Well-posedness for a monotone solver for traffic junctions

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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 ρₕ,ₜ + fₕ(ρₕ)ₓ = 0, for h∈ {1,, 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 diehl, ColomboGoatinConstraint, scontrainte, ACₜransmission, germes.