Comptes Rendus
no. 13-14
Partial Differential Equations
Non-local crowd dynamics
[Dynamique non locale des foules]

Présenté par : the Editorial Board

Rinaldo M. Colombo1 ; Mauro Garavello2 ; Magali Lécureux-Mercier3
1 Dipartimento di Matematica, Università degli studi di Brescia, Via Branze 38, 25123 Brescia, Italy
2 Di.S.T.A., Università del Piemonte Orientale, Viale Teresa Michel 11, 15121 Alessandria, Italy
3 Université dʼOrléans, bâtiment de mathématiques, rue de Chartres, B.P. 6759, 45067 Orléans cedex 2, France
Comptes Rendus. Mathématique, Volume 349 (2011) no. 13-14, pp. 769-772.

Nous présentons ici un nouveau modèle macroscopique de trafic piéton dans lequel chaque individu se dirige vers une cible fixe en déviant du plus court chemin en fonction de la distribution de la population. On obtient une loi de conservation avec flux non local qui génère un semi-groupe de solutions et est stable par rapport aux fonctions et paramètres quʼelle contient. On montre de plus que la densité reste bornée pour tout temps. On sʼintéresse plus particuliérement à deux modèles précis.

We present a new class of macroscopic models for pedestrian flows. Each individual is assumed to move toward a fixed target, deviating from the best path according to the crowd distribution. The resulting equation is a conservation law with a non-local flux. Each equation in this class generates a Lipschitz semigroup of solutions and is stable with respect to the functions and parameters defining it. Moreover, key qualitative properties such as the boundedness of the crowd density are proved. Two specific models in this class are considered.

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Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.07.005
Rinaldo M. Colombo 1 ; Mauro Garavello 2 ; Magali Lécureux-Mercier 3

1 Dipartimento di Matematica, Università degli studi di Brescia, Via Branze 38, 25123 Brescia, Italy
2 Di.S.T.A., Università del Piemonte Orientale, Viale Teresa Michel 11, 15121 Alessandria, Italy
3 Université dʼOrléans, bâtiment de mathématiques, rue de Chartres, B.P. 6759, 45067 Orléans cedex 2, France 
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     author = {Rinaldo M. Colombo and Mauro Garavello and Magali L\'ecureux-Mercier},
     title = {Non-local crowd dynamics},
     journal = {Comptes Rendus. Math\'ematique},
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Rinaldo M. Colombo; Mauro Garavello; Magali Lécureux-Mercier. Non-local crowd dynamics. Comptes Rendus. Mathématique, Volume 349 (2011) no. 13-14, pp. 769-772. doi : 10.1016/j.crma.2011.07.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.07.005/

[1] A. Bressan, R.M. Colombo, P.D.E. models of pedestrian flow, unpublished, 2007.

[2] R.M. Colombo, M. Garavello, M. Lécureux-Mercier, A class of non-local models for pedestrian traffic, preprint, , 2011. | HAL

[3] R.M. Colombo, M. Herty, M. Mercier, Control of the continuity equation with a non-local flow, ESAIM: COCV, 2010.

[4] R.M. Colombo; M. Mercier; M.D. Rosini Stability and total variation estimates on general scalar balance laws, Commun. Math. Sci., Volume 7 (2009) no. 1, pp. 37-65

[5] E. Cristiani; B. Piccoli; A. Tosin Multiscale modeling of granular flows with application to crowd dynamics, Multiscale Model. Simul., Volume 9 (2011) no. 1, pp. 155-182

[6] M. Di Francesco; P.A. Markowich; J.-F. Pietschmann; M.-T. Wolfram On the Hughesʼ model for pedestrian flow: The one-dimensional case, J. Differential Equations, Volume 250 (2011) no. 3, pp. 1334-1362

[7] D. Helbing; I. Farkás; P. Molnár; T. Vicsek Simulation of pedestrian crowds in normal and evacuation situations (M. Schreckenberg; S.D. Sharma, eds.), Pedestrian and Evacuation Dynamics, Springer, Berlin, 2002, pp. 21-58

[8] D. Helbing; A. Johansson Pedestrian, crowd and evacuation dynamics, Encyclopedia of Complexity and Systems Science, 2010, pp. 6476-6495

[9] S. Hoogendoorn; P.H.L. Bovy Simulation of pedestrian flows by optimal control and differential games, Optimal Control Appl. Methods, Volume 24 (2003) no. 3, pp. 153-172

[10] R.L. Hughes A continuum theory for the flow of pedestrians, Transport. Res. Part B: Methodol., Volume 36 (2002) no. 6, pp. 507-535

[11] R.L. Hughes The flow of human crowds, Annu. Rev. Fluid Mech., Volume 35 (2003), pp. 169-182

[12] S.N. Kružkov First order quasilinear equations with several independent variables, Mat. Sb., Volume 81 (1970) no. 123, pp. 228-255

[13] B. Piccoli; A. Tosin Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., Volume 199 (2011), pp. 707-738

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