Communications in Mathematical Physics

, Volume 172, Issue 1, pp 39–55 | Cite as

Long-time effect of relaxation for hyperbolic conservation laws

  • I-Liang Chern


The hyperbolic conservation laws with relaxation appear in many physical systems such as nonequilibrium gas dynamics, flood flow with friction, viscoelasticity, magnetohydrodynamics, etc. This article studies the long-time effect of relaxation when the initial data is a perturbation of an equilibrium constant state. It is shown that in this case the long-time effect of relaxation is equivalent to a viscous effect, or in other words, the Chapman-Enskog expansion is valid. It is also shown that the corresponding solution tends to a diffusion wave time asymptotically. This diffusion wave carries an invariant mass. The convergence rate to this diffusion wave in theL p -sense for 1≦p≦∞ is also obtained and this rate is optimal.


Neural Network Complex System Initial Data Nonlinear Dynamics Convergence Rate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Chern, I-L.: Multiple-mode diffusion waves for viscous nonstrictly hyperbolic conservation laws. Commun. Math. Phys.138, 51–61 (1991)CrossRefGoogle Scholar
  2. 2.
    Chern, I-L., Liu, Tai-Ping: Convergence to diffusion waves of solutions for viscous conservation laws. Commun. Math. Phys.110, 503–517 (1987)120, 525–527 (1989)CrossRefGoogle Scholar
  3. 3.
    Kawashima, S.: Systems of a hyperbolic-parabolic composite type, with application to the equation of magnetohydrodynamics. Doctoral thesis, Kyoto University (1983)Google Scholar
  4. 4.
    Kawashima, S.: Large-time behavior of solutions to hyperbolic-parabolic systems of conservation laws and applications. Proc. R. Soc. Edinburgh106 A, 169–194 (1987)Google Scholar
  5. 5.
    Liu, T.-P.: Hyperbolic conservation laws with relaxation. Commun. Math. Phys.108, 153–175 (1987)CrossRefGoogle Scholar
  6. 6.
    Moser, J.: A rapidly convergent iteration method and nonlinear partial differential equation. I. Ann. Sc. Norm. Super. Pisa20, 265–315 (1966)Google Scholar
  7. 7.
    Vincenti, W., Kruger, C.: Introduction to physical gas dynamics. Melbourne: Robert E. Krieger, 1982Google Scholar
  8. 8.
    Whitham, J.: “Linear and nonlinear waves.” New York: Wiley, 1974Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • I-Liang Chern
    • 1
  1. 1.Department of MathematicsNational Taiwan UniversityTaipeiTaiwan, R.O.C.

Personalised recommendations