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Indian Journal of Pure and Applied Mathematics

, Volume 49, Issue 4, pp 671–688 | Cite as

Existence and Stability of Riemann Solution to the Aw-Rascle Model with Friction

  • Gan YinEmail author
  • Jianjun Chen
Article
  • 38 Downloads

Abstract

This manuscript is concerned with the Riemann problem and the stability of the Riemann solution of the Aw-Rascle (AR) model with a source term. There exists a unique Riemann solution to the AR model with friction. The problem with three-piecewiseconstant initial data is studied ulteriorly to reach the results on interactions of waves without the vacuum. Based on these results, the Riemann solution turns out to be stable for such a small perturbed initial data.

Key words

Riemann problem interaction of waves shock wave rarefaction wave Aw-Rascle model 

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References

  1. 1.
    A. Aw, A. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader model, SIAM J. Appl. Math., 63 (2002), 259–278.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    A. Aw and M. Rascle, Resurrection of second order models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916–938.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    F. Berthelin, P. Degond, M. Delitata and M. Rascle, A model for the formation and evolution of traffic jams, Arch. Rational Mech. Anal., 187 (2008), 185–220.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    T. Chang and L. Hsiao, The Riemann problem and interaction of waves in gas dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics 41, Longman Scientific and Technical, Harlow, UK, 1989.zbMATHGoogle Scholar
  5. 5.
    C. M. Dafermos, Hyperbolic conservation laws in continuum physics, Grundlehren Math. Wiss., Springer-Verlag, Berlin, 2000.CrossRefGoogle Scholar
  6. 6.
    C. Daganzo, Requiem for second order fluid approximations of traffic flow, Transportaion Research Part B, 29 (1995), 277–286.CrossRefGoogle Scholar
  7. 7.
    D. A. E. Daw and M. Nedeljkov, Shadow waves for pressureless gas balance laws, Applied Mathematics Letters, 57 (2016), 54–59.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    G. Faccanoni and A. Mangeney, Exact solution for granular flows, Int. J. Numer. Anal. Meth. Geomech., 37 (2012), 1408–1433.CrossRefGoogle Scholar
  9. 9.
    L. Guo, T. Li and G. Yin, The vanishing pressure limits of Riemann solutions to the Chaplygin gas equations with a source term, Comm. Pure Appl. Anal., 16(1) (2017), 295–309.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    D. J. Korchinski, Solution of a Riemann problem for a 2 × 2 sytem of conservation laws possessing no classical weak solution, Thesis, Adelphi University, 1977.Google Scholar
  11. 11.
    P. Lefloch, An existence and uniqueness result for two nonstrictly hyperbolic systems, in ”Nonlinear evolution equations that change type”, IMA Volumes in Mathematics and its Applications 27, Springer-Verlag, Berlin/New York, 27 (1990), 126–138.Google Scholar
  12. 12.
    S. B. Savage and K. Hutter, The motion of finite mass of granular material down a rough incline, J. Fluid Mech., 199 (1989), 177–215.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    D. Serre, Systems of conservation laws, 1 and 2, Cambridge University Press, Cambridge, UK, 1999 and 2000.zbMATHGoogle Scholar
  14. 14.
    S. F. Shandarin and Ya. B. Zeldovich, The large-scale structure of the universe: Turbulence, intermittency, structures in a self-gravitating medium, Rev. Mod. Phys., 61 (1989), 185–220.MathSciNetCrossRefGoogle Scholar
  15. 15.
    C. Shen, The Riemann problem for the pressureless Euler system with the Coulomb-like friction term, IMA J. appl. Math., 81 (2016), 76–99.MathSciNetzbMATHGoogle Scholar
  16. 16.
    C. Shen, The Riemann problem for the Chaplygin gas equations with a source term, Z. Angew. Math. Mech., 96 (2016), 681–695.MathSciNetCrossRefGoogle Scholar
  17. 17.
    W. Shens and T. Zhang, The Riemann problem for the transport equations in gas dynamics, Mem. Amer. Math. Soc., AMS, Providence, 137(654) (1999).Google Scholar
  18. 18.
    R. Smith, The Riemann problem in gas dynamics, Trans. Amer. Math. Soc., 249 (1979), 1–50.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    J. Smoller, Shock waves and reaction-diffusion equations, Springer-Verlag, New York, 1994.CrossRefzbMATHGoogle Scholar
  20. 20.
    M. Sun, Interactions of elementary waves for the Aw-Rascle model, SIAM J. Appl. Math., 69(6) (2009), 1542–1558.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    M. Sun, A note on the interactions of elementary waves for the AR traffic flow model without vacuum, Acta Math. Sci., 31(B) (2011), 1503–1512.MathSciNetzbMATHGoogle Scholar
  22. 22.
    W. E. Yu, G. Rykov and G. Ya, Sinai, Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Commun. Math. Phys., 177 (1996), 349–380.CrossRefzbMATHGoogle Scholar

Copyright information

© The Indian National Science Academy 2018

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang University of Science & TechnologyHangzhouChina

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