'Society for Industrial & Applied Mathematics (SIAM)'
Doi
Abstract
The problem of branched transportation aims to describe the movement of masses when, due to concavity effects, they have the interest to travel together as much as possible, because the cost for a path of length ℓ covered by a mass m is proportional to mαℓ with 0<α<1. The optimization of this criterion let branched structures appear and is suitable to applications like road systems, blood vessels, river networks\dots Several models have been employed in the literature to present this transport problem, and the present paper looks at a dynamical one, similar to the celebrated Benamou-Brenier formulation of Kantorovitch optimal transport. The movement is represented by a path ρt of probabilities, connecting an initial state μ0 to a final state μ1, satisfying the continuity equation \partial_t\rho+\dive_xq=0 together with a velocity field v (with q=ρv being the momentum). The transportation cost to be minimized is non-convex and finite on atomic measures: ∫01(∫Ωρα−1∣q∣d#(x))dt