Well-posedness for a monotone solver for traffic junctions

Abstract

In this paper we aim at proving well-posedness of solutions obtained as vanishing viscosity limits for the Cauchy problem on a traffic junction where $m$ incoming and $n$ outgoing roads meet. The traffic on each road is governed by a scalar conservation law ${\rho }_{h,t}+f_h({\rho }_h{)}_x=0$, for $h\in \{1,\dots ,m+n\}$. Our proof relies upon the complete description of the set of road-wise constant solutions and its properties, which is of some interest on its own. Then we introduce a family of Kruzhkov-type adapted entropies at the junction and state a definition of admissible solution in the same spirit as in \cite{diehl, ColomboGoatinConstraint, scontrainte, AC_transmission, germes}.