*Accepted Paper*

**Inserted:** 16 apr 2010

**Last Updated:** 17 feb 2011

**Journal:** SIAM J. Math. Anal.

**Year:** 2010

**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 $\ell$ covered by a mass $m$ is proportional to $m^\alpha\ell$ with $0<\alpha<1$. The optimization of this criterion let branched structures appear and is suitable to applications like road systems, blood vessels, river networks... 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 Kantorovich optimal transport. The movement is represented by a path $\rho_t$ of probabilities, connecting an initial state $\mu_0$ to a final state $\mu_1$, satisfying the continuity equation $\partial_t\rho+\mathrm{div}_x q=0$ together with a velocity field $v$ (with $q=\rho v$ being the momentum). The transportation cost to be minimized is non-convex and finite on atomic measures: $\int_0^1\big(\int_\Omega\rho^{\alpha-1}

q

\,d\#(x)\big)\,dt$.

**Keywords:**
Optimal transport, Branched transport, continuity equation, functionals on spaces of measures

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