Communications in Mathematical Sciences

Volume 1 (2003)

Number 3

Entropy methods for kinetic models of traffic flow

Pages: 409 – 421

DOI: http://dx.doi.org/10.4310/CMS.2003.v1.n3.a2

Authors

Jean Dolbeault

Reinhard Illner

Abstract

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.

Keywords

Traffic flow; time-dependent diffusions; drift-diffusion equations; nonlinear friction and diffusion coefficients; entropy method; relative entropy; large time asymptotics

2010 Mathematics Subject Classification

35B40, 35B45, 35K55, 60J60, 60J70, 70F40, 90B20, 92D99, 94A17

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