On entropy weak solutions of Hughes’ model for pedestrian motion

An Erratum to this article was published on 18 September 2014


We consider a generalized version of Hughes’ macroscopic model for crowd motion in the one-dimensional case. It consists in a scalar conservation law accounting for the conservation of the number of pedestrians, coupled with an eikonal equation giving the direction of the flux depending on pedestrian density. As a result of this non-trivial coupling, we have to deal with a conservation law with space–time discontinuous flux, whose discontinuity depends non-locally on the density itself. We propose a definition of entropy weak solution, which allows us to recover a maximum principle. Moreover, we study the structure of the solutions to Riemann-type problems, and we construct them explicitly for small times, depending on the choice of the running cost in the eikonal equation. In particular, aiming at the optimization of the evacuation time, we propose a strategy that is optimal in the case of high densities. All results are illustrated by numerical simulations.

This is a preview of subscription content, access via your institution.


  1. 1

    Amadori D., Di Francesco M.: The one-dimensional Hughes model for pedestrian flow: Riemann-type solutions. Acta Math. Sci. 32(1), 259–280 (2012)

    MathSciNet  MATH  Google Scholar 

  2. 2

    Bardi M., Capuzzo-Dolcetta I.: Optimal control and viscosity solutions of Hamilton–Jacobi–Bellman equations. Systems & Control: Foundations & Applications. Birkhäuser Boston Inc., Boston, MA, With appendices by Maurizio Falcone and Pierpaolo Soravia. (1997)

  3. 3

    Bardos C., Roux A.Y., Nédélec J.-C.: First order quasilinear equations with boundary conditions. Comm. Partial Different. Equ. 4(9), 1017–1034 (1979)

    MATH  Article  Google Scholar 

  4. 4

    Chalons C.: Numerical approximation of a macroscopic model of pedestrian flows. SIAM J. Sci. Comput 29(2), 539–555 (electronic), (2007)

  5. 5

    Colombo R.M., Goatin P., Rosini M.D.: A macroscopic model for pedestrian flows in panic situations. In Current Advances in Nonlinear Analysis and Related Topics, vol. 32 of. GAKUTO International Series in Mathematical Science and Applications, pp. 255–272. Gakkōtosho, Tokyo, (2010)

  6. 6

    Colombo R.M., Rosini M.D.: Pedestrian flows and non-classical shocks. Math. Methods Appl. Sci 28(13), 1553–1567 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7

    Colombo R.M., Rosini M.D.: Existence of nonclassical solutions in a pedestrian flow model. Nonlinear Anal. Real World Appl. 10(5), 2716–2728 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8

    Dafermos C.M.: Hyperbolic Conservation Laws in Continuum Physics Volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Second Edition. Springer, Berlin (2005)

    Google Scholar 

  9. 9

    Di Francesco M., Markowich P.A., Pietschmann J.-F., Wolfram M.-T.: On the Hughes’ model for pedestrian flow: the one-dimensional case. J. Different. Equ. 250(3), 1334–1362 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10

    Hughes R.L.: A continuum theory for the flow of pedestrians. Transp. Res. B 36(6), 507–535 (2002)

    Article  Google Scholar 

  11. 11

    Hughes R.L.: The flow of human crowds. In Annual Review of Fluid Mechanics, Vol. 35, pp. 169–182. Annual Reviews, Palo Alto, CA (2003)

  12. 12

    Karlsen K.H., Towers J.D.: Convergence of the Lax-Friedrichs scheme and stability for conservation laws with a discontinuous space-time dependent flux. Chin. Ann. Math. Ser. B 25(3), 287–318 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13

    Kružhkov S.N.: First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81(123), 228–255 (1970)

    MathSciNet  Google Scholar 

  14. 14

    Mishra S.: Convergence of upwind finite difference schemes for a scalar conservation law with indefinite discontinuities in the flux function. SIAM J. Numer. Anal. 43(2), 559–577 (electronic) (2005)

    Google Scholar 

  15. 15

    Rosini M.D.: Nonclassical interactions portrait in a macroscopic pedestrian flow model. J. Different. Equ. 246(1), 408–427 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16

    Rusanov V.V.: The calculation of the interaction of non-stationary shock waves with barriers. Ž Vyčisl. Mat. i Mat. Fiz 1, 267–279 (1961)

    MathSciNet  Google Scholar 

  17. 17

    Soravia P.: Boundary value problems for Hamilton-Jacobi equations with discontinuous Lagrangian. Indiana Univ. Math. J 51(2), 451–477 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18

    Towers J.D.: Convergence of a difference scheme for conservation laws with a discontinuous flux. J. Numer. Anal 38(2), 681–698 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19

    Vasseur A.: Strong traces for solutions of multidimensional scalar conservation laws. Arch. Ration. Mech. Anal 160(3), 181–193 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20

    Zhao H.: A fast sweeping method for eikonal equations. Math. Comp 74(250), 603–627 (electronic) (2005)

Download references

Author information



Corresponding author

Correspondence to Paola Goatin.

Additional information

This research was supported by the ERC Starting Grant 2010 under the project “TRAffic Management by Macroscopic Models” and the Polonium 2011 (French–Polish cooperation program) under the project “CROwd Motion Modeling and Management”. The third author was partially supported by Narodowe Centrum Nauki, grant 4140.

An erratum to this article is available at http://dx.doi.org/10.1007/s00033-014-0448-z.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

El-Khatib, N., Goatin, P. & Rosini, M.D. On entropy weak solutions of Hughes’ model for pedestrian motion. Z. Angew. Math. Phys. 64, 223–251 (2013). https://doi.org/10.1007/s00033-012-0232-x

Download citation

Mathematics Subject Classification (2000)

  • Primary 35L65
  • Secondary 35F21
  • 90B20


  • Pedestrian flow
  • Conservation laws
  • Eikonal equation
  • Weak entropy solutions
  • Numerical approximations