Skip to main content
SpringerLink
Log in

Limits of Solutions to the Relativistic Euler Equations for Modified Chaplygin Gas by Flux Approximation

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

We study the Riemann problem and flux-approximation limits of solutions to the relativistic Euler equations with the state equation for modified Chaplygin gas. The limits of Riemann solutions for the relativistic modified Chaplygin gas equations and the corresponding flux-approximation system are discussed when the pressure term and flux-perturbation parameters tend to zero. It is rigorously proved that, as the pressure and flux approximations vanish respectively, any two-shock-wave Riemann solution tends to a delta-shock solution to the pressureless relativistic Euler equations, and the intermediate density between them tends to a weighted \(\delta \)-measure that forms a delta shock wave. Correspondingly, any two-rarefaction-wave solution becomes two contact discontinuities connecting the constant states and vacuum states, which form a vacuum solution.

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

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Benaoum, H.B.: Accelerated universe from modified Chaplygin gas and tachyonic fluid. arXiv:hep-th/0205140

  2. Chaplygin, S.: On Gas Jets. Sci. Mem. Moscow Univ. Math. Phys., vol. 21, pp. 1–121 (1904)

    Google Scholar 

  3. Chen, J.: Conservation laws for the relativistic P-system. Commun. Partial Differ. Equ. 20, 1605–1646 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, G., Liu, H.: Formation of delta-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids. SIAM J. Math. Anal. 34, 925–938 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, G., Liu, H.: Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids. Physica D 189, 141–165 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cheng, H., Yang, H.: Riemann problem for the relativistic Chaplygin Euler equations. J. Math. Anal. Appl. 381, 17–26 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Li, J.: Note on the compressible Euler equations with zero temperature. Appl. Math. Lett. 14, 519–523 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Santos, F.C., Bedran, M.L., Soares, V.: On the thermodynamic stability of the modified Chaplygin gas. Phys. Lett. B 646, 215–221 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Setare, M.R.: Interacting holographic generalized Chaplygin gas model. Phys. Lett. B 654, 1–6 (2007). arXiv:0708.0118

    Article  Google Scholar 

  10. Sheng, W., Zhang, T.: The Riemann problem for transportation equation in gas dynamics. Mem. Am. Math. Soc. 137(654), 1–77 (1999)

    MathSciNet  Google Scholar 

  11. Smoller, J., Temple, B.: Global solutions of the relativistic Euler equations. Commun. Math. Phys. 156, 67–99 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  12. Taub, A.H.: Approximate solutions of the Einstein equations for isentropic motions of plane-symmetric distributions of perfect fluids. Phys. Rev. 107, 884–900 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  13. Thompson, K.: The special relativistic shock tube. J. Fluid Mech. 171, 365–373 (1986)

    Article  MATH  Google Scholar 

  14. Yang, H.: Riemann problems for a class of coupled hyperbolic systems of conservation laws. J. Differ. Equ. 159, 447–484 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. Yang, H., Liu, J.: Delta-shocks and vacuums in zero-pressure gas dynamics by the flux approximation. Sci. China Math. 58(11), 2329–2346 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Yang, H., Liu, J.: Concentration and cavitation in the Euler equations for nonisentropic fluids with the flux approximation. Nonlinear Anal. 123–124, 158–177 (2015)

    MathSciNet  MATH  Google Scholar 

  17. Yang, H., Wang, J.: Delta-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations for modified Chaplygin gas. J. Math. Anal. Appl. 413, 800–820 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Yang, H., Wang, J.: Concentration in vanishing pressure limit of solutions to the modified Chaplygin gas equations. J. Math. Phys. 57(11), 111504 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Yang, H., Zhang, Y.: Flux approximation to the isentropic relativistic Euler equations. Nonlinear Anal., Theory Methods Appl. 133, 200–227 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Yin, G., Sheng, W.: Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for polytropic gases. J. Math. Anal. Appl. 355, 594–605 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Yin, G., Sheng, W.: Delta wave formation and vacuum state in vanishing pressure limit for system of conservation laws to relativistic fluid dynamics. Z. Angew. Math. Mech. 95, 49–65 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Yin, G., Song, K.: Vanishing pressure limits of Riemann solutions to the isentropic relativistic Euler system for Chaplygin gas. J. Math. Anal. Appl. 411, 506–521 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhang, Y., Yang, H.: Flux-approximation limits of solutions to the relativistic Euler equations for polytropic gas. J. Math. Anal. Appl. 435, 1160–1182 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhang, Y., Zhang, Y.: Delta-shocks and vacuums in the relativistic Euler equations for isothermal fluids with the flux approximation. J. Math. Phys. 60(1), 011508 (2019)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Yunnan University, Kunming, 650091, P.R. China

    Yu Zhang & Hanchun Yang

  2. Department of Mathematics, Yunnan Normal University, Kunming, 650500, P.R. China

    Yu Zhang

Authors
  1. Yu Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hanchun Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yu Zhang.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is supported by the NSF of China (11361073), Yunnan Applied Basic Research Projects (2018FD015) and the Scientific Research Foundation Project of Yunnan Education Department (2018JS150).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, Y., Yang, H. Limits of Solutions to the Relativistic Euler Equations for Modified Chaplygin Gas by Flux Approximation. Acta Appl Math 169, 1–32 (2020). https://doi.org/10.1007/s10440-019-00286-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-019-00286-w

Keywords

Search

Navigation