Accepted Paper

Inserted: 24 nov 2019
Last Updated: 10 sep 2021

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

Abstract:

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.

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