A simple model to handle the flow of people in emergency evacuation
situations is considered: at every point x, the velocity U(x) that individuals
at x would like to realize is given. Yet, the incompressibility constraint
prevents this velocity field to be realized and the actual velocity is the
projection of the desired one onto the set of admissible velocities. Instead of
looking at a microscopic setting (where individuals are represented by rigid
discs), here the macroscopic approach is investigated, where the unknwon is the
evolution of the density .

The M^\alpha energy which is usually minimized in branched transport problems
among singular 1-dimensional rectifiable vector measures with prescribed
divergence is approximated (and convergence is proved) by means of a sequence
of elliptic energies, defined on more regular vector fields. The procedure
recalls the Modica-Mortola one for approximating the perimeter, and the
double-well potential is replaced by a concave power.

This paper deals with the existence of optimal transport maps for some
optimal transport problems with a convex but non strictly convex cost. We give
a decomposition strategy to address this issue. As part of our strategy, we
have to treat some transport problems, of independent interest, with a convex
constraint on the displacement.