Advances in Differential Equations

Entropy-type conditions for Riemann solvers at nodes

Mauro Garavello and Benedetto Piccoli

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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 a sonic point. First we show that in the case $n \ne m$, 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).

Article information

Adv. Differential Equations Volume 16, Number 1/2 (2011), 113-144.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 90B20: Traffic problems 35L65: Conservation laws


Garavello, Mauro; Piccoli, Benedetto. Entropy-type conditions for Riemann solvers at nodes. Adv. Differential Equations 16 (2011), no. 1/2, 113--144.

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