Entropy Dissipation at the Junction for Macroscopic Traffic Flow Models

A maximum entropy dissipation problem at a traffic junction and the corresponding coupling condition are studied. We prove that this problem is equivalent to a coupling condition introduced by Holden and Risebro. An $L^1$-contraction property of the coupling condition and uniqueness of solutions to the Cauchy problem are proven. Existence is obtained by a kinetic approximation of Bhatnagar-Gross-Krook-type together with a kinetic coupling condition obtained by a kinetic maximum entropy dissipation problem. The arguments do not require $TV$-bounds on the initial data compared to previous results. We also discuss the role of the entropies involved in the macroscopic coupling condition at the traffic junction by studying an example.