  This paper is concerned with six variational problems and their mutual
connections: The quadratic Monge-Kantorovich optimal transport, the
Schr\"odinger problem, Brenier's relaxed model for incompressible fluids, the
so-called Br\"odinger problem recently introduced by M. Arnaudon & al. [3], the
multiphase Brenier model, and the multiphase Br\"odinger problem. All of them
involve the minimization of a kinetic action and/or a relative entropy of some
path measures with respect to the reversible Brownian motion. As the viscosity
parameter $\nu\to 0$ we establish Gamma-convergence relations between the
corresponding problems, and prove the convergence of the associated pressures
arising from the incompressibility constraints. We also present new results on
the time-convexity of the entropy for some of the dynamical interpolations.
Along the way we extend previous results by H. Lavenant [30] and J-D. Benamou &
al. [10].
