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

Models for branched networks are often expressed as the minimization of an energy $M^{\alpha }$ over vector measures concentrated on $1$-dimensional rectifiable sets with a divergence constraint. We study a Modica-Mortola type approximation $M_{\epsilon }^{\alpha }$, 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 $\min {M^\alpha_\varepsilon (u);:;\nabla\cdot u=f^±f^-}$. We prove some uniform estimates on these pseudo-distances which allow us to establish a $\mathrm{\Gamma }$-convergence result for these energies with a divergence constraint.