2009/05/27 by Mauro Garavello, Garavello, Mauro, Benedetto Piccoli +1
Engineering · Environmental Science · Mathematics · #35L65 #90B20 #Analysis of PDEs (math.AP) #Ecosystem dynamics and resilience #FOS: Mathematics #Traffic control and management #math.AP #msc:35L65 #msc:90B20
paper · pdf · doi:10.48550/arxiv.0905.4408
35 pages
arxiv created 2009/05/27 · openalex publication_date 2009/05/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper deals with conservation laws on networks, represented by graphs. Entropy-type conditions are considered to determine dynamics at nodes. Since entropy dispersion is a local concept, we consider a network composed by a single node J with n incoming and m outgoing arcs. We extend at J the classical Kružkov entropy obtaining two conditions, denoted by (E1) and (E2): the first requiring entropy condition for all Kružkov entropies, the second only for the value corresponding to sonic point. First we show that in case n ≠ m, no Riemann solver can satisfy the strongest condition. Then we characterize all the Riemann solvers at J satisfying the strongest condition (E1), in the case of nodes with at most two incoming and two outgoing arcs. Finally we focus three different Riemann solvers, introduced in previous papers. In particular, we show that the Riemann solver introduced for data networks is the only one always satisfying (E2).