Advances in Differential Equations

Entropy-type conditions for Riemann solvers at nodes

Mauro Garavello and Benedetto Piccoli

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Abstract

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

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

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355854332

Mathematical Reviews number (MathSciNet)
MR2766896

Zentralblatt MATH identifier
1217.90056

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

Citation

Garavello, Mauro; Piccoli, Benedetto. Entropy-type conditions for Riemann solvers at nodes. Adv. Differential Equations 16 (2011), no. 1/2, 113--144. https://projecteuclid.org/euclid.ade/1355854332.


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