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A macroscopic crowd motion model of gradient flow type

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Date
2010
Link to item file
http://hal.archives-ouvertes.fr/hal-00418511/fr/
Dewey
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Sujet
Continuity Equation; Crowd Motion; Gradient Flow; Wasserstein Distance
Journal issue
Mathematical Models and Methods in Applied Sciences
Volume
20
Number
10
Publication date
2010
Article pages
1787-1821
Publisher
World Scientific
DOI
http://dx.doi.org/10.1142/S0218202510004799
URI
https://basepub.dauphine.fr/handle/123456789/3516
Collections
  • CEREMADE : Publications
Metadata
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Author
Santambrogio, Filippo
Roudneff-Chupin, Aude
Maury, Bertrand
Type
Article accepté pour publication ou publié
Abstract (EN)
A simple model to handle the flow of people in emergency evacuation situations is considered: at every point x, the velocity U(x) that individuals at x would like to realize is given. Yet, the incompressibility constraint prevents this velocity field to be realized and the actual velocity is the projection of the desired one onto the set of admissible velocities. Instead of looking at a microscopic setting (where individuals are represented by rigid discs), here the macroscopic approach is investigated, where the unknwon is the evolution of the density . If a gradient structure is given, say U is the opposite of the gradient of D where D is, for instance, the distance to the exit door, the problem is presented as a Gradient Flow in the Wasserstein space of probability measures. The functional which gives the Gradient Flow is neither finitely valued (since it takes into account the constraints on the density), nor geodesically convex, which requires for an ad-hoc study of the convergence of a discrete scheme.

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