Skip to main content
Log in

The Riemann Problem for Chaplygin Gas Flow in a Duct with Discontinuous Cross-Section

  • Published:
Chinese Annals of Mathematics, Series B Aims and scope Submit manuscript


The fluid flows in a variable cross-section duct are nonconservative because of the source term. Recently, the Riemann problem and the interactions of the elementary waves for the compressible isentropic gas in a variable cross-section duct were studied. In this paper, the Riemann problem for Chaplygin gas flow in a duct with discontinuous cross-section is studied. The elementary waves include rarefaction waves, shock waves, delta waves and stationary waves.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Andrianov, N. and Warnecke, G., On the solution to the Riemann problem for the compressible duct flow, SIAM J. Appl. Math., 64, 2004, 878–901.

    Article  MathSciNet  Google Scholar 

  2. Baer, M. R. and Nunziato, J. W., A two-phase mixture theory for the deflagration-todetonation transition (DDT) in reactive granular materials, Int. J. Multiphase Flows, 12, 1986, 861–889.

    Article  Google Scholar 

  3. Brenier, Y., Solutions with concentration to the Riemann problem for one-dimensional Chaplygin gas equations, J. Math. Fluid Mech., 7, 2005, 326–331.

    Article  MathSciNet  Google Scholar 

  4. Chaplygin, S., On gas jets, Sci. Mem. Moscow Univ. Math. Phys., 21, 1904, 1–121.

    Google Scholar 

  5. Dal Maso, G., LeFloch, P. G. and Murat, F., Definition and weak stability of nonconservative products, J. Math. Pures Appl., 74(9), 1995, 483–548.

    MathSciNet  MATH  Google Scholar 

  6. Guo, L. H., Sheng, W. C. and Zhang, T., The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system, Commun. Pure Appl. Anal., 9(2), 2010, 431–458.

    Article  MathSciNet  Google Scholar 

  7. Kong, D. X. and Wang, Y. Z., Global existence of smooth solutions to two-dimensional compressible isentropic Euler equations for Chaplygin gases, Sci. China Math., 53(3), 2010, 719–738.

    Article  MathSciNet  Google Scholar 

  8. Kroner, D., LeFloch, P. G. and Thanh, M. D., The minimum entropy principle for fluid flows in a nozzle with discontinuous cross-section, M2AN Math. Model Numer. Anal., 42, 2008, 425–442.

    Article  MathSciNet  Google Scholar 

  9. Kroner, D. and Thanh, M. D., Numerical solutions to compressible flows in a nozzle with variable cross-section, SIAM J. Numer. Anal., 43, 2005, 796–824.

    Article  MathSciNet  Google Scholar 

  10. Lai, G., Sheng, W. C. and Zheng, Y. X., Simple waves and pressure delta waves for a Chaplygin gas in multi-dimensions, Discrete Contin. Dyn. Syst., 31(2), 2011, 489–523.

    Article  MathSciNet  Google Scholar 

  11. Lax, P. D., Shock waves and entropy, Contributions to Functional Analysis, Zarantonello, E. A. (ed.), Academic Press, New York, 1971, 603–634.

  12. LeFloch, P. G., Shock Waves for Nonlinear Hyperbolic Systems in Nonconservative Form, Institute for Mathematics and its Application, Univ. of Minnesota, 593, preprint, 1989.

  13. LeFloch, P. G. and Thanh, M. D., The Riemann problem for fluid flows in a nozzle with discontinuous cross-section, Commun. Math. Sci., 1, 2003, 763–797.

    Article  MathSciNet  Google Scholar 

  14. Marchesin, D. and Paes-Leme, P. J., A Riemann problem in gas dynamics with bifurcation, Comput. Math. Appl. Part A, 12, 1986, 433–455.

    MathSciNet  MATH  Google Scholar 

  15. Serre, D., Multidimensional shock interaction for a Chaplygin gas, Arch. Ration. Mech. Anal., 191, 2009, 539–577.

    Article  MathSciNet  Google Scholar 

  16. Sheng, W. C. and Zhang, Q. L., Interaction of the elementary waves of isentropic flow in a variable cross-section duct, Commun. Math. Sci., 16(6), 2018, 1659–1684.

    Article  MathSciNet  Google Scholar 

  17. Sheng, W. C. and Zhang, T., The Riemann problem for transportation equations in gas dynamics, Mem. Amer. Math. Soc., 137 (654), 1999, viii+77pp.

  18. Thanh, M. D., The Riemann problem for a nonisentropic fluid in a nozzle with discontinuous cross-sectional area, SIAM J. Appl. Math., 69, 2009, 1501–1519.

    Article  MathSciNet  Google Scholar 

  19. Tsien, H. S., Two dimensional subsonic flow of compressible fluids, J. Aeronaut. Sci., 6, 1939, 399–407.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Wancheng Sheng.

Additional information

This work was supported by the National Natural Science Foundation of China (Nos. 11371240, 11771274).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dang, L., Sheng, W. The Riemann Problem for Chaplygin Gas Flow in a Duct with Discontinuous Cross-Section. Chin. Ann. Math. Ser. B 41, 531–546 (2020).

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI:


2000 MR Subject Classification