  We study the multiphasic formulation of the incompressible Euler equation
introduced by Brenier: infinitely many phases evolve according to the
compressible Euler equation and are coupled through a global in-compressibility
constraint. We are able to prove that the entropy, when averaged over all
phases, is a convex function of time, a result that was conjectured by Brenier.
The novelty in our approach consists in introducing a time-discretization that
allows us to import a flow interchange inequality previously used by Matthes,
McCann and Savar{\'e} to study first order in time PDE, namely the JKO scheme
associated with non-linear parabolic equations.
