Flows on networks: recent results and perspectives
type: Item
instance of: scholarly article
title: Flows on networks: recent results and perspectives
zbMATH Open document ID: 1301.35193
DOI: 10.4171/EMSS/2
author: Alberto Bressan
author: Sunčica Čanić
author: Mauro Garavello
author: B. Piccoli
author: Michael Herty
published in: EMS Surveys in Mathematical Sciences
publication date: 2014-07-10T00:00:00Z
review text: This well--written survey paper is on flows on networks and is organized as follows. Section 2 presents results on existence of solutions and includes both the case of a single conservation law and systems of conversation laws. Section 3 provides various models to which the theory can be applied: vehicular traffic, gas pipelines, supply chains, data and telecommunication networks, irrigation channals and blood flows. Section 4 considers more complex solutions where a coupling of different solutions of equations (ODE--PDE or PDE--PDE) is needed. Also in this case various examples are provided, such as piston problems, sewer systems, supply chains and blood flow. Section 5 considers controlled systems, where the control acts within the coupling condition at the vertex \(\Psi\). The main examples are provided by vehicular traffic and gas pipelines. Section 6 presents some numerical methods for conservation laws on networks. Section 7 presents the specific applications of the developed methods: the study of optimal shape of vascular stent and vehicular traffic monitoring using GPS data from mobile sensors. The final section, section 8, presents open problems and future perspectives of this research. The ample bibliography contains 125 items.
reviewed by: Wiesław Kotarski
Mathematics Subject Classification ID: 35R02
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 34H05
Mathematics Subject Classification ID: 93C20
Mathematics Subject Classification ID: 93C15
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 49K15
Mathematics Subject Classification ID: 49K20
zbMATH DE Number: 6316076
zbMATH Keywords: networks
zbMATH Keywords: balance laws
zbMATH Keywords: control problems, coupled PDE--ODE and PDE--PDE systems
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/emss/2
OpenAlex ID: W1970599443
description: scientific article