The Li-Yau inequalities, notably the more robust Hamilton's matrix inequality, offer a lower-bound assessment for the Hessian of the pressure in solutions to the heat equation. This proves valuable as it facilitates the derivation of additional estimates for the solution, including the Harnack inequality for the heat flow.
When exploring the discretization of the heat equation through the JKO method, a natural question arises concerning the persistence of these inequalities. In this presentation, we will delve into a paper by P. L.W Lee that provides insights into deriving a second-order estimate on the torus, and I will discuss possible generalizations of this result.