# American Institute of Mathematical Sciences

June  2017, 12(2): 191-215. doi: 10.3934/nhm.2017008

## 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  May 2017

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.

Citation: Fabio Camilli, Raul De Maio, Andrea Tosin. Transport of measures on networks. Networks and Heterogeneous Media, 2017, 12 (2) : 191-215. doi: 10.3934/nhm.2017008
##### References:
 [1] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. [2] V. I. Bogachev, Measure Theory, 1st edition, Springer-Verlag, Berlin Heidelberg, 2007.  doi: 10.1007/978-3-540-34514-5. [3] G. Bretti, M. Briani and E. Cristiani, An easy-to-use algorithm for simulating traffic flow on networks: Numerical experiments, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 379-394.  doi: 10.3934/dcdss.2014.7.379. [4] M. Briani and E. Cristiani, An easy-to-use algorithm for simulating traffic flow on networks: Theoretical study, Netw. Heterog. Media, 9 (2014), 519-552.  doi: 10.3934/nhm.2014.9.519. [5] F. Camilli and C. Marchi, Stationary mean field games systems defined on networks, SIAM J. Control. Optim., 54 (2016), 1085-1103.  doi: 10.1137/15M1022082. [6] J. A. Cañizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Models Methods Appl. Sci., 21 (2011), 515-539.  doi: 10.1142/S0218202511005131. [7] E. Cristiani, B. Piccoli and A. Tosin, Multiscale Modeling of Pedestrian Dynamics, vol. 12 of MS & A: Modeling, Simulation and Applications, Springer International Publishing, 2014. doi: 10.1007/978-3-319-06620-2. [8] E. Cristiani and S. Sahu, On the micro-to-macro limit for first-order traffic flow models on networks, Netw. Heterog. Media, 11 (2016), 395-413.  doi: 10.3934/nhm.2016002. [9] C. D'Apice, R. Manzo and B. Piccoli, Modelling supply networks with partial differential equations, Quart. Appl. Math., 67 (2009), 419-440.  doi: 10.1090/S0033-569X-09-01129-1. [10] K.-J. Engel, M. K. Fijavž, R. Nagel and E. Sikolya, Vertex control of flows in networks, Netw. Heterog. Media, 3 (2008), 709-722.  doi: 10.3934/nhm.2008.3.709. [11] J. H. M. Evers, S. C. Hille and A. Muntean, Mild solutions to a measure-valued mass evolution problem with flux boundary conditions, J. Differential Equations, 259 (2015), 1068-1097.  doi: 10.1016/j.jde.2015.02.037. [12] J. H. M. Evers, S. C. Hille and A. Muntean, Measure-valued mass evolution problems with flux boundary conditions and solution-dependent velocities, SIAM J. Math. Anal., 48 (2016), 1929-1953.  doi: 10.1137/15M1031655. [13] L. Fermo and A. Tosin, A fully-discrete-state kinetic theory approach to traffic flow on road networks, Math. Models Methods Appl. Sci., 25 (2015), 423-461.  doi: 10.1142/S0218202515400023. [14] M. Garavello and B. Piccoli, Traffic Flow on Networks -Conservation Laws Models, AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences, Springfield, MO, 2006. [15] P. Gwiazda, G. Jamróz and A. Marciniak-Czochra, Models of discrete and continuous cell differentiation in the framework of transport equation, SIAM J. Math. Anal., 44 (2012), 1103-1133.  doi: 10.1137/11083294X. [16] D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Understanding Complex Systems, Springer International Publishing, 2014. doi: 10.1007/978-3-319-04621-1. [17] B. Piccoli and F. Rossi, Generalized Wasserstein distance and its application to transport equations with source, Arch. Ration. Mech. Anal., 211 (2014), 335-358.  doi: 10.1007/s00205-013-0669-x. [18] Y. V. Pokornyi and A. V. Borovskikh, Differential equations on networks, J. Math. Sci. (N. Y.), 119 (2004), 691-718.  doi: 10.1023/B:JOTH.0000012752.77290.fa. [19] D. T. H. Worm, Semigroups on Spaces of Measures, PhD thesis, Leiden University, 2010.

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##### References:
 [1] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. [2] V. I. Bogachev, Measure Theory, 1st edition, Springer-Verlag, Berlin Heidelberg, 2007.  doi: 10.1007/978-3-540-34514-5. [3] G. Bretti, M. Briani and E. Cristiani, An easy-to-use algorithm for simulating traffic flow on networks: Numerical experiments, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 379-394.  doi: 10.3934/dcdss.2014.7.379. [4] M. Briani and E. Cristiani, An easy-to-use algorithm for simulating traffic flow on networks: Theoretical study, Netw. Heterog. Media, 9 (2014), 519-552.  doi: 10.3934/nhm.2014.9.519. [5] F. Camilli and C. Marchi, Stationary mean field games systems defined on networks, SIAM J. Control. Optim., 54 (2016), 1085-1103.  doi: 10.1137/15M1022082. [6] J. A. Cañizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Models Methods Appl. Sci., 21 (2011), 515-539.  doi: 10.1142/S0218202511005131. [7] E. Cristiani, B. Piccoli and A. Tosin, Multiscale Modeling of Pedestrian Dynamics, vol. 12 of MS & A: Modeling, Simulation and Applications, Springer International Publishing, 2014. doi: 10.1007/978-3-319-06620-2. [8] E. Cristiani and S. Sahu, On the micro-to-macro limit for first-order traffic flow models on networks, Netw. Heterog. Media, 11 (2016), 395-413.  doi: 10.3934/nhm.2016002. [9] C. D'Apice, R. Manzo and B. Piccoli, Modelling supply networks with partial differential equations, Quart. Appl. Math., 67 (2009), 419-440.  doi: 10.1090/S0033-569X-09-01129-1. [10] K.-J. Engel, M. K. Fijavž, R. Nagel and E. Sikolya, Vertex control of flows in networks, Netw. Heterog. Media, 3 (2008), 709-722.  doi: 10.3934/nhm.2008.3.709. [11] J. H. M. Evers, S. C. Hille and A. Muntean, Mild solutions to a measure-valued mass evolution problem with flux boundary conditions, J. Differential Equations, 259 (2015), 1068-1097.  doi: 10.1016/j.jde.2015.02.037. [12] J. H. M. Evers, S. C. Hille and A. Muntean, Measure-valued mass evolution problems with flux boundary conditions and solution-dependent velocities, SIAM J. Math. Anal., 48 (2016), 1929-1953.  doi: 10.1137/15M1031655. [13] L. Fermo and A. Tosin, A fully-discrete-state kinetic theory approach to traffic flow on road networks, Math. Models Methods Appl. Sci., 25 (2015), 423-461.  doi: 10.1142/S0218202515400023. [14] M. Garavello and B. Piccoli, Traffic Flow on Networks -Conservation Laws Models, AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences, Springfield, MO, 2006. [15] P. Gwiazda, G. Jamróz and A. Marciniak-Czochra, Models of discrete and continuous cell differentiation in the framework of transport equation, SIAM J. Math. Anal., 44 (2012), 1103-1133.  doi: 10.1137/11083294X. [16] D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Understanding Complex Systems, Springer International Publishing, 2014. doi: 10.1007/978-3-319-04621-1. [17] B. Piccoli and F. Rossi, Generalized Wasserstein distance and its application to transport equations with source, Arch. Ration. Mech. Anal., 211 (2014), 335-358.  doi: 10.1007/s00205-013-0669-x. [18] Y. V. Pokornyi and A. V. Borovskikh, Differential equations on networks, J. Math. Sci. (N. Y.), 119 (2004), 691-718.  doi: 10.1023/B:JOTH.0000012752.77290.fa. [19] D. T. H. Worm, Semigroups on Spaces of Measures, PhD thesis, Leiden University, 2010.
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
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
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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