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Pag 2 d i c h i a r a z i o n e s o s t i t u t i v a d i c e r t i f i c a z i o n e(art. 46 D.P.R. 445 del 28.12.2000)La/il sottoscritta/o……………………………………………………………………………………… nata/o a .

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.

We consider a hyperbolic conservation law with discontinuous flux. Such a partial differential equation arises in different applications, in particu- lar we are motivated by a model of traffic flow. We provide a new formulation in terms of Riemann Solvers. Moreover, we determine the class of Riemann Solvers which provide existence and uniqueness of the corresponding weak en- tropic solutions

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We construct a model of traffic flow with sources and destinations on a roads network. The model is based on a conservation law for the density of traffic and on semilinear equations for traffic-type functions, i.e. functions describing paths for cars.We propose a definition of solution at junctions, which depends on the traffic-type functions. Finally we prove, for every positive time T , existence of entropic solutions on the whole network for perturbations of constant initial data.Our method is based on the wave-front tracking approach.

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