JKO estimates in linear and non-linear Fokker–Planck equations, and Keller–Segel: Lp and Sobolev bounds

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 LP and L1 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 particular to some specific BV and Sobolev estimates, where the JKO scheme together with the so-called "five gradients inequality" allows us to recover some estimates 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 chemotaxis model.