WELL-POSEDNESS FOR VANISHING VISCOSITY SOLUTIONS OF SCALAR CONSERVATION LAWS ON A NETWORK

摘要

We provide a complete study of the model investigated in [Coclite, Garavello, SIAM J. Math. Anal., 2010]. We prove well-posedness of solutions obtained as vanishing viscosity limits for the Cauchy problem for scalar conservation laws ρ_(h,t) + f_h(ρ_h)_x = 0, for h ∈ {1,... ,m + n}, on a junction where m incoming and n outgoing edges meet. Our analysis and the definition of the admissible solution rely upon the complete description of the set of edge-wise constant solutions and its properties, which is of some interest on its own. The Riemann solver at the junction is characterized. In order to prove uniqueness, we introduce a family of Kruzhkov-type adapted entropies at the junction. Existence is justified both by the vanishing viscosity method and via the proof of convergence of a monotone well-balanced finite volume discretization. Beyond the classical vanishing viscosity framework, the numerical procedure and the uniqueness argument can be applied to general junction solvers enjoying the crucial order-preservation property.