Archive for Rational Mechanics and Analysis

, Volume 216, Issue 1, pp 221–259 | Cite as

Long Time Existence of Entropy Solutions to the One-Dimensional Non-isentropic Euler Equations with Periodic Initial Data

Article
First Online:
Received:
Accepted:

Abstract

The non-isentropic Euler system with periodic initial data in \({{\mathbb{R}}^1}\) is studied by analyzing wave interactions in a framework of specially chosen Riemann invariants, generalizing Glimm’s functionals and applying the method of approximate conservation laws and approximate characteristics. An \({{\mathcal O}(\varepsilon^{-2})}\) lower bound is established for the life span of the entropy solutions with initial data that possess \({\varepsilon}\) variation in each period.

Keywords

Cauchy Problem Entropy Solution Riemann Invariant Compressible Euler Equation Wave Strength 
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.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bianchini S., Colombo R.M.: Monti, F.: 2 × 2 systems of conservation laws with \({L^\infty}\) data. J. Differ. Equ. 249(12), 3466–3488 (2010)ADSCrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bressan A.: Hyperbolic Systems of Conservation Laws: The One Dimensional Cauchy Problem Oxford Lecture Series in Mathematics and its Applications, vol. 20. Oxford University Press, Oxford (2000)Google Scholar
  3. 3.
    Bressan A., LeFloch P.G.: Uniqueness of weak solutions to systems of conservation laws. Arch. Ration. Mech. Anal. 140(4), 301–317 (1997)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Chen G.-Q., Dafermos C.M.: The vanishing viscosity method in one-dimensional thermoelasticity. Trans. Am. Math. Soc. 347(2), 531–541 (1995)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Cheverry C.: Systèmes de lois de conservation et stabilité BV. Mémoires de la Société Mathématique de France. Nouvelle Série 75, 1–103 (1998)MATHGoogle Scholar
  6. 6.
    Dafermos C.M.: Large time behavior of periodic solutions of hyperbolic systems of conservation laws. J. Differ. Equ. 121, 183–202 (1995)ADSCrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Dafermos C.M.: Hyperbolic Conservation Laws in Continuum Physics.Grundlehren der Mathematischen Wissenschaften, vol. 325. 3rd edn. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  8. 8.
    DiPerna, R.J.: The Structure of Solutions to Hyperbolic Conservation Laws. Research Notes in Mathematics, Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, vol. IV, pp. 1–16. Pitman, Boston, 1979Google Scholar
  9. 9.
    Glimm J.: Solutions in the large for nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math. 18, 697–715 (1965)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Glimm J., Lax P.D.: Decay of solutions of systems of nonlinear hyperbolic conservation laws. Mem. Am. Math. Soc. 101, 1–112 (1970)MathSciNetGoogle Scholar
  11. 11.
    LeFloch P.G., Xin, Z.: Formation of singularities in periodic solutions to gas dynamics equations (1993, unpublished)Google Scholar
  12. 12.
    Li T.-T., Kong D.-X.: Blow up of periodic solutions to quasilinear hyperbolic systems. Nonlinear Anal. Theory Methods Appl. 26(11), 1779–1789 (1996)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Liu T.-P.: Development of singularities in the nonlinear waves for quasi-linear hyperbolic partial differential equations. J. Differ. Equ. 33(1), 92–111 (1979)ADSCrossRefMATHGoogle Scholar
  14. 14.
    Liu T.-P.: Admissible solutions of hyperbolic conservation laws. Mem. Am. Math. Soc. 30(240), 1–78 (1981)Google Scholar
  15. 15.
    Pego R.L.: Some explicit resonating waves in weakly nonlinear gas dynamics. Stud. Appl. Math. 79(3), 263–270 (1988)MATHMathSciNetGoogle Scholar
  16. 16.
    Smoller J.: Shock Waves and Reaction–Diffusion Equations. Grundlehren der Mathematischen Wissenschaften, vol. 258. Springer, New York (1983)Google Scholar
  17. 17.
    Temple B., Young R.: The large time stability of sound waves. Commun. Math. Phys. 179(2), 417–466 (1996)ADSCrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Temple B., Young R.: A paradigm for time-periodic sound wave propagation in the compressible Euler equations. Methods Appl. Anal. 16(3), 341–364 (2009)MATHMathSciNetGoogle Scholar
  19. 19.
    Temple B., Young R.: Time-periodic linearized solutions of the compressible Euler equations and a problem of small divisors. SIAM J. Math. Anal. 43(1), 1–49 (2011)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Xiao, J.: Some topics on hyperbolic conservation laws. Master thesis, The Chinese University of Hong Kong, Hong Kong, June 2008Google Scholar
  21. 21.
    Young R.: Sup-norm stability for Glimm’s scheme. Commun. Pure Appl. Math. 46(6), 903–948 (1993)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.The Institute of Mathematical SciencesThe Chinese University of Hong KongHong KongHong Kong

Personalised recommendations