Existence in the large for Riemann problems for systems of conservation laws
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- by Michael Sever PDF
- Trans. Amer. Math. Soc. 292 (1985), 375-381 Request permission
Abstract:
An existence theorem in the large is obtained for the Riemann problem for nonlinear systems of conservation laws. Our 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 refraction 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.References
- 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. MR 466993, DOI 10.1016/0022-0396(78)90062-1 —, The Riemann problem for a class of hyperbolic conservation laws exhibiting a parabolic degeneracy, preprint.
- P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 93653, DOI 10.1002/cpa.3160100406
- 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. MR 0393870
- Tai Ping Liu, The Riemann problem for general systems of conservation laws, J. Differential Equations 18 (1975), 218–234. MR 369939, DOI 10.1016/0022-0396(75)90091-1
- Tai Ping Liu, The Riemann problem for general $2\times 2$ conservation laws, Trans. Amer. Math. Soc. 199 (1974), 89–112. MR 367472, DOI 10.1090/S0002-9947-1974-0367472-1
- Tai Ping Liu, Existence and uniqueness theorems for Riemann problems, Trans. Amer. Math. Soc. 212 (1975), 375–382. MR 380135, DOI 10.1090/S0002-9947-1975-0380135-2
- M. S. Mock, Discrete shocks and genuine nonlinearity, Michigan Math. J. 25 (1978), no. 2, 131–146. MR 483572 —, A difference scheme employing fourth-order viscosity to enforce an entropy inequality, Proc. Bat-Sheva Conf., Tel-Aviv Univ., 1977.
- M. S. Mock, Systems of conservation laws of mixed type, J. Differential Equations 37 (1980), no. 1, 70–88. MR 583340, DOI 10.1016/0022-0396(80)90089-3
- M. S. Mock, A topological degree for orbits connecting critical points of autonomous systems, J. Differential Equations 38 (1980), no. 2, 176–191. MR 597799, DOI 10.1016/0022-0396(80)90003-0
- J. A. Smoller, Contact discontinuities in quasi-linear hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 791–801. MR 267273, DOI 10.1002/cpa.3160230507
- 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. MR 247283
- J. A. Smoller, A uniqueness theorem for Riemann problems, Arch. Rational Mech. Anal. 33 (1969), 110–115. MR 237961, DOI 10.1007/BF00247755 —, Shock waves and reaction-diffusion equations, Springer-Verlag, New York, 1963.
- Burton Wendroff, The Riemann problem for materials with nonconvex equations of state. I. Isentropic flow, J. Math. Anal. Appl. 38 (1972), 454–466. MR 328387, DOI 10.1016/0022-247X(72)90103-5
- Burton Wendroff, The Riemann problem for materials with nonconvex equations of state. II. General flow, J. Math. Anal. Appl. 38 (1972), 640–658. MR 313655, DOI 10.1016/0022-247X(72)90075-3
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 292 (1985), 375-381
- MSC: Primary 35L65; Secondary 76N10
- DOI: https://doi.org/10.1090/S0002-9947-1985-0805969-5
- MathSciNet review: 805969