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29 Sep 2006
A viscosity approximation to a system of conservation laws with no classical Riemann solution
 Barbara Lee Keyfitz,
 Herbert C. Kranzer
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Abstract
There are examples of systems of conservation laws which are strictly hyperbolic and genuinely nonlinear but for which the Riemann problem can be solved only for states which are sufficiently close together. For one such example, we introduce a particular type of artificial viscosity and show how it suggests a possible definition of "generalized" solution to the Riemann problem.
 J. F. Colombeau and A. Y. LeRoux, "Numerical techniques in elastoplasticity," in Nonlinear Hyperbolic Problems, (ed Carasso, Raviart and Serre), Lecture Notes in Math. 1270 (1987), SpringerVerlag, Berlin, 103–114.
 C. M. Dafermos, "Solution of the Riemann problem for a class of hyperbolic systems of conservation laws by the viscosity method," Arch. Rat. Mech. Anal. 52 (1973), 1–9. CrossRef
 C. M. Dafermos and R. J. DiPerna, "The Riemann problem for certain classes of hyperbolic systems of conservation laws," Jour. Diff. Eqns. 20 (1976), 90–114. CrossRef
 B. L. Keyfitz and H. C. Kranzer, "Existence and uniqueness of entropy solutions to the Riemann problem for hyperbolic systems of two nonlinear conservation laws," Jour. Diff. Eqns. 27 (1978), 444–476. CrossRef
 B. L. Keyfitz and H. C. Kranzer, "A system of conservation laws with no classical Riemann solution," preprint (1988).
 Ph. Le Floch, "Nonlinear hyperbolic systems under nonconservative form," to appear in Comm in PDE (1988).
 P. D. Lax, "Shock waves and entropy," in Contributions to Nonlinear Functional Analysis, (ed Zarantonello), Academic Press, New York (1971), 603–634.
 M. Slemrod, "A limiting ‘viscosity’ approach to the Riemann problem for materials exhibiting change of phase," Univ. of Wisconsin preprint (1987).
 V. A. Tupciev, "On the method of introducing viscosity in the study of problems involving decay of a discontinuity," Dokl. Akad. Nauk. SSR 211 (1973), 55–58; translated in Soviet Math. Dokl. 14.
 Title
 A viscosity approximation to a system of conservation laws with no classical Riemann solution
 Book Title
 Nonlinear Hyperbolic Problems
 Book Subtitle
 Proceedings of an Advanced Research Workshop held in Bordeaux, France, June 13–17, 1988
 Pages
 pp 185197
 Copyright
 1989
 DOI
 10.1007/BFb0083875
 Print ISBN
 9783540517467
 Online ISBN
 9783540468004
 Series Title
 Lecture Notes in Mathematics
 Series Volume
 1402
 Series ISSN
 00758434
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag
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 Editors
 Authors

 Barbara Lee Keyfitz ^{(1)}
 Herbert C. Kranzer ^{(2)}
 Author Affiliations

 1. Department of Mathematics, University of Houston, 77004, Houston, Texas
 2. Department of Mathematics, Adelphi University, 11530, Garden City, New York
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