Time-convexity of the entropy in the multiphasic formulation of the incompressible Euler equation |
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Authors: | Email author" target="_blank">Hugo?LavenantEmail author |
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Institution: | 1.Laboratoire de Mathématiques d’Orsay,Univ. Paris-Sud, CNRS, Université Paris-Saclay,Orsay Cedex,France |
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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 incompressibility constraint. In a convex domain, 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. |
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