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arXiv · 2102.06400

Well-posedness theory for nonlinear scalar conservation laws on networks

Ulrik Skre Fjordholm·Markus Musch·Nils Henrik Risebro
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Abstract

We consider nonlinear scalar conservation laws posed on a network. We 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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Ulrik Skre Fjordholm, Markus Musch, Nils Henrik Risebro. 2021-02-12. Well-posedness theory for nonlinear scalar conservation laws on networks. https://arxiv.org/abs/2102.06400

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