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.
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.