We will start with a paper by Lee (https://arxiv.org/abs/1408.3180) whose vague title "on the JKO scheme" suggests that it can be of interest for us but does not provide hints on the content: in such a paper it is proven that the Lipschitz constant of \log\rho+V can be controlled along the iterations of the JKO scheme when the functional is the entropy + the potential energy given by V. Lee considers the case of the torus. I will show the main tools to prove the bound, and explain how to adapt to the case of the ball and of more general convex domains (results that I obtained with V. Ferrari some years ago), and which are the computational consequences of this estimate, besides regularity. If time permits, I will also show how to obtain a Hölder bound, how to recover the Lipschitz bound from the Hölder one, and why the Hölder estimates are also interesting.
