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Uniform estimates for a Modica-Mortola type approximation of branched transportation

Models for branched networks are often expressed as the minimization of an energy $M^α$ over vector measures concentrated on $1$-dimensional rectifiable sets with a divergence constraint. We study a Modica-Mortola type approximation $M^α_\varepsilon$, introduced by Edouard Oudet and Filippo Santambrogio, which is defined over $H^1$ vector measures. These energies induce some pseudo-distances between $L^2$ functions obtained through the minimization problem $\min \{M^α_\varepsilon (u)\;:\;\nabla\cdot u=f^+-f^-\}$. We prove some uniform estimates on these pseudo-distances which allow us to establish a $Γ$-convergence result for these energies with a divergence constraint.