An Aronson-B\'enilan / Li-Yau estimate in the JKO scheme in small dimension

TL;DR

This paper establishes Aronson-Bénilan and Li-Yau estimates within the JKO scheme for porous-medium, heat, and fast-diffusion equations in low dimensions, leading to uniform bounds and optimality conditions.

Contribution

It introduces a novel maximum principle approach for Hessian determinants of Brenier potentials, applicable in small dimensions, to derive key estimates in the JKO scheme.

Findings

Obtained local $L^ abla$ bounds on density uniform in time step.

Derived optimality conditions for fast-diffusion equations.

Extended estimates to simple domains like cubes and half-spaces.

Abstract

We derive an Aronson-B\'enilan / Li-Yau estimate in the JKO scheme associated to the porous-medium, heat, and fast-diffusion equations, in dimensions $1$ and $2$, and on simple domains (cubes, quarter-space, half-spaces, whole space, and the torus). Our method is based on a maximum principle for the determinant of the Hessian of Brenier potentials, iterated as a one-step improvement along the scheme. As a consequence, we obtain local $L^\infty$ bounds on the density, uniform in the time step, consistent with the continuous-time result. As a byproduct, we rigorously derive the optimality conditions in the fast-diffusion case, filling a gap in the literature.