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 ap- propriate 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.

Transport of measures on networks / Camilli, Fabio; DE MAIO, Raul; Tosin, Andrea. - In: NETWORKS AND HETEROGENEOUS MEDIA. - ISSN 1556-1801. - STAMPA. - 12:2(2017), pp. 191-215. [10.3934/nhm.2017008]

### 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 ap- propriate 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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Network, transport equation, measure valued solutions, distribution conditions
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Transport of measures on networks / Camilli, Fabio; DE MAIO, Raul; Tosin, Andrea. - In: NETWORKS AND HETEROGENEOUS MEDIA. - ISSN 1556-1801. - STAMPA. - 12:2(2017), pp. 191-215. [10.3934/nhm.2017008]
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11573/953531`
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