Skip to main content
SpringerLink
Log in

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

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

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.

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

Access this article

Log in via an institution

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. 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)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  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. Bressan A., LeFloch P.G.: Uniqueness of weak solutions to systems of conservation laws. Arch. Ration. Mech. Anal. 140(4), 301–317 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chen G.-Q., Dafermos C.M.: The vanishing viscosity method in one-dimensional thermoelasticity. Trans. Am. Math. Soc. 347(2), 531–541 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cheverry C.: Systèmes de lois de conservation et stabilité BV. Mémoires de la Société Mathématique de France. Nouvelle Série 75, 1–103 (1998)

    MATH  Google Scholar 

  6. Dafermos C.M.: Large time behavior of periodic solutions of hyperbolic systems of conservation laws. J. Differ. Equ. 121, 183–202 (1995)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. Dafermos C.M.: Hyperbolic Conservation Laws in Continuum Physics.Grundlehren der Mathematischen Wissenschaften, vol. 325. 3rd edn. Springer, Heidelberg (2010)

    Book  Google Scholar 

  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, 1979

  9. Glimm J.: Solutions in the large for nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math. 18, 697–715 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  10. Glimm J., Lax P.D.: Decay of solutions of systems of nonlinear hyperbolic conservation laws. Mem. Am. Math. Soc. 101, 1–112 (1970)

    MathSciNet  Google Scholar 

  11. LeFloch P.G., Xin, Z.: Formation of singularities in periodic solutions to gas dynamics equations (1993, unpublished)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  ADS  MATH  Google Scholar 

  14. Liu T.-P.: Admissible solutions of hyperbolic conservation laws. Mem. Am. Math. Soc. 30(240), 1–78 (1981)

    Google Scholar 

  15. Pego R.L.: Some explicit resonating waves in weakly nonlinear gas dynamics. Stud. Appl. Math. 79(3), 263–270 (1988)

    MATH  MathSciNet  Google Scholar 

  16. Smoller J.: Shock Waves and Reaction–Diffusion Equations. Grundlehren der Mathematischen Wissenschaften, vol. 258. Springer, New York (1983)

    Google Scholar 

  17. Temple B., Young R.: The large time stability of sound waves. Commun. Math. Phys. 179(2), 417–466 (1996)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  20. Xiao, J.: Some topics on hyperbolic conservation laws. Master thesis, The Chinese University of Hong Kong, Hong Kong, June 2008

  21. Young R.: Sup-norm stability for Glimm’s scheme. Commun. Pure Appl. Math. 46(6), 903–948 (1993)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Hong Kong, Hong Kong

    Peng Qu & Zhouping Xin

Authors
  1. Peng Qu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhouping Xin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peng Qu.

Additional information

Communicated by T.-P. Liu

This research is partially supported by the Zheng Ge Ru Foundation, Hong Kong RGC Earmarked Research Grants CUHK-4041/11P and CUHK-4048/13P, a Focus Area Grant from the Chinese University of Hong Kong, and a grant from the Croucher Foundation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qu, P., Xin, Z. Long Time Existence of Entropy Solutions to the One-Dimensional Non-isentropic Euler Equations with Periodic Initial Data. Arch Rational Mech Anal 216, 221–259 (2015). https://doi.org/10.1007/s00205-014-0807-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-014-0807-0

Keywords

Search

Navigation