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

Hugo Lavenant (1)

1 LMO - Laboratoire de Mathématiques d'Orsay

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 Savaré to study first order in time PDE, namely the JKO scheme associated with non-linear parabolic equations.

Mots clés: Flow interchange; Euler equations; Wasserstein space

Domaines: Equations aux dérivées partielles [math.AP]

Hugo Lavenant. Time-convexity of the entropy in the multiphasic formulation of the incompressible euler equation. 2017. ⟨hal-01442103v3⟩