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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Notes
Let us precise that this idea is adapted from an ongoing work [14] with Filippo Santambrogio where the same kind of technique is used to provide regularity for solutions of quadratic Mean Field Games.
Strictly speaking, in [12], it is required that \(\Omega \) has a piecewise \(C^1\) boundary, but this assumption is only used to prove that the Minkowski functional of \(\Omega \) is Lipschitz. If \(\Omega \) is convex, then its Minkowski functional is convex, hence Lipschitz. Thus, one can drop the assumption of a piecewise \(C^1\) boundary if \(\Omega \) is convex.
One may worry about the non uniqueness of the geodesic and hence of the fact that the extension operator \(E_N\) is ill-defined. However, it is a classical result of optimal transport that the constant-speed geodesic joining two measures is unique as soon as one of the two measures is absolutely continuous w.r.t. \(\mathcal {L}\). Moreover, for a traffic plan \(Q \in \mathcal {P}(\Gamma _{T^N})\), if \(H_Q(t) < + \infty \) for \(t \in T^N\), then Q-a.e. \(\rho \) is absolutely continuous w.r.t. \(\mathcal {L}\) at time t. Thus as long as we work with \(W_2\)-traffic pans Q such that \(H_Q(k \tau ) < + \infty \) for any \(k \in \{ 1,2, \ldots , N-1 \}\) (and we leave it to the reader to check that it is the case), the operator \(E_N\) is well defined.
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Acknowledgements
The author acknowledges the support of ANR project ISOTACE (ANR-12-MONU-0013). He also thanks Filippo Santambrogio, Aymeric Baradat, Paul Pegon and Yann Brenier for fruitful discussions and advice.
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Communicated by L. Ambrosio.
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Lavenant, H. Time-convexity of the entropy in the multiphasic formulation of the incompressible Euler equation. Calc. Var. 56, 170 (2017). https://doi.org/10.1007/s00526-017-1262-1
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DOI: https://doi.org/10.1007/s00526-017-1262-1