Geodesic
A continuous model of transportation revisited
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXII, Tome 411 (2013), pp. 5-37

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We review two models of optimal transport, where congestion effects during the transport can be possibly taken into account. The first model is Beckmann's one, where the transport activities are modeled by vector fields with given divergence. The second one is the model by Carlier et al. (SIAM J. Control Optim 47: 1330–1350, 2008), which in turn is the continuous reformulation of Wardrop's model on graphs. We discuss the extensions of these models to their natural functional analytic setting and show that they are indeed equivalent, by using Smirnov decomposition theorem for normal $1$-currents. 
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     author = {L. Brasco and M. Petrache},
     title = {A continuous model of transportation revisited},
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     volume = {411},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a0/}
}
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L. Brasco; M. Petrache. A continuous model of transportation revisited. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXII, Tome 411 (2013), pp. 5-37. http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a0/