Analysis of a system of nonlocal balance laws with weighted work in progress

Abstract

We consider a system of nonlocal balance laws, modeling a multi-commodity supply edge. The coupling of the different goods is realized via a “weighted work in progress” (WWIP), such that the velocity of every balance law also depends on the “load” of the other commodities. This WWIP represents nonlocal impact and due to it, the proposed model offers it to weight goods which are close to the “entrance” different from goods which leave the supply edge. It also allows the nonlocal term to have an impact on the solution only on a subset of the space domain, changing over time. Incorporating weights and source terms in the right-hand side, we demonstrate the existence of weak solutions for the resulting model and, under some restrictions on the weight and the inhomogeneity, we ensure the uniqueness of weak solutions. Furthermore, we consider source terms which depend pointwise on the solution and prove existence of a solution under suitable conditions.