Calculus of Variations and Geometric Measure Theory

S. Di Marino - F. Santambrogio

JKO estimates in linear and non-linear Fokker-Planck equations, and Keller-Segel: $L^p$ and Sobolev bounds

created by santambro on 24 Nov 2019
modified on 10 Sep 2021


Accepted Paper

Inserted: 24 nov 2019
Last Updated: 10 sep 2021

Journal: Annales de l'Institut Henri Poincaré C, Analyse Non Linéaire
Year: 2021


We analyze some parabolic PDEs with different drift terms which are gradient flows in the Wasserstein space and consider the corresponding discrete-in-time JKO scheme. We prove with optimal transport techniques how to control the $L^p$ and $L^\infty$ norms of the iterated solutions in terms of the previous norms, essentially recovering well-known results obtained on the continuous-in-time equations. Then we pass to higher order results, and in particulat to some specific BV and Sobolev estimates, where the JKO scheme together with the so-called ``five gradients inequality'' allows to recover some inequalities that can be deduced from the Bakry-Emery theory for diffusion operators, but also to obtain some novel ones, in particular for the Keller-Segel chemiotaxis model.