Transport of measures on networks

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.

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Cited by in corpus (5)

• Generalized Dynamic Programming Principle and Sparse Mean-Field Control Problems
• Superposition principle and schemes for Measure Differential Equations
• Measure-valued solutions to nonlocal transport equations on networks
• A measure theoretic approach to traffic flow optimization on networks
• A time-dependent switching mean-field game on networks motivated by optimal visiting problems