Book ChapterDOI
Crowd Dynamics Through Conservation Laws
Rinaldo M. Colombo,Magali Lécureux-Mercier,Mauro Garavello +2 more
- Vol. 47, Iss: 1, pp 83-110
TLDR
In this paper, several macroscopic models, based on systems of conservation laws, were considered for the study of crowd dynamics. But none of the models considered here contain nonlocal terms, usually obtained through convolutions with smooth functions used to reproduce the visual horizon of each individual.Abstract:
We consider several macroscopic models, based on systems of conservation laws, for the study of crowd dynamics. All the systems considered here contain nonlocal terms, usually obtained through convolutions with smooth functions, used to reproduce the visual horizon of each individual. We classify the various models according to the physical domain (the whole space \({\mathbb {R}}^N\) or a bounded subset), to the terms affected by the nonlocal operators, and to the number of different populations we aim to describe. For all these systems, we present the basic well posedness and stability results.read more
Citations
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On the singular local limit for conservation laws with nonlocal fluxes
TL;DR: In this article, the authors give an answer to a question posed in Amorim et al. (ESAIM Math Model Numer Anal 49(1):19-37), which can loosely speaking, be formulated as follows: consider a family of continuity equations where the velocity depends on the solution via the convolution by a regular kernel.
Posted Content
A general result on the approximation of local conservation laws by nonlocal conservation laws: The singular limit problem for exponential kernels
TL;DR: In this paper, the problem of approximating a scalar conservation law by a conservation law with nonlocal flux was studied, and it was shown that the (unique) weak solution of the nonlocal problem converges strongly in O(L √ n) to the entropy solution of local conservation law.
Journal ArticleDOI
On existence and uniqueness of weak solutions to nonlocal conservation laws with BV kernels
TL;DR: In this paper , the existence and uniqueness of weak solutions to conservation laws with nonlocal flux was shown to be true under the condition that the nonlocal term is given by a convolution.
Journal ArticleDOI
A general result on the approximation of local conservation laws by nonlocal conservation laws: The singular limit problem for exponential kernels
TL;DR: In this paper , the problem of approximating a scalar conservation law by a conservation law with nonlocal flux was studied, and it was shown that the (unique) weak solution of the nonlocal problem converges strongly in O(L √ n) to the entropy solution of local conservation law.
Posted Content
Non Local Conservation Laws in Bounded Domains
Elena Rossi,Rinaldo M. Colombo +1 more
TL;DR: The well posedness for a class of non local systems of conservation laws in a bounded domain is proved and various stability estimates are provided.
References
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Journal ArticleDOI
Nonlocal Crowd Dynamics Models for Several Populations
TL;DR: In this article, the authors developed the basic analytical theory related to some recently introduced crowd dynamics models, where well posedness was known only locally in time, it was here extended to all of
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Rigorous Derivation of Nonlinear Scalar Conservation Laws from Follow-the-Leader Type Models via Many Particle Limit
TL;DR: In this article, it was shown that the Kržkov entropy solution to a scalar nonlinear conservation law with strictly monotone velocity and nonnegative initial condition can be rigorously obtained as the large particle limit of a microscopic follow-the-leader type model, interpreted as the discrete Lagrangian approximation of the nonlinear scalar conservation law.
Journal ArticleDOI
Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients
TL;DR: In this paper, the well-posedness of entropy solutions to degenerate parabolic equations with explicit Lipschitz continuous dependence was studied, and a wellposedness theory for the Cauchy problem was established.
Journal ArticleDOI
Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow
TL;DR: In this paper, the authors prove existence and uniqueness of measure solutions for the Cauchy problem associated to the (vectorial) continuity equation with a non-local flow and give a stability result with respect to various parameters.
Journal ArticleDOI
Macroscopic modeling and simulations of room evacuation
TL;DR: In this article, the authors analyze numerically two macroscopic models of crowd dynamics: the classical Hughes model and the second order model being an extension to pedestrian motion of the Payne-Whitham vehicular traffic model.