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Wavefronts in Traffic Flows and Crowds Dynamics

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Anomalies in Partial Differential Equations

Part of the book series: Springer INdAM Series ((SINDAMS,volume 43))

Abstract

In this paper we give an overview of some recent results concerning partial differential equations modeling collective movements, namely, vehicular traffic flows and crowds dynamics. The focus is on traveling-wave solutions to degenerate parabolic equations in one space dimension, even if we briefly discuss models based on different equations. The case of networks is also taken into consideration. The parabolic degeneration opens the possibilities of several different behaviors of the traveling-wave solutions, which are investigated in details.

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References

  1. Almond, J.: Proceedings of the Second Symposium on the Theory of Road Traffic Flow. Organisation for economic co-operation and development, Paris (1965)

    Google Scholar 

  2. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. The Clarendon Press/Oxford University Press, New York (2000)

    MATH  Google Scholar 

  3. Ambroso, A., Chalons, C., Coquel, F., Godlewski, E., Lagoutière, F., Raviart, P.-A., Seguin, N.: The coupling problem of different thermal-hydraulic models arising in two-phase flow codes for nuclear reactors. In: Coupled Problems 2009, pp. 1–5. CIMNE, Barcelona (2009)

    Google Scholar 

  4. Andreu, F., Caselles, V., Mazón, J.M.: The Cauchy problem for a strongly degenerate quasilinear equation. J. Eur. Math. Soc. (JEMS) 7(3), 361–393 (2005)

    Google Scholar 

  5. Aronson, D.G., Weinberger, H.F.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30(1), 33–76 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  6. Aw, A., Rascle, M.: Resurrection of “second order” models of traffic flow. SIAM J. Appl. Math. 60(3), 916–938 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bagnerini, P. Colombo, R.M., Corli, A.: On the role of source terms in continuum traffic flow models. Math. Comput. Modelling 44(9–10), 917–930 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bando, M., Hasebe, K., Nakayama, A., Shibata, A., Sugiyama, Y.: Dynamical model of traffic congestion and numerical simulation. Phys. Rev. E 51, 1035 (1995)

    Article  Google Scholar 

  9. Bao, L., Zhou, Z.: Traveling wave in backward and forward parabolic equations from population dynamics. Discrete Contin. Dyn. Syst. Ser. B 19(6), 1507–1522 (2014)

    MathSciNet  MATH  Google Scholar 

  10. Bao, L., Zhou, Z.: Traveling wave solutions for a one dimensional model of cell-to-cell adhesion and diffusion with monostable reaction term. Discrete Contin. Dyn. Syst. Ser. S 10(3), 395–412 (2017)

    MathSciNet  Google Scholar 

  11. Bellomo, N., Coscia, V.: First order models and closure of the mass conservation equation in the mathematical theory of vehicular traffic flow. C.R. Mecanique 333, 843–851 (2005)

    Article  MATH  Google Scholar 

  12. Bellomo, N., Dogbe, C.: On the modeling of traffic and crowds: a survey of models, speculations, and perspectives. SIAM Rev. 53(3), 409–463 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. 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)

    Article  MATH  Google Scholar 

  14. Bertsch, M., Dal Passo, R.: Hyperbolic phenomena in a strongly degenerate parabolic equation. Arch. Rational Mech. Anal. 117(4), 349–387 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bonheure, D., Habets, P., Obersnel, F., Omari, P.: Classical and non-classical solutions of a prescribed curvature equation. J. Differ. Equ. 243(2), 208–237 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bressan, A.: Hyperbolic Systems of Conservation Laws. Oxford University, Oxford (2000)

    MATH  Google Scholar 

  17. Bressan, A., Colombo, R.: P.D.E. models of Pedestrian Flow, Unpublished (2007)

    Google Scholar 

  18. 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)

    Article  MathSciNet  MATH  Google Scholar 

  19. Calvo, J., Campos, J., Caselles, V., Sánchez, O., Soler, J.: Flux-saturated porous media equations and applications. EMS Surv. Math. Sci. 2(1), 131–218 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Calvo, J., Campos, J., Caselles, V., Sánchez, O., Soler, J.: Qualitative behaviour for flux-saturated mechanisms: travelling waves, waiting time and smoothing effects. J. Eur. Math. Soc. (JEMS) 19(2), 441–472 (2017)

    Google Scholar 

  21. Campos, J., Soler, J.: Qualitative behavior and traveling waves for flux-saturated porous media equations arising in optimal mass transportation. Nonlinear Anal. 137, 266–290 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Campos, J., Corli, A., Malaguti, L.: Saturated fronts in crowds dynamics, Preprint (2020)

    Google Scholar 

  23. Carrillo, J.: Entropy solutions for nonlinear degenerate problems. Arch. Ration. Mech. Anal. 147(4), 269–361 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. Caselles, V.: An existence and uniqueness result for flux limited diffusion equations. Discrete Contin. Dyn. Syst. 31(4), 1151–1195 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Caselles, V.: On the entropy conditions for some flux limited diffusion equations. J. Differ. Equ. 250(8), 3311–3348 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Caselles, V.: Flux limited generalized porous media diffusion equations. Publ. Mat. 57(1), 155–217 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Chertock, A., Kurganov, A., Rosenau, P.: Formation of discontinuities in flux-saturated degenerate parabolic equations. Nonlinearity 16(6), 1875–1898 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  28. Chertock, A., Kurganov, A., Rosenau, P.: On degenerate saturated-diffusion equations with convection. Nonlinearity 18(2), 609–630 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Chowdhury, D., Santen, L., Schadschneider, A.: Statistical physics of vehicular traffic and some related systems. Phys. Rep. 329(4–6), 199–329 (2000)

    Article  MathSciNet  Google Scholar 

  30. Coclite, G.M., Garavello, M.: Vanishing viscosity for traffic on networks. SIAM J. Math. Anal. 42(4), 1761–1783 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Coclite, G.M., Garavello, M., Piccoli, B.: Traffic flow on a road network. SIAM J. Math. Anal. 36(6), 1862–1886 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. Colombo, R.M., Garavello, M., Lécureux-Mercier, M.: A class of nonlocal models for pedestrian traffic. Math. Models Methods Appl. Sci. 22(4), 1150023, 34 (2012)

    Google Scholar 

  34. 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)

    Article  MathSciNet  MATH  Google Scholar 

  35. Corli, A., Malaguti, L.: Viscous profiles in models of collective movement with negative diffusivity. Z. Angew. Math. Phys. 70(2), Art. 47, 22 (2019)

    Google Scholar 

  36. Corli, A., di Ruvo, L., Malaguti, L.: Sharp profiles in models of collective movements. NoDEA Nonlinear Differ. Equ. Appl. 24(4), Art. 40, 31 (2017)

    Google Scholar 

  37. 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)

    Article  MathSciNet  MATH  Google Scholar 

  38. Coscia, V., Canavesio, C.: First-order macroscopic modelling of human crowd dynamics. Math. Models Methods Appl. Sci. 18(suppl), 1217–1247 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. Cristiani, E., Piccoli, B., Tosin, A.: Multiscale modeling of pedestrian dynamics. In: Modeling, Simulations and Applications. Springer, Cham (2014)

    Google Scholar 

  40. Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, 3rd edn. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  41. Daganzo, C.: Requiem for second-order fluid approximation of traffic flow. Transp. Res. B 29(4), 277–286 (1995)

    Article  Google Scholar 

  42. Dáger, R., Zuazua, E.: Wave propagation, observation and control in 1-d flexible multi-structures. In: Mathématiques and Applications (Berlin) [Mathematics and Applications], vol. 50. Springer, Berlin (2006)

    Google Scholar 

  43. Dal Passo, R.: Uniqueness of the entropy solution of a strongly degenerate parabolic equation. Commun. Partial Differ. Equ. 18(1–2), 265–279 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  44. DiCarlo, D.A., Juanes, R., Tara, L., Witelski, T.P.: Nonmonotonic traveling wave solutions of infiltration into porous media. Water Resour. Res. 44, 1–12 (2008)

    Article  Google Scholar 

  45. Evans, L.C., Portilheiro, M.: Irreversibility and hysteresis for a forward-backward diffusion equation. Math. Models Methods Appl. Sci. 14(11), 1599–1620 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  46. Ferracuti, L., Marcelli, C., Papalini, F.: Travelling waves in some reaction-diffusion-aggregation models. Adv. Dyn. Syst. Appl. 4(1), 19–33 (2009)

    MathSciNet  Google Scholar 

  47. Flynn, M.R., Kasimov, A.R., Nave, J.-C., Rosales, R.R., Seibold, B.: Self-sustained nonlinear waves in traffic flow. Phys. Rev. E (3) 79(5), 056113, 13 (2009)

    Google Scholar 

  48. Garavello, M., Han, K., Piccoli, B.: Models for Vehicular Traffic on Networks. American Institute of Mathematical Sciences (AIMS), Springfield (2016)

    Google Scholar 

  49. Garavello, M., Piccoli, B.: Traffic flow on networks. In: AIMS Series on Applied Mathematics, vol. 1. American Institute of Mathematical Sciences (AIMS), Springfield (2006)

    Google Scholar 

  50. Garrione, M., Sanchez, L.: Monotone traveling waves for reaction-diffusion equations involving the curvature operator. Bound. Value Probl. 2015, 45, 31 (2015)

    Google Scholar 

  51. Gilding, B.H., Kersner, R.: The characterization of reaction-convection-diffusion processes by travelling waves. J. Differ. Equ. 124(1), 27–79 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  52. Gilding, B.H., Kersner, R.: Travelling Waves in Nonlinear Diffusion-convection Reaction. Birkhäuser, Basel (2004)

    Google Scholar 

  53. Goodman, J., Kurganov, A., Rosenau, P.: Breakdown in Burgers-type equations with saturating dissipation fluxes. Nonlinearity 12(2), 247–268 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  54. Haberman, R.: Mathematical Models. Prentice-Hall, Inc., Englewood Cliffs, (1977). Mechanical vibrations, population dynamics, and traffic flow, An introduction to applied mathematics

    Google Scholar 

  55. Helbing, D.: Traffic and related self-driven many-particle systems. Rev. Mod. Phys. 73, 1067–1141 (2001)

    Article  MathSciNet  Google Scholar 

  56. Herty, M., Puppo, G., Roncoroni, S., Visconti, G.: The BGK approximation of kinetic models for traffic. Kin. Rel. Mod. 13(2), 279–307 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  57. Höllig, K.: Existence of infinitely many solutions for a forward backward heat equation. Trans. Am. Math. Soc. 278(1), 299–316 (1983)

    Article  MathSciNet  Google Scholar 

  58. Horstmann, D., Painter, K.J., Othmer, H.G.: Aggregation under local reinforcement: from lattice to continuum. European J. Appl. Math. 15(5), 546–576 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  59. Kalashnikov, A.S.: Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations. Russian Mat. Surv. 42(2(254)), 169–222 (1987)

    Google Scholar 

  60. Kerner, B.S.: Experimental features of self-organization in traffic flow. Phys. Rev. Lett. 81, 3797–3800 (1998)

    Article  MATH  Google Scholar 

  61. Kerner, B.S., Konhäser, P.: Structure and parameters of clusters in traffic flow. Phys. Rev. E 50, 54 (1994)

    Article  Google Scholar 

  62. Kerner, B.S., Osipov, V.V.: Autosolitons. Kluwer Academic Publishers Group, Dordrecht (1994)

    Book  Google Scholar 

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

    Google Scholar 

  64. Kurganov, A., Rosenau, P.: Effects of a saturating dissipation in Burgers-type equations. Commun. Pure Appl. Math. 50(8), 753–771 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  65. Kurganov, A., Rosenau, P.: On reaction processes with saturating diffusion. Nonlinearity 19(1), 171–193 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  66. Kuzmin, M., Ruggerini, S.: Front propagation in diffusion-aggregation models with bi-stable reaction. Discrete Contin. Dyn. Syst. Ser. B 16(3), 819–833 (2011)

    MathSciNet  MATH  Google Scholar 

  67. Lagnese, J.E., Leugering, G., Schmidt, E.J.P.G.: Modeling, analysis and control of dynamic elastic multi-link structures. In: Systems and Control: Foundations and Applications. Birkhäuser Boston Inc., Boston (1994)

    Google Scholar 

  68. Lax, P.D.: Hyperbolic systems of conservation laws. II. Commun. Pure Appl. Math. 10, 537–566 (1957)

    Article  MATH  Google Scholar 

  69. LeFloch, P.G.: Hyperbolic systems of conservation laws. In: Lectures in Mathematics ETH Zürich. Birkhäuser, Basel (2002)

    Google Scholar 

  70. Lighthill, M.J., Whitham, G.B.: On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. R. Soc. London. Ser. A 229, 317–345 (1955)

    MATH  Google Scholar 

  71. Maini, P.K., Malaguti, L., Marcelli, C., Matucci, S.: Diffusion-aggregation processes with mono-stable reaction terms. Discrete Contin. Dyn. Syst. Ser. B 6(5), 1175–1189 (2006)

    MathSciNet  MATH  Google Scholar 

  72. Maini, P.K., Malaguti, L., Marcelli, C., Matucci, S.: Aggregative movement and front propagation for bi-stable population models. Math. Models Methods Appl. Sci. 17(9), 1351–1368 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  73. Malaguti, L., Marcelli, C.: Finite speed of propagation in monostable degenerate reaction-diffusion-convection equations. Adv. Nonlinear Stud. 5(2), 223–252 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  74. Malaguti, L., Marcelli, C., Matucci, S.: Continuous dependence in front propagation of convective reaction-diffusion equations. Commun. Pure Appl. Anal. 9(4), 1083–1098 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  75. Mascia, C., Terracina, A., Tesei, A.: Two-phase entropy solutions of a forward-backward parabolic equation. Arch. Ration. Mech. Anal. 194(3), 887–925 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  76. Mugnolo, D., Rault, J.-F.: Construction of exact travelling waves for the Benjamin-Bona-Mahony equation on networks. Bull. Belg. Math. Soc. Simon Stevin 21(3), 415–436 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  77. Murray, J.D.: Mathematical biology: II, 3rd edn. Springer, New York (2003)

    Book  Google Scholar 

  78. Nelson, P.: Synchronized traffic flow from a modified Lighthill-Whitham model. Phys. Review E 61, R6052–R6055 (2000)

    Article  Google Scholar 

  79. Nelson, P.: Traveling-wave solutions of the diffusively corrected kinematic-wave model. Math. Comput. Modelling 35(5–6), 561–579 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  80. Newell, G.F.: Theories of instability in dense highway traffic. Oper. Res. Soc. Jap. 5, 9–54 (1962)

    MathSciNet  Google Scholar 

  81. Oleinik, O.A.: Discontinuous solutions of non-linear differential equations. Uspehi Mat. Nauk (N.S.) 12(3(75)), 3–73 (1957)

    Google Scholar 

  82. Oleinik, O.A.: Discontinuous solutions of non-linear differential equations. Am. Math. Soc. Transl. (2) 26, 95–172 (1963)

    Google Scholar 

  83. Payne, H.J.: Models of freeway traffic and control. Simul. Council Proc. 1, 51–61 (1971)

    Google Scholar 

  84. Plotnikov, P.I.: Passage to the limit with respect to viscosity in an equation with a variable direction of parabolicity. Differentsialnye Uravneniya 30(4), 665–674, 734 (1994)

    Google Scholar 

  85. Pokornyi, Y.V., Borovskikh, A.V.: Differential equations on networks (geometric graphs). J. Math. Sci. (N. Y.) 119(6), 691–718 (2004). Differential equations on networks

    Google Scholar 

  86. Richards, P.I.: Shock waves on the highway. Oper. Res. 4, 42–51 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  87. Ridder, J., Shen, W.: Traveling waves for nonlocal models of traffic flow. Discrete Contin. Dyn. Syst. 39(7), 4001–4040 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  88. Rosini, M.D.: Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications. Springer, Heidelberg (2013)

    Book  MATH  Google Scholar 

  89. Rykov, Y.G.: Discontinuous solutions of some strongly degenerate parabolic equations. Russ. J. Math. Phys. 7(3), 341–356 (2000)

    MathSciNet  MATH  Google Scholar 

  90. Schönhof, M., Helbing, D.: Empirical features of congested traffic states and their implications for traffic modeling. Transport. Sci. 41(2), 135–166 (2007)

    Article  Google Scholar 

  91. Seibold, B., Flynn, M.R., Kasimov, A.R., Rosales, R.R.: Constructing set-valued fundamental diagrams from jamiton solutions in second order traffic models. Netw. Heterog. Media 8(3), 745–772 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  92. Shen, W.: Traveling wave profiles for a follow-the-leader model for traffic flow with rough road condition. Netw. Heterog. Media 13(3), 449–478 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  93. Shen, W.: Traveling waves for conservation laws with nonlocal flux for traffic flow on rough roads. Netw. Heterog. Media 14(4), 709–732 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  94. Smarrazzo, F., Tesei, A.: Degenerate regularization of forward-backward parabolic equations: the regularized problem. Arch. Ration. Mech. Anal. 204(1), 85–139 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  95. Smarrazzo, F., Tesei, A.: Degenerate regularization of forward-backward parabolic equations: the vanishing viscosity limit. Math. Ann. 355(2), 551–584 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  96. Smoller, J.: Shock Waves and Reaction-Diffusion Equations, 2nd edn. Springer, New York (1994)

    Book  MATH  Google Scholar 

  97. Terracina, A.: Non-uniqueness results for entropy two-phase solutions of forward-backward parabolic problems with unstable phase. J. Math. Anal. Appl. 413(2), 963–975 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  98. Terracina, A.: Two-phase entropy solutions of forward-backward parabolic problems with unstable phase. Interfaces Free Bound. 17(3), 289–315 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  99. Vázquez, J.L.: The porous medium equation. The Clarendon Press/Oxford University, Oxford (2007)

    MATH  Google Scholar 

  100. Volpert, A.I., Hudjaev, S.I.: The Cauchy problem for second order quasilinear degenerate parabolic equations. Mat. Sb. (N.S.) 78(120), 374–396 (1969)

    Google Scholar 

  101. von Below, J.: Parabolic Network Equations. Tübinger Universitätsverlag, Tübingen (1994)

    Google Scholar 

  102. von Below, J.: Front propagation in diffusion problems on trees. In: Calculus of Variations, Applications and Computations (Pont-à-Mousson, 1994). Pitman Research Notes in Mathematics Series, vol. 326, pp. 254–265. Longman Science Technology, Harlow (1995)

    Google Scholar 

  103. Witelski, T.P.: The structure of internal layers for unstable nonlinear diffusion equations. Stud. Appl. Math. 97(3), 277–300 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  104. Zhang, H.M.: A non-equilibrium traffic model devoid of gas-like behavior. Transp. Res. B 36, 275–290 (2002)

    Article  Google Scholar 

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Acknowledgements

The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and acknowledge financial support from this institution.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Ferrara, Ferrara, Italy

    Andrea Corli

  2. Department of Sciences and Methods for Engineering, University of Modena and Reggio Emilia, Modena, Italy

    Luisa Malaguti

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  1. Andrea Corli
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  2. Luisa Malaguti
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Correspondence to Andrea Corli .

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Editors and Affiliations

  1. Dipartimento di Matematica, Università di Bologna, Bologna, Italy

    Massimo Cicognani

  2. Dipartimento di Matematica e Geoscienze, Università di Trieste, Trieste, Italy

    Daniele Del Santo

  3. Dipartimento di Matematica, Università di Bologna, Bologna, Italy

    Alberto Parmeggiani

  4. Institut für Angewandte Analysis, TU Bergakademie Freiberg, Freiberg, Germany

    Michael Reissig

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Corli, A., Malaguti, L. (2021). Wavefronts in Traffic Flows and Crowds Dynamics. In: Cicognani, M., Del Santo, D., Parmeggiani, A., Reissig, M. (eds) Anomalies in Partial Differential Equations. Springer INdAM Series, vol 43. Springer, Cham. https://doi.org/10.1007/978-3-030-61346-4_8

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