Well-posedness theory for nonlinear scalar conservation laws on networks

TL;DR

This paper develops a comprehensive well-posedness framework for nonlinear scalar conservation laws on networks, establishing stability, uniqueness, and convergence of solutions using finite volume methods and stationary states.

Contribution

It introduces a novel approach centered on stationary states to prove existence, stability, and uniqueness for conservation laws on networks, including for monotone fluxes with upwind schemes.

Findings

Proved $L^1$ stability and uniqueness of solutions.

Established convergence of finite volume methods to the entropy solution.

Demonstrated the theory through numerical experiments.

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.