Li-Yau-Hamilton Inequality on the JKO Scheme for the Granular-Medium Equation

We establish a version of the Li–Yau–Hamilton inequality for the Granular-Medium equation on the torus, both at the PDE level and for its time-discrete approximation given by the JKO scheme. We then apply this estimate to derive further quantitative results for the continuous and discrete JKO flows, including Lipschitz and $L^{\mathrm{\infty }}$ bounds, as well as a quantitative Harnack inequality. Finally, we use the regularity provided by this estimate to show that the JKO scheme for the Fokker–Planck equation converges in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2((0,+\mathrm{\infty });H^2({\mathbb{T}}^d))$.