Skip to main content
SpringerLink
Log in
Book cover

Nonlinear Hyperbolic Problems pp 185–197Cite as

A viscosity approximation to a system of conservation laws with no classical Riemann solution

  • Hyperbolic P.D.E. Theory
  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1402))

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.

Research supported in part by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR 86-0088, and by a grant from the Energy Laboratory, University of Houston. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   46.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

V. References

  1. 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.

    Chapter  Google Scholar 

  2. 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.

    Article  MathSciNet  MATH  Google Scholar 

  3. 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.

    Article  MathSciNet  MATH  Google Scholar 

  4. 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.

    Article  MathSciNet  MATH  Google Scholar 

  5. B. L. Keyfitz and H. C. Kranzer, "A system of conservation laws with no classical Riemann solution," preprint (1988).

    Google Scholar 

  6. Ph. Le Floch, "Nonlinear hyperbolic systems under nonconservative form," to appear in Comm in PDE (1988).

    Google Scholar 

  7. P. D. Lax, "Shock waves and entropy," in Contributions to Nonlinear Functional Analysis, (ed Zarantonello), Academic Press, New York (1971), 603–634.

    Google Scholar 

  8. M. Slemrod, "A limiting ‘viscosity’ approach to the Riemann problem for materials exhibiting change of phase," Univ. of Wisconsin preprint (1987).

    Google Scholar 

  9. 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.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Houston, 77004, Houston, Texas

    Barbara Lee Keyfitz

  2. Department of Mathematics, Adelphi University, 11530, Garden City, New York

    Herbert C. Kranzer

Authors
  1. Barbara Lee Keyfitz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Herbert C. Kranzer
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Claude Carasso Pierre Charrier Bernard Hanouzet Jean-Luc Joly

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Springer-Verlag

About this paper

Cite this paper

Keyfitz, B.L., Kranzer, H.C. (1989). A viscosity approximation to a system of conservation laws with no classical Riemann solution. In: Carasso, C., Charrier, P., Hanouzet, B., Joly, JL. (eds) Nonlinear Hyperbolic Problems. Lecture Notes in Mathematics, vol 1402. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0083875

Download citation

  • DOI: https://doi.org/10.1007/BFb0083875

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51746-7

  • Online ISBN: 978-3-540-46800-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation