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
On Traffic Flow with Nonlocal Flux: a Relaxation Representation
Alberto Bressan,Wen Shen +1 more
TL;DR: In this paper, the authors considered a conservation law model of traffic flow, where the velocity of each car depends on a weighted average of the traffic density ahead, and obtained uniform BV bounds on the solution.
Journal ArticleDOI
Improved stability estimates for general scalar conservation laws
TL;DR: In this article, the authors considered the general scalar balance law ∆t u + Div f(t, x, u) = F (t, X, u), and improved the results of Colombo, Mercier and Rosini who gave an estimate of the dependence of the solutions in terms of the flux f and the source F.
Journal ArticleDOI
A discrete mathematical model for the dynamics of a crowd of gazing pedestrians with and without an evolving environmental awareness
TL;DR: A microscopic-discrete mathematical model describing crowd dynamics in no panic conditions focuses on pedestrian flow in two-dimensional buildings with several structural elements and demonstrates the potential of the mathematical model in addressing different engineering problems, allowing for optimization issues as well.
Journal ArticleDOI
Definitions of solutions to the ibvp for multi-dimensional scalar balance laws
TL;DR: The main result is the proof of the equivalence among the presented definitions of solution of the initial-boundary value problem for a scalar balance laws in several space dimensions.