Conservation laws on a star-shaped network

Hyperbolic conservation laws defined on oriented graphs are widely used in the modeling of a variety of phenomena such as vehicular and pedestrian traffic, irrigation channels, blood circulation, gas pipelines, structured population dynamics.
From the point of view of the mathematical analysis each of these situations demands for a different definition of admissible solution, encoding in particular the node coupling between incoming and outgoing edges which is the most coherent with physical observations.
A comprehensive study of the necessary and sufficient properties of the coupling conditions which lead to well-posedness of the corresponding admissible solutions is available in the framework of conservation laws with discontinuous flux, which can be seen as a simple $1-1$ network, see (2). A similar theory for conservation laws on star-shaped graph is at its beginning.
In particular, the characterization of family of solutions obtained as limits of regularizing approximations, such as vanishing viscosity limits, is still a partially open problem.
In this talk we’ll provide a general introduction to the topic, an overview of the most recent results and some explicit examples.