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Entropy methods for kinetic models of traffic flow

Dolbeault, Jean; Illner, Reinhard (2003), Entropy methods for kinetic models of traffic flow, Communications in Mathematical Sciences, 1, 3, p. 409-421. http://projecteuclid.org/euclid.cms/1250880093

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Type
Article accepté pour publication ou publié
Date
2003
Journal name
Communications in Mathematical Sciences
Volume
1
Number
3
Publisher
International Press
Pages
409-421
Publication identifier
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Author(s)
Dolbeault, Jean cc
Illner, Reinhard
Abstract (EN)
In these notes we first introduce logarithmic entropy methods for time-dependent drift-diffusion equations and then consider a kinetic model of Vlasov-Fokker-Planck type for traffic flows. In the spatially homogeneous case the model reduces to a special type of nonlinear driftdiffusion equation which may permit the existence of several stationary states corresponding to the same density. Then we define general convex entropies and prove a convergence result for large times to steady states, even if more than one exists in the considered range of parameters, provided that some entropy estimates are uniformly bounded.
Subjects / Keywords
Traffic flow; time-dependent diffusions; drift-diffusion equations; nonlinear friction and diffusion coefficients; entropy method; relative entropy; large time asymptotics