Flows on networks: recent results and perspectives

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, 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. 1 The author acknowledges partial support of 2013 GNAMPA project "Leggi di Conservazione: Teoria e Applicazioni". 2 The author acknowledges support by BMBF KinOpt, DFG Cluster of Excellence EXC128 and DAAD 54365630, 55866082. 3 The author acknowledges partial support of NSF Research Network in the Mathematical Sciences KI-Net "Kinetic description of emerging challenges in multiscale problems of natural sciences" Grant # : 1107444.

Existence of BV solutions for a non-conservative constrained Aw–Rascle–Zhang model for vehicular traffic

“…Macroscopic models of traffic flows are nowadays a consolidated and nonetheless continuously in ferment field of mathematical research from both theoretical and applied points of view, as the surveys [5,17,18,20] and the books [7,13,19] demonstrate.…”

Explicit formulae for limit periodic flows on networks

“…[1,3,5,6,9,13,15] and also the monograph [16]. Here we focus on transport on networks, modelled by a system of first order transport equations on the edges of the digraph representing the network.…”

Mode Selection in Compressible Active Flow Networks

“…Identifying generic self-organization principles [6,7] that control the dynamics of these biological or artificial far-fromequilibrium systems remains one of the foremost challenges of modern statistical physics. Despite promising experimental [3,[8][9][10] and theoretical [1,4,[11][12][13] advances over the past decade, it is not well understood how the interactions between local energy input, dissipation, and network topology determine the coordinated global behaviors of cells [8], plasmodia [3], or tissues [14]. Further progress requires analytically tractable models that help clarify the underlying nonequilibrium mode-selection principles [15].…”

“…Identifying generic self-organization principles [6,7] that control the dynamics of these biological or artificial far-fromequilibrium systems remains one of the foremost challenges of modern statistical physics. Despite promising experimental [3,[8][9][10] and theoretical [1,4,[11][12][13] advances over the past decade, it is not well understood how the interactions between local energy input, dissipation, and network topology determine the coordinated global behaviors of cells [8] [19], the theory accounts for network activity through a nonlinear friction [19][20][21]. We work in a fully compressible framework allowing accumulated matter at vertices to affect flow through network pressure gradients, generalizing previous work on incompressible pseudoequilibrium active flow networks [22,23], as suited to the many biological systems exhibiting flexible network geometry [3] or variations in the density of active components [7].…”