Advertisement

Vanishing Viscosity for Traffic on Networks with Degenerate Diffusivity

  • Giuseppe Maria CocliteEmail author
  • Lorenzo di Ruvo
Article
  • 14 Downloads

Abstract

We consider the vanishing viscosity approximation of the traffic model LWR with degenerate diffusivity on a networks composed by a single junction with n incoming and m outgoing roads. We prove that the solution of the parabolic approximation exists and, as the viscosity vanishes, the solution of the parabolic problem converges to a solution of the original problem.

Keywords

Degenerate diffusivity vanishing viscosity traffic model networks compensated compactness conservation law 

Mathematics Subject Classification

90B20 35L65 

Notes

References

  1. 1.
    Ancona, F., Cesaroni, A., Coclite, G.M., Garavello, M.: On the optimization of conservation law models at a junction. SIAM J. Control Optim. 56(5), 3370–3403 (2018)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Andreianov, B.P., Coclite, G.M., Donadello, C.: Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network. Discrete Contin. Dyn. Syst. 37(11), 5913–5942 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bellomo, N., Delitala, M., Coscia, V.: On the mathematical theory of vehicular traffic flow. I. Fluid dynamic and kinetic modelling. Math. Models Methods Appl. Sci. 12(12), 1801–1843 (2002)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bonzani, I.: Hydrodynamic models of traffic flow: drivers’ behaviour and nonlinear diffusion. Math. Comput. Model. 31, 1–8 (2000)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bonzani, I., Mussone, L.: Stochastic modelling of traffic flow. Math. Comput. Model. 36, 109–119 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bruno, L., Tosin, A., Tricerri, P., Venuti, F.: Non-local first-order modelling of crowd dynamics: a multidimensional framework with applications. Appl. Math. Model. 35(1), 426–445 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Coclite, G. M., Corli, A., di Ruvo, L.: Vanishing viscosity limits of scalar equations with degenerate diffusivity (Submitted)Google Scholar
  8. 8.
    Coclite, G.M., Garavello, M.: Vanishing viscosity for traffic on networks. SIAM J. Math. Anal. 42(4), 1761–1783 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Coclite, G.M., Garavello, M., Piccoli, B.: Traffic flow on a road network. SIAM J. Math. Anal. 36(6), 1862–1886 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Corli, A., di Ruvo, L., Malaguti, L.: Sharp profiles in models of collective movements. Nonlinear Differ. Equ. Appl. 24, 40 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Corli, A., di Ruvo, L., Malaguti, L., Rosini, M.D.: Traveling waves for degenerate diffusive equations on networks. Netw. Heterog. Media 12(3), 339–370 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Corli, A., Malaguti, L.: Semi-wavefront solutions in models of collective movements with density-dependent diffusivity. Dyn. Partial Differ. Equ. 13(4), 297–331 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    De Angelis, E.: Nonlinear hydrodynamic models of traffic flow modelling and mathematical problems. Math. Comput. Model. 29, 83–95 (1999)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Garavello, M., Piccoli, B.: Traffic Flow on Networks: Conservation Laws Models. AIMS Ser. Appl. Math. 1, American Institute of Mathematical Sciences, Springfield, MO (2006)Google Scholar
  15. 15.
    Garavello, M., Piccoli, B.: Entropy-type conditions for Riemann solvers at nodes. Adv. Differ. Equ. 16, 113–144 (2011)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Lighthill, M .J., Whitham, G .B.: On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. R. Soc. Lond. Ser. A. 229, 317–345 (1955)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Murat, F.: L’injection du cône positif de \({H}^{-1}\) dans \({W}^{-1,\,q}\) est compacte pour tout \(q<2\). J. Math. Pures Appl. (9) 60(3), 309–322 (1981)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Nelson, P.: Synchronized traffic flow from a modified Lighthill-Whitham model. Phys. Rev. E 61, R6052–R6055 (2000)CrossRefGoogle Scholar
  19. 19.
    Nelson, P.: Traveling-wave solutions of the diffusively corrected kinematic-wave model. Math. Comput. Model. 35, 561–579 (2002)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Payne, H.J.: Models of freeway traffic and control. Simul. Counc. Proc. 1, 51–61 (1971)Google Scholar
  21. 21.
    Richards, P.I.: Shock waves on the highway. Oper. Res. 4, 42–51 (1956)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Tartar, L.: Compensated compactness and applications to partial differential equations. In: Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, vol. IV, pp. 136–212. Pitman, Boston, Mass (1979)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Dipartimento di Meccanica Matematica e ManagementPolitecnico di BariBariItaly
  2. 2.Dipartimento di MatematicaUniversità di BariBariItaly

Personalised recommendations