January/February 2011 Entropy-type conditions for Riemann solvers at nodes
Mauro Garavello, Benedetto Piccoli
Adv. Differential Equations 16(1/2): 113-144 (January/February 2011). DOI: 10.57262/ade/1355854332


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).


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Mauro Garavello. Benedetto Piccoli. "Entropy-type conditions for Riemann solvers at nodes." Adv. Differential Equations 16 (1/2) 113 - 144, January/February 2011. https://doi.org/10.57262/ade/1355854332


Published: January/February 2011
First available in Project Euclid: 18 December 2012

zbMATH: 1217.90056
MathSciNet: MR2766896
Digital Object Identifier: 10.57262/ade/1355854332

Primary: 35L65 , 90B20

Rights: Copyright © 2011 Khayyam Publishing, Inc.


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Vol.16 • No. 1/2 • January/February 2011
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