We construct a model of traffic flow with sources and destinations on a roads network. The model is based on a conservation law for the density of traffic and on semilinear equations for traffic-type functions, i.e. functions describing paths for cars.
We propose a definition of solution at junctions, which depends on the traffic-type functions. Finally we prove, for every positive time T, existence of entropic solutions on the whole network for perturbations of constant initial data.
Our method is based on the wave-front tracking approach.
"Source-Destination Flow on a Road Network." Commun. Math. Sci. 3 (3) 261 - 283, September 2005.