Transport of measures on networks

Fabio Camilli; Raul De Maio; Andrea Tosin

doi:10.3934/nhm.2017008

Article Contents

2017, Volume 12, Issue 2: 191-215. Doi: 10.3934/nhm.2017008

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Transport of measures on networks

1.
Dipartimento di Scienze di Base e Applicate per l'Ingegneria, "Sapienza" Università di Roma, Via Scarpa 16, 00161 Rome, Italy
2.
Department of Mathematical Sciences "G. L. Lagrange", Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Turin, Italy

Received: October 2016

Revised: February 2017

Published: October 2017

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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 and 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.

Keywords:

Mathematics Subject Classification: 35R02, 35Q35, 28A50.

Citation:

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Figure 1. Sketch of the characteristics of problem (17) in the two cases $\tau(0)=\sigma(0)<T$ (left) and $\tau(0)=\sigma(0)>T$ (right). For pictorial purposes we imagine a constant velocity field, so that the characteristics are straight lines in the space-time

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Figure 2. The 1-2 junction with a sketch of the characteristics along which the solution to the example of Section 6.1 propagates in the space-time of the network

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Figure 3. The 2-1 junction with a sketch of the characteristics along which the solution to the example of Section 6.3 propagates in the space-time of the network

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