Flows on networks: recent results and perspectives
A. Bressan, S. Čanić, M. Garavello, M. Herty and B. Piccoli
Abstract: 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 [MR], and Modeling, simulation, and optimization of supply chains, SIAM, Philadelphia, 2010 [MR]. 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 present 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 appeared subsequently to the publication of the aforementioned books.