Transport of measures on networks

TL;DR

This paper develops a measure-valued linear transport theory on networks, providing explicit solutions for arc-level problems and constructing global solutions by connecting arcs, enabling multiscale flow analysis.

Contribution

It introduces a novel measure-valued framework for transport on networks, including explicit solutions and a method to assemble global solutions from arc solutions.

Findings

Explicit solution formula for measure-valued transport on an interval.

A method to assemble global network solutions from arc solutions.

Framework suitable for multiscale flows with microscopic and macroscopic phases.

Abstract

In this paper we formulate a theory of measure-valued linear transport equations on networks. The building block of our approach is the initial/boundary-value problem for the measure-valued linear transport equation on a bounded interval, which is the prototype of an arc of the network. For this problem we give an explicit representation formula of the solution, which also considers the total mass flowing out of the interval. Then we construct the global solution on the network by gluing all the measure-valued solutions on the arcs by means of appropriate distribution rules at the vertexes. The measure-valued approach makes our framework suitable to deal with multiscale flows on networks, with the microscopic and macroscopic phases represented by Lebesgue-singular and Lebesgue-absolutely continuous measures, respectively, in time and space.