Crowd Dynamics Through Conservation Laws
01 Jan 2020-Vol. 47, Iss: 1, pp 83-110
TL;DR: 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.
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01 Jan 2017
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.
Abstract: We give an answer to a question posed in Amorim et al. (ESAIM Math Model Numer Anal 49(1):19–37, 2015), 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. In the singular limit where the convolution kernel is replaced by a Dirac delta, one formally recovers a conservation law. Can we rigorously justify this formal limit? We exhibit counterexamples showing that, despite numerical evidence suggesting a positive answer, one does not in general have convergence of the solutions. We also show that the answer is positive if we consider viscous perturbations of the nonlocal equations. In this case, in the singular local limit the solutions converge to the solution of the viscous conservation law.
18 citations
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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.
Abstract: We deal with the problem of approximating a scalar conservation law by a conservation law with nonlocal flux. As convolution kernel in the nonlocal flux, we consider an exponential-type approximation of the Dirac distribution. This enables us to obtain a total variation bound on the nonlocal term. By using this, we prove that the (unique) weak solution of the nonlocal problem converges strongly in $C(L^{1}_{\text{loc}})$ to the entropy solution of the local conservation law. We conclude with several numerical illustrations which underline the main results and, in particular, the difference between the solution and the nonlocal term.
10 citations
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.
Abstract: Abstract In this note, we extend the known results on the existence and uniqueness of weak solutions to conservation laws with nonlocal flux. In case the nonlocal term is given by a convolution $$\gamma *q$$ γ ∗ q , we weaken the standard assumption on the kernel $$\gamma \in L^\infty \big ((0,T); W^{1,\infty }({\mathbb {R}})\big )$$ γ ∈ L ∞ ( ( 0 , T ) ; W 1 , ∞ ( R ) ) to the substantially more general condition $$\gamma \in L^\infty ((0,T); BV({\mathbb {R}}))$$ γ ∈ L ∞ ( ( 0 , T ) ; B V ( R ) ) , which allows for discontinuities in the kernel.
6 citations
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.
5 citations
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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.
Abstract: 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. This construction is motivated by the modelling of crowd dynamics, which also leads to define a non local operator adapted to the presence of a boundary. Numerical integrations show that the resulting model provides qualitatively reasonable solutions.
4 citations
References
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TL;DR: In this paper, the authors deal with a system of conservation laws coupled with an ordinary differential equation, where the former is defined on a domain with boundary and the coupling is provided by the boundary condition.
Abstract: Motivated by applications to the piston problem, to a manhole model, to blood flow and to supply chain dynamics, this paper deals with a system of conservation laws coupled with a system of ordinary differential equations. The former is defined on a domain with boundary and the coupling is provided by the boundary condition. For each of the examples considered, numerical integrations are provided.
53 citations
TL;DR: In this paper, a hyperbolic conservation law that models a highly re-entrant manufacturing system as encountered in semi-conductor production is studied, and the authors prove the existence and uniqueness of solutions for $L 1$-data and study their regularity properties.
Abstract: This article studies a hyperbolic conservation law that models
a highly re-entrant manufacturing system as encountered in
semi-conductor production.
Characteristic features are the nonlocal character of the velocity
and that the influx and outflux constitute the control and output
signal, respectively.
We prove the existence and uniqueness of solutions for $L^1$-data,
and study their regularity properties.
We also prove the existence of optimal controls
that minimizes in the $L^2$-sense the mismatch between the actual
and a desired output signal.
Finally, the time-optimal control for a step between equilibrium states is
identified and proven to be optimal.
52 citations
TL;DR: It is proved the well-posedness of entropy weak solutions for a class of scalar conservation laws with non-local flux arising in traffic modeling with respect to the initial data through the doubling of variable technique.
Abstract: We prove the well-posedness of entropy weak solutions for a class of scalar conservation laws with non-local flux arising in traffic modeling. We approximate the problem by a Lax-Friedrichs scheme and we provide L ∞ and BV estimates for the sequence of approximate solutions. Stability with respect to the initial data is obtained from the entropy condition through the doubling of variable technique. The limit model as the kernel support tends to infinity is also studied.
51 citations
27 Feb 2002
TL;DR: In this paper, the authors consider the dependence of the entropic solution of a hyperbolic system of conservation laws on the flux function f and prove that the solution is Lipschitz continuous w.r.t. the C 0 norm of the derivative of the perturbation of f.
Abstract: We consider the dependence of the entropic solution of a hyperbolic system of conservation laws foruma math. on the flux function f. We prove that the solution is Lipschitz continuous w.r.t. the C 0 norm of the derivative of the perturbation of f. We apply this result to prove the convergence of the solution of the relativistic Euler equation to the classical limit.
48 citations
Posted Content•
TL;DR: In this article, a hyperbolic conservation law that models a highly reentrant manufacturing system as encountered in semi-conductor production is studied, and a time-optimal control for a step between equilibrium states is identified and proven to be optimal.
Abstract: This article studies a hyperbolic conservation law that models a highly re-entrant manufacturing system as encountered in semi-conductor production. Characteristic features are the nonlocal character of the velocity and that the influx and outflux constitute the control and output signal, respectively. We prove the existence and uniqueness of solutions for $L^1$-data, and study their regularity properties. We also prove the existence of optimal controls that minimizes in the $L^2$-sense the mismatch between the actual and a desired output signal. Finally, the time-optimal control for a step between equilibrium states is identified and proven to be optimal.
42 citations