From discrete to continuous Wardrop equilibria
Jean-Bernard Baillon Guillaume Carlier
Networks & Heterogeneous Media 2012, 7(2): 219-241 doi: 10.3934/nhm.2012.7.219
The notion of Wardrop equilibrium in congested networks has been very popular in congested traffic modelling since its introduction in the early 50's, it is also well-known that Wardrop equilibria may be obtained by some convex minimization problem. In this paper, in the framework of $\Gamma$-convergence theory, we analyze what happens when a cartesian network becomes very dense. The continuous model we obtain this way is very similar to the continuous model of optimal transport with congestion of Carlier, Jimenez and Santambrogio [6] except that it keeps track of the anisotropy of the network.
keywords: $\Gamma$-convergence eikonal equation. Wardrop equilibria traffic congestion
Numerical approximation of continuous traffic congestion equilibria
Fethallah Benmansour Guillaume Carlier Gabriel Peyré Filippo Santambrogio
Networks & Heterogeneous Media 2009, 4(3): 605-623 doi: 10.3934/nhm.2009.4.605
Starting from a continuous congested traffic framework recently introduced in [8], we present a consistent numerical scheme to compute equilibrium metrics. We show that equilibrium metric is the solution of a variational problem involving geodesic distances. Our discretization scheme is based on the Fast Marching Method. Convergence is proved via a $\Gamma$-convergence result and numerical results are given.
keywords: Fast Marching Method. traffic congestion eikonal equation subgradient descent Wardrop equilibria
Remarks on a class of kinetic models of granular media: Asymptotics and entropy bounds
Martial Agueh Guillaume Carlier Reinhard Illner
Kinetic & Related Models 2015, 8(2): 201-214 doi: 10.3934/krm.2015.8.201
We obtain new a priori estimates for spatially inhomogeneous solutions of a kinetic equation for granular media, as first proposed in [3] and, more recently, studied in [1]. In particular, we show that a family of convex functionals on the phase space is non-increasing along the flow of such equations, and we deduce consequences on the asymptotic behaviour of solutions. Furthermore, using an additional assumption on the interaction kernel and a ``potential for interaction'', we prove a global entropy estimate in the one-dimensional case.
keywords: global in time estimates Kinetic granular media asymptotic behavior entropy bounds.

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