Skip to main content
SpringerLink
Log in

Riemann Problem for a Class of Non-strictly Hyperbolic Systems of Conservation Laws

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

This paper studies the Riemann problem for a class of non-strictly hyperbolic systems of conservation laws modeling polymer flooding in enhanced oil recovery. The Riemann solutions contain two kinds. One consists of two contact discontinuities and a vacuum state, the other involves the delta shock wave. The generalized Rankine-Hugoniot relations and entropy condition are proposed to solve delta shock wave. Besides, by the vanishing viscosity approach, the stability of the Riemann solution involving the delta shock wave is proved.

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. Aw, A., Rascle, M.: Resurrection of “second order” models of traffic flow. SIAM J. Appl. Math. 60, 916–938 (2000)

    Article  MathSciNet  Google Scholar 

  2. Barkve, T.: The Riemann problem for a nonstrictly hyperbolic system modeling nonisothermal two-phase flow in a porous medium. SIAM J. Appl. Math. 49, 784–798 (1989)

    Article  MathSciNet  Google Scholar 

  3. Dafermos, C.: Solution of the Riemann problem for a class of hyperbolic systems of conservation laws by the viscosity method. Arch. Ration. Mech. Anal. 52, 1–9 (1973)

    Article  MathSciNet  Google Scholar 

  4. Dafermos, C., Diperna, R.: The Riemann problem for certain classes of hyperbolic systems of conservation laws. J. Differ. Equ. 20, 90–114 (1976)

    Article  MathSciNet  Google Scholar 

  5. Fayers, F.: Some theoretical results concerning the displacement of a viscous oil by a hot fluid in a porous medium. J. Fluid Mech. 13, 65–76 (1962)

    Article  Google Scholar 

  6. Hu, J.: A limiting viscosity approach to Riemann solutions containing delta-shock waves for nonstrictly hyperbolic conservation laws. Q. Appl. Math. 55, 361–373 (1997)

    Article  MathSciNet  Google Scholar 

  7. James, F., Peng, Y., Perthame, B.: Kinetic formulation for chromatography and some other hyperbolic systems. J. Math. Pures Appl. 74, 367–385 (1995)

    MathSciNet  MATH  Google Scholar 

  8. Johansen, T., Winther, R.: The solution of the Riemann problem for a hyperbolic system of conservation laws modeling polymer flooding. SIAM J. Math. Anal. 19, 541–566 (1988)

    Article  MathSciNet  Google Scholar 

  9. Kalasnikov, S.: Construction of generalized solutions of quasi-linear equations of first order without convexity conditions as limits of solutions of parabolic equations with a small parameter. Dokl. Akad. Nauk SSSR 127, 27–30 (1959)

    MathSciNet  MATH  Google Scholar 

  10. Keyfitz, B., Kranzer, H.: A system of nonstrictly hyperbolic conservation laws arising in elasticity. Arch. Ration. Mech. Anal. 72, 219–241 (1980)

    Article  MathSciNet  Google Scholar 

  11. Korchinski, D.: Solution of a Riemann problem for a \(2\times 2\) system of conservation laws possessing no classical weak solution. Thesis, Adelphi University (1977)

  12. Kranzer, H., Keyfitz, B.: A strictly hyperbolic system of conservation laws admitting singular shock. In: Nonlinear Evolution Equations That Change Type. IMA Vol. Math. Appl., vol. 27. Springer, Berlin (1990)

    Chapter  Google Scholar 

  13. Le Floch, P.: An existence and uniqueness result for two nonstrictly hyperbolic systems. In: Nonlinear Evolution Equations That Change Type. IMA Vol. Math. Appl., vol. 27, pp. 126–138. Springer, New York (1990)

    Chapter  Google Scholar 

  14. Li, J., Zhang, T.: Generalized Rankine-Hugoniot relations of delta-shocks in solutions of transportation equations. In: Chen, G.Q. (ed.) Nonlinear PDE and Related Areas, pp. 219–232. World Scientific, Singapore (1998)

    Google Scholar 

  15. Li, J., Zhang, T., Yang, S.: The Two-Dimensional Riemann Problem in Gas Dynamics. Pitman Monogr. Surv. Pure Appl. Math., vol. 98. Longman, Harlow (1998)

    MATH  Google Scholar 

  16. Lu, Y.: Existence of global bounded weak solutions to a non-symmetric system of Keyfitz-Kranzer type. J. Funct. Anal. 261, 2797–2815 (2011)

    Article  MathSciNet  Google Scholar 

  17. Lu, Y.: Existence of global bounded weak solutions to a symmetric system of Keyfitz-Kranzer type. Nonlinear Anal., Real World Appl. 13, 235–240 (2012)

    Article  MathSciNet  Google Scholar 

  18. Lu, Y.: Existence of global entropy solutions to general system of Keyfitz-Kranzer type. J. Funct. Anal. 264, 2457–2468 (2013)

    Article  MathSciNet  Google Scholar 

  19. Lu, Y.: Existence of global weak entropy solutions to some nonstrictly hyperbolic systems. SIAM J. Math. Anal. 45, 3592–3610 (2013)

    Article  MathSciNet  Google Scholar 

  20. Shelkovich, V.: The Riemann problem admitting \(\delta \), \(\delta '\)-shocks, and vacuum states (the vanishing viscosity approach). J. Differ. Equ. 231, 459–500 (2006)

    Article  MathSciNet  Google Scholar 

  21. Sheng, W., Zhang, T.: The Riemann Problem for Transportation Equation in Gas Dynamics. Mem. Amer. Math. Soc. (1999)

    Book  Google Scholar 

  22. Slemrod, M.: A limiting “viscosity” approach to the Riemann problem for materials exhibiting change of phase. Arch. Ration. Mech. Anal. 41, 327–366 (1989)

    Article  MathSciNet  Google Scholar 

  23. Slemrod, M., Tzavaras, A.: A limiting viscosity approach for the Riemann problem in isentropic gas dynamics. Indiana Univ. Math. J. 38, 1047–1074 (1989)

    Article  MathSciNet  Google Scholar 

  24. Tan, D., Zhang, T.: Two-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws I. Four-J cases, II. Initial data involving some rarefaction waves. J. Differ. Equ. 111, 203–282 (1994)

    Article  Google Scholar 

  25. Tan, D., Zhang, T., Zheng, Y.: Delta shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws. J. Differ. Equ. 112, 1–32 (1994)

    Article  MathSciNet  Google Scholar 

  26. Tupciev, A.: On the method of introducing viscosity in the study of problems involving the decay of discontinuity. Dokl. Akad. Nauk SSSR 211, 55–58 (1973)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  28. Yang, H., Zhang, Y.: New developments of delta shock waves and its applications in systems of conservation laws. J. Differ. Equ. 252, 5951–5993 (2012)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of science, Henan University of Engineering, Zhengzhou, 451191, P.R. China

    Shiwei Li

Authors
  1. Shiwei Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shiwei Li.

Additional information

Publisher’s Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, S. Riemann Problem for a Class of Non-strictly Hyperbolic Systems of Conservation Laws. Acta Appl Math 177, 3 (2022). https://doi.org/10.1007/s10440-022-00464-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10440-022-00464-3

Keywords

Search

Navigation