Skip to main content
SpringerLink
Log in
Book cover

Hyperbolic Conservation Laws in Continuum Physics pp 37–47Cite as

Hyperbolic Systems of Balance Laws

  • Chapter

  • 743 Accesses

Part of the book series: Grundlehren der mathematischen Wissenschaften ((GL,volume 325))

Abstract

The ambient space for the system of balance laws, introduced in Chapter 1, will be visualized here as space-time, and the central notion of hyperbolicity in the time direction will be motivated and defined. Companions to the flux, considered in Section 1.5, will now be realized as entropy-entropy flux pairs.

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   74.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

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.

Notes

  1. Bethe, H.: Report on the theory of shock waves for an arbitrary equation of state. Report No. PB-32189. Clearinghouse for Federal Scientific and Technical Information, U.S. Dept. of Commerce, Washington DC, 1942.

    Google Scholar 

  2. von Neumann, J.: Theory of shock waves. Collected Works, Vol. VI, pp. 178–202. Oxford: Pergamon Press, 1963.

    Google Scholar 

  3. Weyl, H.: Shock waves in arbitrary fluids. Comm. Pure Appl. Math. 2 (1949), 103–122.

    MathSciNet  MATH  Google Scholar 

  4. Friedrichs, K.O. and P.D. Lax: Systems of conservation equations with a convex extension. Proc. Natl. Acad. Sc1. USA 68 (1971), 1686–1688.

    Article  MathSciNet  MATH  Google Scholar 

  5. Lax, P.D.: Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. CBMS Regional Conference Series in Mathematics No. 11. Philadelphia: SIAM, 1973.

    Google Scholar 

  6. Smoller, J.: Shock Waves and Reaction-Diffusion Equations ( Second Edition ). New York: Springer, 1994.

    Google Scholar 

  7. Serre, D.: Systèmes de Lois de Conservation, Vols. I-I1. Paris: Diderot, 1996. English translation: Systems of Conservation Laws, Vols. 1–2. Cambridge: Cambridge University Press, 1999.

    Google Scholar 

  8. Rozdestvenskii, B.L. and N.N. Janenko: Systems of Quasilinear Equations and Their Applications to Gas Dynamics. Moscow: Nauka, 1978. English translation: Providence: American Mathematical Society, 1983.

    Google Scholar 

  9. Jeffrey, A.: Magnetohydrodynamics. Edinburgh: Oliver and Boyd, 1966.

    MATH  Google Scholar 

  10. Bressan, A.: Lecture Notes on Systems of Conservation Laws. Trieste: SISSA, 1995.

    Google Scholar 

  11. Majda, A.: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. New York: Springer, 1984.

    Book  MATH  Google Scholar 

  12. Hörmander, L.: Lectures on Nonlinear Hyperbolic Differential Equations. Paris: Springer, 1997.

    MATH  Google Scholar 

  13. Taylor, M.E.: Partial Differential Equations II1. New York: Springer, 1996.

    Book  Google Scholar 

  14. Leveque, R.J.: Numerical Methods for Conservation Laws. Basel: Birkhäuser, 1990.

    MATH  Google Scholar 

  15. Godlewski, E. and P.-A. Raviart: Hyperbolic Systems of Conservation Laws. Paris: Ellipses, 1991.

    MATH  Google Scholar 

  16. Godlewski, E. and P.-A. Raviart: Numerical Approximation of Hyperbolic Systems of Conservation Laws. New York: Springer, 1996.

    MATH  Google Scholar 

  17. Tadmor, E.: Approximate Solutions of Nonlinear Conservation Laws. Lecture Notes in Math. 1697 (1998), 1–149. Berlin: Springer.

    Google Scholar 

  18. Whitham, G.B.: Linear and Nonlinear Waves. New York: Wiley-Interscience, 1974.

    MATH  Google Scholar 

  19. Zeldovich, Ya. and Yu. Raizer: Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Vols. I—I1. New York: Academic Press, 1966–1967.

    Google Scholar 

  20. Lax, P.D.: Shock waves and entropy. Contributions to Functional Analysis pp. 603–634, ed. E.A. Zarantonello. New York: Academic Press, 1971.

    Google Scholar 

  21. Whitham, G.B.: Linear and Nonlinear Waves. New York: Wiley-Interscience, 1974.

    MATH  Google Scholar 

  22. Noll, W.: A mathematical theory of the mechanical behavior of continuous media. Arch. Rational Mech. Anal. 2 (1958), 197–226.

    MATH  Google Scholar 

  23. Lions, P.-L.: Mathematical Topics in Fluid Mechanics Vols. I—I1. Oxford: Oxford University Press, 1996–1998.

    Google Scholar 

  24. Müller, 1. and T. Ruggeri: Rational Extended Thermodynamics. ( Second Edition ). New York: Springer, 1998.

    Google Scholar 

  25. Boillat, G. and T. Ruggeri: Hyperbolic principal subsystems: entropy convexity and subcharacteristic conditions. Arch. Rational Mech. Anal. 137 (1997), 305–320.

    MathSciNet  MATH  Google Scholar 

  26. Coleman, B.D. and E.H. Dill: Thermodynamic restrictions on the constitutive equations of electromagnetic theory. ZAMP 22 (1971), 691–702.

    Article  MATH  Google Scholar 

  27. Grot, R.A.: Relativistic continuum physics: electromagnetic interactions. Continuum Physics, Vol. III, pp. 129–219, ed. A.C. Eringen. New York: Academic Press, 1976.

    Google Scholar 

  28. Bloom, F.: Mathematical Problems of Classical Nonlinear Electromagnetic Theory. Harlow: Longman, 1993.

    MATH  Google Scholar 

  29. Cabannes, H.: Theoretical Magnetofluiddynamics. New York: Academic Press, 1970. Caginalp, G.

    Google Scholar 

  30. Jeffrey, A.: Magnetohydrodynamics. Edinburgh: Oliver and Boyd, 1966.

    MATH  Google Scholar 

  31. Taub, A.H.: Relativistic Rankine-Hugoniot equations. Phys. Rev. 74 (1948), 328–334.

    Article  MathSciNet  MATH  Google Scholar 

  32. Friedrichs, K.O.: On the laws of relativistic electro-magneto-fluid dynamics. Comm. Pure Appl. Math. 27 (1974), 749–808.

    MathSciNet  MATH  Google Scholar 

  33. Ruggeri, T.: Galilean invariance and entropy principle for systems of balance laws. Cont. Mech. Therm. 1 (1989), 3–20.

    Article  MathSciNet  MATH  Google Scholar 

  34. Ruggeri, T.: Convexity and symmetrization in relativistic theories. Cont. Mech. Therm. 2 (1990), 163–177.

    Article  MathSciNet  Google Scholar 

  35. Temple, B.: Systems of conservation laws with invariant submanifolds. Trans. AMS 280 (1983), 781–795.

    Article  MathSciNet  MATH  Google Scholar 

  36. Temple, B.: No LI-contractive metric for systems of conservation laws. Trans. AMS 288 (1985), 471–480.

    MathSciNet  MATH  Google Scholar 

  37. Pant, V. Global entropy solutions for isentropic relativistic fluid dynamics. Comm. PDE 21 (1996), 1609–1641.

    Article  MathSciNet  MATH  Google Scholar 

  38. Chen, J.: Conservation laws for the relativistic p-system. Comm. PDE 20 (1995), 1605–1646.

    Article  MATH  Google Scholar 

  39. Rhee, H.-K., Aris, R. and N.R. Amundson: First-Order Partial Differential Equations, Vols. 1-I1. Englewood Cliffs: Prentice-Hall, 1986–1989.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Division of Applied Mathematics, Brown University, 02912, Providence, RI, USA

    Constantine M. Dafermos

Authors
  1. Constantine M. Dafermos
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Dafermos, C.M. (2000). Hyperbolic Systems of Balance Laws. In: Hyperbolic Conservation Laws in Continuum Physics. Grundlehren der mathematischen Wissenschaften, vol 325. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-22019-1_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-22019-1_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-22021-4

  • Online ISBN: 978-3-662-22019-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation