The broad research thematic of flows on networks was addressed in recent years by many researchers, in the area of applied mathematics, with new models based on partial differential equations. The latter brought a significant innovation in a field previously dominated by more classical techniques from discrete mathematics or methods based on ordinary differential equations. In particular, a number of results, mainly dealing with vehicular traffic, supply chains and data networks, were collected in two monographs: Traffic flow on networks, AIMSciences, Springfield, 2006, and Modeling, simulation, and optimization of supply chains, SIAM, Philadelphia, 2010. The field continues to flourish and a considerable number of papers devoted to the subject is published every year, also because of the wide and increasing range of applications: from blood flow to air traffic management. The aim of the present survey paper is to provide a view on a large number of themes, results and applications related to this broad research direction. The authors cover different expertise (modeling, analysis, numeric, optimization and other) so to provide an overview as extensive as possible. The focus is mainly on developments which appeared subsequently to the publication of the aforementioned books.
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Alberto Bressan, Sunčica Čanić, Mauro Garavello, Michael Herty, Benedetto Piccoli, Flows on networks: recent results and perspectives. EMS Surv. Math. Sci. 1 (2014), no. 1, pp. 47–111