Archive for Rational Mechanics and Analysis

, Volume 189, Issue 1, pp 97–130 | Cite as

Continuous Dependence of Entropy Solutions to the Euler Equations on the Adiabatic Exponent and Mach Number

  • Gui-Qiang Chen
  • Cleopatra Christoforou
  • Yongqian Zhang


We establish the L1-estimates for continuous dependence of entropy solutions to the full Euler equations away from the vacuum on two physical parameters: the adiabatic exponent γ → 1 that passes from the non-isentropic to isothermal Euler equations and the Mach number \(\tt{M} \to 0\) that passes from the compressible to incompressible Euler equations. Our analysis involves the effective approach developed in our earlier work and additional new techniques that generalize this approach to the setting of the full Euler equations.


Mach Number Euler Equation Shock Front Continuous Dependence Entropy Solution 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Asakura F.: Wave-front tracking for the equations of non-isentropic gas dynamics (preprint)Google Scholar
  2. 2.
    Bianchini S., Colombo R. (2002) On the stability of the standard Riemann semigroup. Proc. Am. Math. Soc. 130, 1961–1973MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bressan A. (2000) Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem. Oxford University Press, OxfordMATHGoogle Scholar
  4. 4.
    Bressan A. (1995) The unique limit of the Glimm scheme. Arch. Ration. Mech. Anal. 130, 105–230MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bressan A., Liu T.-P., Yang T. (1999) L 1 stability estimates for n ×  n conservation laws. Arch. Ration. Mech. Anal. 149, 1–22MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chen, G.-Q.: Euler Equations and Related Hyperbolic Conservation Laws. Chapter 1, In: Handbook on Differential Equations, Vol. 2. Elsevier/North-Holland, Amsterdam, 1–104, 2005Google Scholar
  7. 7.
    Chen G.-Q., Christoforou C., Zhang Y. (2007) Dependence of entropy solutions with large oscillations to the Euler equations on nonlinear flux functions. Indiana Univ. Math. J. 56, 2535–2568MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chen G.-Q., Wagner D. (2003) Global entropy solutions to exothermically reacting, compressible Euler equations. J. Differ. Equ. 191, 277–322ADSMathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Colombo R.M., Risebro N.H. (1998) Continuous dependence in the large for some equations of gas dynamics. Commun. Partial Differ. Equ. 23, 1693–1718MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Courant R., Friedrichs K.O. (1948) Supersonic Flow and Shock Waves. Interscience, New YorkMATHGoogle Scholar
  11. 11.
    Dafermos C.M. (2005) Hyperbolic Conservation Laws in Continuum Physics, 2nd edn. Springer, BerlinCrossRefMATHGoogle Scholar
  12. 12.
    Glimm J. (1965) Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math. 18, 95–105MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Holden H., Risebro N.H. (2002) Front Tracking for Hyperbolic Conservation Laws. Springer, New YorkCrossRefMATHGoogle Scholar
  14. 14.
    Klainerman S., Majda A. (1982) Compressible and incompressible fluids. Comm. Pure Appl. Math. 35, 629–653ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Lax P.D. (1957) Hyperbolic systems of conservation laws II. Comm. Pure Appl. Math. 10, 537–566MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    LeFloch Ph. (2002) Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves. Birkhäuser, BaselCrossRefMATHGoogle Scholar
  17. 17.
    Liu T.-P. (1977) Solutions in the large for the equations of nonisentropic gas dynamics. Indiana Univ. Math. J. 26, 147–177ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Majda A. (1984) Compressible Fluid Flow and Systems of Conservation Law in Several Space Variables. Springer, New YorkCrossRefMATHGoogle Scholar
  19. 19.
    Metivier G., Schochet S. (2001) The incompressible limit of the non-isentropic Euler equations. Arch. Ration. Mech. Anal. 158, 61–90MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Nishida T. (1968) Global solution for an initial-boundary-value problem of a quasilinear hyperbolic system. Proc. Jap. Acad. 44, 642–646MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Nishida T., Smoller J. (1973) Solutions in the large for some nonlinear hyperbolic conservation laws. Comm. Pure Appl. Math. 26, 183–200MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Saint-Raymond L. (2000) Isentropic approximation of the compressible Euler system in one space dimension. Arch. Ration. Mech. Anal. 155, 171–199MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Schochet S. (2005) The mathematical theory of low Mach number flows. ESAIM: Math. Model. Numer. Anal. 39, 441–458MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Serre D. (1999) Systems of Conservation Laws I & II. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  25. 25.
    Smoller J. (1994) Shock Waves and Reaction–Diffusion Equations. Springer, New YorkCrossRefMATHGoogle Scholar
  26. 26.
    Temple B. (1981) Solutions in the large for the nonlinear hyperbolic conservation laws of gas dynamics. J. Differ. Equ. 41, 96–161ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Gui-Qiang Chen
    • 1
    • 2
  • Cleopatra Christoforou
    • 2
    • 3
  • Yongqian Zhang
    • 4
  1. 1.School of Mathematical SciencesFudan UniversityShanghaiPeople’s Republic of China
  2. 2.Department of MathematicsNorthwestern UniversityEvanstonUSA
  3. 3.Department of MathematicsUniversity of HoustonHoustonUSA
  4. 4.School of Mathematical Sciences and Key Laboratory of Mathematics for Nonlinear Sciences (Ministry of Education)Fudan UniversityShanghaiPeople’s Republic of China

Personalised recommendations