Well-posedness of general 1D Initial Boundary Value Problems for scalar balance laws

TL;DR

This paper establishes the well-posedness and stability of solutions for a broad class of 1D scalar balance law initial boundary value problems, using a Lax-Friedrichs scheme and Kruzhkov's techniques.

Contribution

It proves well-posedness and stability under general conditions, extending previous results to more complex flux and source functions.

Findings

Solutions exist and are unique under broad assumptions.

Solutions depend continuously on flux and source variations.

The convergence of a numerical scheme is demonstrated.

Abstract

We focus on the initial boundary value problem for a general scalar balance law in one space dimension. Under rather general assumptions on the flux and source functions, we prove the well-posedness of this problem and the stability of its solutions with respect to variations in the flux and in the source terms. For both results, the initial and boundary data are required to be bounded functions with bounded total variation. The existence of solutions is obtained from the convergence of a Lax-Friedrichs type algorithm with operator splitting. The stability result follows from an application of Kru\v{z}kov's doubling of variables technique, together with a careful treatment of the boundary terms.