Measure-valued solutions to nonlocal transport equations on networks

TL;DR

This paper develops a measure-theoretic framework for nonlinear nonlocal transport equations on networks, establishing existence, uniqueness, and continuous dependence of solutions, with a representation formula and an example of nonlocal velocity.

Contribution

It introduces a novel measure-theoretic approach for solving nonlinear nonlocal transport equations on networks, including a representation formula and an example fitting the framework.

Findings

Proved existence and uniqueness of solutions.

Established continuous dependence on initial data.

Provided a representation formula for solutions.

Abstract

We study a nonlinear transport equation defined on an oriented network where the velocity field depends not only on the state variable, but also on the solution itself. We prove existence, uniqueness and continuous dependence results for the solution of the problem intended in a suitable measure-theoretic sense. We also provide a representation formula in terms of the push-forward of the initial and boundary data along the network and discuss an example of nonlocal velocity field fitting our framework.