# Time-convexity of the entropy in the multiphasic formulation of the incompressible euler equation

## TL;DR

This paper proves that the averaged entropy in Brenier's multiphasic formulation of the incompressible Euler equations is convex over time, using a novel time-discretization approach inspired by the JKO scheme.

## Contribution

It introduces a new time-discretization method to prove the convexity of entropy in the multiphasic Euler formulation, confirming Brenier's conjecture.

## Key findings

- Entropy averaged over phases is convex in time.
- A new approach using flow interchange inequalities is developed.
- The method adapts techniques from nonlinear parabolic PDEs.

## Abstract

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.