Skip to main content
SpringerLink
Log in

Hyperbolic conservation laws with relaxation

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

The effect of relaxation is important in many physical situations. It is present in the kinetic theory of gases, elasticity with memory, gas flow with thermo-non-equilibrium, water waves, etc. The governing equations often take the form of hyperbolic conservation laws with lower-order terms. In this article, we present and analyze a simple model of hyperbolic conservation laws with relaxation effects. Dynamic subcharacteristics governing the propagation of disturbances over strong wave forms are identified. Stability criteria for diffusion waves, expansion waves and traveling waves are found and justified nonlinearly. Time-asymptotic expansion and the energy method are used in the analysis. For dissipative waves, the expansion is similar in spirit to the Chapman-Enskog expansion in the kinetic theory. For shock waves, however, a different approach is needed.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Caflish, R., Liu, T.-P.: Nonlinear stability of shock waves for the Broadwell model (to appear)

  2. Cercignani, C.: Theory and application of the Boltzman equation. Scottish Academic Press 1975

  3. Dafermos, C., Nohel, J.: A nonlinear hyperbolic Volterra equation in viscoelasticity, Contribution to Analysis and Geometry, pp. 87–116. Blatimore, MD.: John Hopkins University Press 1981

    Google Scholar 

  4. Greenberg, J., Hsiao, L.: The Riemann problem for systemu t+σ x =0 and (σ−f(w) t)+(σ−μf(u))=0. Arch. Ration. Mech. Anal.82, 87–108 (1983)

    Google Scholar 

  5. Liu, T.-P.: The Riemann problem for general system of conservation laws. J. Differ. Equations18, 218–234 (1975)

    Google Scholar 

  6. Liu, T.-P.: Nonlinear stability of shock waves for viscous conservation laws. Mem. Am. Math. Soc.328, No. 328 (1985)

    Google Scholar 

  7. Whitham, J.: Linear and nonlinear waves. New York: Wiley 1974

    Google Scholar 

  8. Vicenti, W., Kruger, C.: Introduction to physical gas dynamics. Melbourne: Robert E. Krieger 1982

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Maryland, 20742, College Park, MD, USA

    Tai-Ping Liu

Authors
  1. Tai-Ping Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by A. Jaffe

The paper was written at Mittag-Leffler Institute; the author wants to thank the Institute for the visiting position in 1986. This work was supported in part by an NSF grant

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, TP. Hyperbolic conservation laws with relaxation. Commun.Math. Phys. 108, 153–175 (1987). https://doi.org/10.1007/BF01210707

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01210707

Keywords

Search

Navigation