A viscosity approximation to a system of conservation laws with no classical Riemann solution
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), Springer-Verlag, 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.
- 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
- pp 185-197
- Print ISBN
- Online ISBN
- Series Title
- Lecture Notes in Mathematics
- Series Volume
- Series ISSN
- Springer Berlin Heidelberg
- Copyright Holder
- Additional Links
- Industry Sectors
- eBook Packages
To view the rest of this content please follow the download PDF link above.