Concentration Phenomenon of Riemann Solutions for the Relativistic Euler Equations with the Extended Chaplygin Gas

Abstract

The solutions of the Riemann problem for the isentropic relativistic Euler equations with the extended Chaplygin gas are constructed completely for all the possible cases. The asymptotic limits of solutions to the Riemann problem for the relativistic Euler equations are captured in detail when the equation of state of extended Chaplygin gas becomes the one of Chaplygin gas. It is shown that the formations of delta shock wave solution and two-contact-discontinuity solution are derived and analyzed rigorously during the limiting process.

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

Fig. 1
Fig. 2

References

  1. 1.

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

    MathSciNet  Article  Google Scholar 

  2. 2.

    Marti, J.M., Muller, E.: The analytical solution of the Riemann problem in relativistic hydrodynamics. J. Fluid Mech. 258, 317–333 (1994)

    MathSciNet  Article  Google Scholar 

  3. 3.

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

    MathSciNet  Article  Google Scholar 

  4. 4.

    Chen, G., Li, Y.: Stability of Riemann solutions with large oscillation for the relativistic Euler equations. J. Differ. Equ. 202, 332–353 (2004)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Li, Y., Feng, D., Wang, Z.: Global entropy solutions to the relativistic Euler equations for a class of large initial data. Z. Angew. Math. Phys. 56, 239–253 (2005)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Pourhassan, B., Kahya, E.O.: Extended Chaplygin gas model. Results Phys. 4, 101–102 (2014)

    Article  Google Scholar 

  7. 7.

    Pourhassan, B.: Extended Chaplygin gas in Horava-Lifshitz gravity. Phys. Dark Universe 13, 132–138 (2016)

    Article  Google Scholar 

  8. 8.

    Geng, Y., Li, Y.: Non-relativistic global limits of entropy solutions to the extremely relativistic Euler equations. Z. Angew. Math. Phys. 61, 201–220 (2010)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Ding, M., Li, Y.: Non-relativistic limits of rarefaction wave to the 1-D piston problem for the isentropic relativistic Euler equations. J. Math. Phys. 58, 081510 (2017)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Guo, L., Li, T., Yin, G.: The vanishing pressure limits of Riemann solutions to the Chaplygin gas equations with a source term. Commun. Pure Appl. Anal. 16, 295–309 (2017)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Guo, L., Zhang, Y., Yin, G.: Interactions of delta shock waves for the relativistic Chaplygin Euler equations with split delta functions. Math. Methods Appl. Sci. 38, 2132–2148 (2015)

    MathSciNet  Article  Google Scholar 

  12. 12.

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

    MathSciNet  Article  Google Scholar 

  13. 13.

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

    MathSciNet  Article  Google Scholar 

  14. 14.

    Chen, G.Q., 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)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Mitrovic, D., Nedeljkov, M.: Delta-shock waves as a limit of shock waves. J. Hyperbolic Differ. Equ. 4, 629–653 (2007)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Shen, C., Sun, M.: The singular limits of Riemann solutions to a chemotaxis model with flux perturbation. Z. Angew. Math. Mech. 99, e201800046 (2019)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Shen, C., Sun, M.: Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed Aw-Rascle model. J. Differ. Equ. 249, 3024–3051 (2010)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Sen, A., Raja Sekhar, T.: Structural stability of the Riemann solution for a strictly hyperbolic system of conservation laws with flux approximation. Commun. Pure Appl. Anal. 18, 931–942 (2018)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Sun, M.: Concentration and cavitation phenomena of Riemann solutions for the isentropic Euler system with the logarithmic equation of state. Nonlinear Anal., Real World Appl. 53, 103068 (2020)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Yin, G., Sheng, W.: Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations. Chin. Ann. Math., Ser. B 29, 611–622 (2009)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Yin, G., Sheng, W.: Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for the polytropic gases. J. Math. Anal. Appl. 335, 594–605 (2009)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Li, H., Shao, Z.: Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for generalized Chaplygin gas. Commun. Pure Appl. Anal. 15, 2373–2400 (2016)

    MathSciNet  Article  Google Scholar 

  23. 23.

    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)

    MathSciNet  Article  Google Scholar 

  24. 24.

    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)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Nedeljkov, M.: Higher order shadow waves and delta shock blow up in the Chaplygin gas. J. Differ. Equ. 256, 3859–3887 (2014)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Nedeljkov, M., Ruzicic, S.: On the uniqueness of solution to generalized Chaplygin gas. Discrete Contin. Dyn. Syst. 37, 4439–4460 (2017)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Shen, C.: The Riemann problem for the Chaplygin gas equations with a source term. Z. Angew. Math. Mech. 96, 681–695 (2016)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Zhang, Y., Sun, M., Lin, X.: Solutions with concentration and cavitation to the Riemann problem for the isentropic relativistic Euler system for the extended Chaplygin gas. Open Math. 17, 220–241 (2019)

    MathSciNet  Article  Google Scholar 

  29. 29.

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

    MathSciNet  Article  Google Scholar 

  30. 30.

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

    MathSciNet  Article  Google Scholar 

  31. 31.

    Tong, M., Shen, C.: The limits of Riemann solutions for the isentropic Euler system with extended Chaplygin gas. Appl. Anal. 98, 2668–2687 (2019)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Sun, M.: Singular solutions to the Riemann problem for a macroscopic production model. Z. Angew. Math. Mech. 97, 916–931 (2017)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Danilov, V.G., Shelkovich, V.M.: Dynamics of propagation and interaction of \(\delta \)-shock waves in conservation law systems. J. Differ. Equ. 211, 333–381 (2005)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Danilov, V.G., Mitrovic, D.: Delta shock wave formation in the case of triangular hyperbolic system of conservation laws. J. Differ. Equ. 245, 3704–3734 (2008)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Danilov, V.G.: Nonsmooth nonoscillating exponential-type asymptotics for linear parabolic PDE. SIAM J. Math. Anal. 49, 3550–3572 (2017)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Ibrahim, M., Liu, F., Liu, S.: Concentration of mass in the pressureless limit of Euler equations for power law. Nonlinear Anal., Real World Appl. 47, 224–235 (2019)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Sheng, S., Shao, Z.: Concentration of mass in the pressureless limit of the Euler equations of one-dimensional compressible fluid flow. Nonlinear Anal., Real World Appl. 52, 103039 (2020)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the two anonymous referees for their very valuable comments and suggestions, which improves the original manuscript greatly. This work is partially supported by Natural Science Foundation of Shandong Province (ZR2019MA019).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Meina Sun.

Additional information

Publisher’s Note

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

This work is partially supported by Shandong Provincial Natural Science Foundation (ZR2019MA019).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zhang, Y., Sun, M. Concentration Phenomenon of Riemann Solutions for the Relativistic Euler Equations with the Extended Chaplygin Gas. Acta Appl Math (2020). https://doi.org/10.1007/s10440-020-00345-7

Download citation

Keywords

  • Isentropic relativistic Euler equations
  • Extended Chaplygin gas
  • Delta shock wave
  • Riemann problem

Mathematics Subject Classification (2010)

  • 35L65
  • 35L67
  • 76N15