This paper is concerned with six variational problems and their mutual connections: The quadratic Monge-Kantorovich optimal transport, the Schrödinger problem, Brenier's relaxed model for incompressible fluids, the so-called Brödinger problem recently introduced by M. Arnaudon & al. [3], the multiphase Brenier model, and the multiphase Brödinger 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 ν → 0 we establish Gammaconvergence 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].
In the Brenier variational model for perfect fluids, the datum is the joint law of the initial and final positions of the particles. In this paper, we show that both the optimal action and the pressure field are Hölder continuous with respect to this datum metrized in Monge-Kantorovic distance.
In [4], Arnaudon, Cruzeiro, Léonard and Zambrini introduced an entropic regularization of the Brenier model for perfect incompressible fluids. We show that as in the original setting, there exists a scalar pressure field which is interpreted as the Lagrange multiplier associated to the incompressibility constraint. The proof goes through a reformulation of the problem in PDE terms. 1 If X is a polish space, P(X ) stands for the set of Borel probability measures on X .
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