@article{Musch2022, 
author = {Markus Musch and Ulrik Skre Fjordholm and Nils Henrik Risebro},
title = {Well-posedness theory for nonlinear scalar conservation laws on networks},
year = {2022},
journal = {Networks and Heterogeneous Media},
volume = {17},
number = {1},
pages = {101-128},
keywords = {Hyperbolic conservation laws, networks, finite volume methods, well-posedness, convergence},
url = {https://www.sciopen.com/article/10.3934/nhm.2021025},
doi = {10.3934/nhm.2021025},
abstract = {We consider nonlinear scalar conservation laws posed on a network. We define an entropy condition for scalar conservation laws on networks and establish        L    1   stability, and thus uniqueness, for weak solutions satisfying the entropy condition. We apply standard finite volume methods and show stability and convergence to the unique entropy solution, thus establishing existence of a solution in the process. Both our existence and stability/uniqueness theory is centred around families of stationary states for the equation. In one important case – for monotone fluxes with an upwind difference scheme – we show that the set of (discrete) stationary solutions is indeed sufficiently large to suit our general theory. We demonstrate the method's properties through several numerical experiments.}
}