Advertisement

Archive for Rational Mechanics and Analysis

, Volume 219, Issue 2, pp 679–699 | Cite as

Long-time Stability in Systems of Conservation Laws, Using Relative Entropy/Energy

  • Denis SerreEmail author
Article

Abstract

We study the long-time stability of shock-free solutions of hyperbolic systems of conservation laws, under an arbitrarily large initial disturbance in L 2L . We use the relative entropy method, a robust tool which allows us to consider rough and large disturbances. We display practical examples in several space dimensions, for scalar equations as well as isentropic gas dynamics. For full gas dynamics, we use a trick from Chen [1], in which the estimate is made in terms of the relative mechanical energy instead of the relative mathematical entropy.

Keywords

Entropy Relative Entropy Rarefaction Wave Entropy Solution Euler System 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Chen G.-Q.: Vacuum states and global stability of rarefaction waves for compressible flow. Methods Appl. Anal., 7, 337–361 (2000)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Chen, G.-Q., Chen, J.: Stability of rarefaction waves and vacuum states for the multidimensional Euler equations. J. Hyperbolic Differential Equations, 4(1), 105–122 (2007)Google Scholar
  3. 3.
    Chen, G.-Q., Frid, H.: Uniqueness and asymptotic stability of Riemann solutions for the compressible Euler equations. Trans. Am. Math. Soc., 353, 1103–1117 (2001)Google Scholar
  4. 4.
    Chen, G.-Q., Li, Y.: Stability of Riemann solutions with large oscillation for the relativistic Euler equations. J. Differ. Eq., 202(2), 332–353 (2004)Google Scholar
  5. 5.
    Dafermos, C.M.: The second law of thermodynamics and stability. Arch. Rational Mech. Anal., 70(2), 167–179 (1979)Google Scholar
  6. 6.
    Dafermos, C.M.: Entropy and the stability of classical solutions of hyperbolic systems of conservation laws. In Recent mathematical methods in nonlinear wave propagation (Montecatini Terme, 1994), volume 1640 of Lecture Notes in Math., pages 48–69. Springer, Berlin (1996)Google Scholar
  7. 7.
    Dafermos, C.M.: Hyperbolic conservation laws in continuum physics, vol. 325. Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (2000)Google Scholar
  8. 8.
    DiPerna R.J.: Uniqueness of solutions to hyperbolic conservation laws. Indiana Univ. Math. J., 28, 137–188 (1979)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Grassin M.: Global smooth solutions to Euler equations for a perfect gas. Indiana Univ. Math. J., 47, 1397–1432 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Hoff, D.: The sharp form of Oleĭnik’s entropy condition in several space variables. Trans. Am. Math. Soc., 276, 707–714 (1983)Google Scholar
  11. 11.
    Serre D.: Solutions classiques globales des équations d’Euler pour un fluide parfait compressible. Ann. Inst. Fourier, 47, 139–153 (1997)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.UMPA, UMR CNRS–ENS Lyon # 5669, École Normale Supérieure de LyonLyon Cedex 07France

Personalised recommendations