Sever, Michael Existence in the large for Riemann problems for system of conservation laws. (English) Zbl 0591.35037 Trans. Am. Math. Soc. 292, 375-381 (1985). Author’s abstract. ”An existence theorem in the large is obtained for the Riemann problem for nonlinear systems of conservation laws. His principal assumptions are strict hyperbolicity, genuine nonlinearity in the strong sense, and the existence of a convex entropy function. The entropy inequality is used to obtain an a priori estimate of the strengths of the shocks and rarefaction waves forming a solution; existence of such a solution then follows by an application of finite-dimensional degree theory. The case of a single degenerate field is also included, with an additional assumption on the existence of Riemann invariants.” Reviewer: M.Tsuji Cited in 4 Documents MSC: 35L65 Hyperbolic conservation laws 35L60 First-order nonlinear hyperbolic equations 35L67 Shocks and singularities for hyperbolic equations 58J20 Index theory and related fixed-point theorems on manifolds Keywords:Riemann problem; nonlinear systems of conservation laws; entropy function; entropy inequality; shocks; rarefaction waves PDFBibTeX XMLCite \textit{M. Sever}, Trans. Am. Math. Soc. 292, 375--381 (1985; Zbl 0591.35037) Full Text: DOI References: [1] Barbara L. Keyfitz and Herbert C. Kranzer, Existence and uniqueness of entropy solutions to the Riemann problem for hyperbolic systems of two nonlinear conservation laws, J. Differential Equations 27 (1978), no. 3, 444 – 476. · Zbl 0364.35036 [2] -, The Riemann problem for a class of hyperbolic conservation laws exhibiting a parabolic degeneracy, preprint. · Zbl 0521.35035 [3] P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537 – 566. · Zbl 0081.08803 [4] Peter Lax, Shock waves and entropy, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) Academic Press, New York, 1971, pp. 603 – 634. · Zbl 0268.35014 [5] Tai Ping Liu, The Riemann problem for general systems of conservation laws, J. Differential Equations 18 (1975), 218 – 234. · Zbl 0297.76057 [6] Tai Ping Liu, The Riemann problem for general 2\times 2 conservation laws, Trans. Amer. Math. Soc. 199 (1974), 89 – 112. · Zbl 0289.35063 [7] Tai Ping Liu, Existence and uniqueness theorems for Riemann problems, Trans. Amer. Math. Soc. 212 (1975), 375 – 382. · Zbl 0317.35062 [8] M. S. Mock, Discrete shocks and genuine nonlinearity, Michigan Math. J. 25 (1978), no. 2, 131 – 146. · Zbl 0397.35044 [9] -, A difference scheme employing fourth-order viscosity to enforce an entropy inequality, Proc. Bat-Sheva Conf., Tel-Aviv Univ., 1977. [10] M. S. Mock, Systems of conservation laws of mixed type, J. Differential Equations 37 (1980), no. 1, 70 – 88. · Zbl 0413.34017 [11] M. S. Mock, A topological degree for orbits connecting critical points of autonomous systems, J. Differential Equations 38 (1980), no. 2, 176 – 191. · Zbl 0417.34053 [12] J. A. Smoller, Contact discontinuities in quasi-linear hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 791 – 801. · Zbl 0195.39201 [13] J. A. Smoller, On the solution of the Riemann problem with general step data for an extended class of hyperbolic systems, Michigan Math. J. 16 (1969), 201 – 210. · Zbl 0185.34501 [14] J. A. Smoller, A uniqueness theorem for Riemann problems, Arch. Rational Mech. Anal. 33 (1969), 110 – 115. · Zbl 0176.09402 [15] -, Shock waves and reaction-diffusion equations, Springer-Verlag, New York, 1963. [16] Burton Wendroff, The Riemann problem for materials with nonconvex equations of state. I. Isentropic flow, J. Math. Anal. Appl. 38 (1972), 454 – 466. · Zbl 0264.76054 [17] Burton Wendroff, The Riemann problem for materials with nonconvex equations of state. II. General flow, J. Math. Anal. Appl. 38 (1972), 640 – 658. · Zbl 0287.76049 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.