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First we present two classical models of Branched Transport: the Lagrangianmodel introduced by Bernot, Caselles, Morel, Maddalena, Solimini, and theEulerian model introduced by Xia. An emphasis is put on the Lagrangian model,for which we give a complete proof of existence of minimizers in a--hopefully-- simplified manner. We also treat in detail some$\sigma$-finiteness and rectifiability issues to yield rigorously the energyformula connecting the irrigation cost I$\alpha$ to the Gilbert EnergyE$\alpha$. Our main purpose is to use this energy formula and exploit a Smirnovdecomposition of vector flows, which was proved via the Dacorogna-Moserapproach by Santambrogio, to establish the equivalence between the Lagrangianand Eulerian models.

On the Lagrangian branched transport model and the equivalence with its Eulerian formulation