Some estimates on the JKO scheme

Abstract:
When a parabolic PDE is a gradient flow in the Wasserstein space W_2 one can study it via the so-called JKO scheme. This is the case for instance of the linear Fokker-Planck equation. Some quantities such as suitable L^p, Sobolev, Lipschitz or Holder norms can be estimated along the iterations via various tools (five-gradients-inequality, flow interchange, Monge-Ampère equation...) studied in the last years. I will give an informal overview of these results and techniques, mainly concentrating on the easiest case of the Fokker-Planck equation, and show some applications (for instance when one wants to improve the convergence of the JKO scheme to the solution of the PDE, obtaining strong convergence in Sobolev spaces instead of weak as measures).