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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Book
Hyberbolic Conservation Laws in Continuum Physics
TL;DR: In this paper, the authors present a masterly exposition and an encyclopedic presentation of the theory of hyperbolic conservation laws, with a focus on balance laws with dissipative source, modeling relaxation phenomena.
Hyperbolic systems of conservation laws : the one-dimensional Cauchy problem
TL;DR: A self-contained introduction to the mathematical theory of hyperbolic systems of conservation laws, with particular emphasis on the study of discontinuous solutions, characterized by the appearance of shock waves, is given in this paper.
Book
Front Tracking for Hyperbolic Conservation Laws
Helge Holden,Nils Henrik Risebro +1 more
TL;DR: In this paper, the authors presented the solution of the Riemann problem for the 3 3 Euler equations of gas dynamics, and the main existence theorem in Chapt. 4, Theorem 4.3, now resembles the one-dimensional case.