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# 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

#### 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