Speaker
Description
The Li–Yau and Aronson–Bénilan estimates provide explicit lower bounds on the Laplacian of the pressure associated with solutions to the heat and porous medium equations. A key feature of these estimates is their independence from the initial datum. They can therefore be used to establish various regularization effects, quantitative Harnack inequalities, and long-time asymptotics for these flows. Motivated by these applications, we investigate analogous estimates for their gradient flow discretization: the JKO scheme.
This question was first studied by P. W. Lee in 2015, who proved that the stronger Hamilton matrix inequality holds for the JKO scheme for the heat equation on the torus. We will show how to extend this result, still on the torus, to more general equations of granular medium type, and present some applications of these estimates at the discrete level. We will then explain how to establish an Aronson–Bénilan-type estimate for the JKO scheme in dimensions one and two.