Optimal transp.& rearrang. tools for hamiltonian PDEs - Part 2 - Yann Brenier

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partie 2/2
Optimal transport and rearrangement tools for some hamiltonian PDEs with dissipation par Yann Brenier (CNRS Nice et UPMC) est le quatrième rendez-vous du Séminaire d'analyse IHP-FSMP, organisé par Jean-Yves Chemin, Sergiu Klainerman et Cédric Villani.

Résumé :
There are important hamiltonian PDEs that may produce singularities in finite time.
Sometimes, their solutions can be extended beyond singularities to the expense of some dissipation mechanism. A classical example is the so-called inviscid Burgers equation with the concepts of shock waves and entropy solutions.
A very ambitious goal would be to address the 3D Euler equations of incompressible fluid mechanics in a similar way, following Kolmogorov analysis of turbulence.
In this lecture, we present a somewhat simpler example where such a strategy can be followed using tools borrowed from rearrangement and optimal transport theories. This is related to the Vlasov-Poisson system with mono-kinetic distributions, and the related problem of "reconstructing the early universe", in cosmology, following Zeldovich, Peebles, and, more recently, Uriel Frisch.

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