Well-posedness theory for nonlinear scalar conservation laws on networks

Markus Musch; Ulrik Skre Fjordholm; Nils Henrik Risebro; Markus Musch; Ulrik Skre Fjordholm; Nils Henrik Risebro

doi:10.3934/nhm.2021025

Networks and Heterogeneous Media

2022, Volume 17, Issue 1: 101-128. doi: 10.3934/nhm.2021025

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Well-posedness theory for nonlinear scalar conservation laws on networks

Department of Mathematics, University of Oslo, Postboks 1053, Blindern, 0316 Oslo, Norway

Received: 01 August 2021 Revised: 01 October 2021
Primary: 65M12, 35L65; Secondary: 65M08

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.
- Hyperbolic conservation laws,
- networks,
- finite volume methods,
- well-posedness,
- convergence
Citation: Markus Musch, Ulrik Skre Fjordholm, Nils Henrik Risebro. Well-posedness theory for nonlinear scalar conservation laws on networks[J]. Networks and Heterogeneous Media, 2022, 17(1): 101-128. doi: 10.3934/nhm.2021025

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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.

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