Open AccessWell-posedness theory for nonlinear scalar conservation laws on networks
Author(s) -
Ulrik Skre Fjordholm,
Markus Musch,
Nils Henrik Risebro
Publication year - 2022
Publication title -
networks and heterogeneous media
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.732
H-Index - 34
eISSN - 1556-181X
pISSN - 1556-1801
DOI - 10.3934/nhm.2021025
Subject(s) - conservation law , uniqueness , mathematics , monotone polygon , nonlinear system , entropy (arrow of time) , scalar (mathematics) , mathematical analysis , physics , geometry , quantum mechanics
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.