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, using an {\it ad hoc} generalization of Smirnov decomposition theorem for flat $1-$currents, which may fail to be normal.
A continuous model of transportation revisited
BRASCO, Lorenzo;
2013
Abstract
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, using an {\it ad hoc} generalization of Smirnov decomposition theorem for flat $1-$currents, which may fail to be normal.I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
