keywords:class of appropriate viscosity matrices; strictly hyperbolic systems; one space dimension; admissible matrices; small amplitude shock wave as limits of smooth traveling wave; parabolic system; linearized stability requirement; Cauchy problem; shock profiles; center manifold Citations:Zbl 0512.76068 PDFBibTeX XMLCite \textit{A. Majda} and \textit{R. L. Pego}, J. Differ. Equations 56, 229--262 (1985; Zbl 0512.76067) Full Text: DOI References: [1] Conley, C.; Smoller, J., Viscosity matrices for two-dimensional nonlinear hyperbolic systems, Comm. Pure Appl. Math., 23, 876-884 (1970) · Zbl 0204.11303 [2] Conley, C.; Smoller, J., Shock waves as limits of progressive wave solutions of higher order equations, II, Comm. Pure Appl. Math., 25, 133-146 (1972) · Zbl 0225.35067 [3] Conlon, J., A theorem in ordinary differential equations with an application to hyperbolic conservation laws, Adv. in Math., 35, 1-18 (1980) · Zbl 0426.35068 [4] Foy, L., Steady state solutions of conservation laws with viscosity terms, Comm. Pure Appl. Math., 17, 177-188 (1964) · Zbl 0178.11902 [5] Gelfand, I. M., Amer. Math. Soc. Trans. Ser. 2, No. 29 (1963), English transl.: [6] Kelley, A., The stable, center-stable, center, center-unstable, and unstable manifolds, (Abraham, R.; Robbin, J., Transversal Mappings and Flows (1967), Benjamin: Benjamin New York) · Zbl 0173.11001 [7] Kelley, A., Stability of the center-stable manifold, J. Math. Anal. Appl., 18, 336-344 (1967) · Zbl 0166.08304 [8] Kopell, N.; Howard, L. N., Bifurcations and trajectories joining critical points, Adv. in Math., 18, 306-358 (1975) · Zbl 0361.34026 [9] Kreiss, H. O., Über matrizen die beschränkte halbgruppen erzeuzen, Math. Scand., 7, 71-80 (1959) · Zbl 0090.09801 [10] Lax, P. D., Hyperbolic systems of conservation laws, II, Comm. Pure Appl. Math., 10, 537-566 (1957) · Zbl 0081.08803 [11] Lax, P. D., Shock waves and entropy, (Contributions to Nonlinear Functional Analysis. Contributions to Nonlinear Functional Analysis, Proc. Symp. Math. Res. Ctr. Univ. Wisconsin, Madison, 1971 (1971), Academic Press: Academic Press New York), 603-634 [12] Liu, T.-P, The entropy condition and the admissibility of shocks, J. Math. Anal. Appl., 53, 78-88 (1976) · Zbl 0332.76051 [13] Mock, M. S., A topological degree for orbits connecting critical points of autonomous systems, J. Differential Equations, 38, 176-191 (1980) · Zbl 0417.34053 [14] Pego, R., Stable viscosities and shock profiles for systems of conservation laws, Trans. Amer. Math. Soc., 282, 749-763 (1984) · Zbl 0512.76068 [15] Richtmyer, R. D.; Morton, K. W., Difference Methods for Initial-Value Problems (1967), Wiley-Interscience: Wiley-Interscience New York · Zbl 0155.47502 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.