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

Fanch Coudreuse

TL;DR

The paper develops a Li–Yau–Hamilton-type Hessian bound for the Granular–Medium equation on the torus and transfers this estimate to the JKO time-discrete scheme. It introduces a Hessian-based pressure variable $u[\rho]$ and proves asymptotic Hessian lower bounds that yield quantitative Lipschitz, $L^{\infty}$, and Harnack estimates, both in the continuous setting and along the JKO scheme. A discrete comparison principle and one-step semi-convexity improvement bridge the continuous and discrete analyses, yielding sharp asymptotics for the induced Hessian bounds as the time step $\tau\to0$. These estimates then enable local-in-time convergence results for the JKO flow to the continuous Granular–Medium (and, in the Fokker–Planck specialization, to $L^2_{\text{loc}}(H^2)$ convergence). The work therefore provides quantitative regularity and convergence tools for diffusion-type gradient flows in periodic settings, with potential extensions to related nonlinear diffusions.

Abstract

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 $L^\infty$ 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 $L^2_{\mathrm{loc}}((0,+\infty); H^2(\mathbb{T}^d))$.

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

TL;DR

The paper develops a Li–Yau–Hamilton-type Hessian bound for the Granular–Medium equation on the torus and transfers this estimate to the JKO time-discrete scheme. It introduces a Hessian-based pressure variable and proves asymptotic Hessian lower bounds that yield quantitative Lipschitz, , and Harnack estimates, both in the continuous setting and along the JKO scheme. A discrete comparison principle and one-step semi-convexity improvement bridge the continuous and discrete analyses, yielding sharp asymptotics for the induced Hessian bounds as the time step . These estimates then enable local-in-time convergence results for the JKO flow to the continuous Granular–Medium (and, in the Fokker–Planck specialization, to convergence). The work therefore provides quantitative regularity and convergence tools for diffusion-type gradient flows in periodic settings, with potential extensions to related nonlinear diffusions.

Abstract

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 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 .

Paper Structure

This paper contains 29 sections, 42 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Organization of the paper
  4. Notations and Conventions
  5. Acknowledgment
  6. Li-Yau-Hamilton Estimate for The Granular-Medium
  7. Notion of Solution
  8. Li-Yau-Hamilton Inequality for Granular-media equation
  9. First consequence: Lipschitz and $L^\infty$-bounds
  10. Quantitative Harnack Inequality
  11. Preliminaries on Periodic Optimal Transport and on the JKO Scheme
  12. Periodic Optimal Transport
  13. The JKO Scheme
  14. Asymptotic Li-Yau-Hamilton Estimate for the JKO Scheme
  15. One-Step Improvement of Semi-Convexity
  16. ...and 14 more sections

Key Result

Lemma 1.1

Let $u : \mathbb{R}^d \to \mathbb{R}$ be such that $D^2 u \succeq -\lambda \rm{I}_d$ in the weak-sense, with $\lambda \geq 0$ (i.e. $u$ is $-\lambda$-convex). Then one has $||\nabla u(x)||_\infty \leq \frac{1}{2} \lambda$ for any $i=1,\ldots,d$ (where $\nabla u(x)$ is understood as any element of $\

Theorems & Definitions (91)

  • Lemma 1.1: Gradient Estimate for Semi-Convex Periodic Function
  • proof
  • Theorem 1.2: Asymptotic Li-Yau-Hamilton estimate
  • Theorem 1.3: Li-Yau-Hamilton estimate
  • Theorem 1.4
  • Theorem 2.1: Gradient-Flow solutions AGS
  • Remark 1
  • Proposition 2.2: Regularity of gradient flow solution
  • proof
  • Definition 2.3: Pressure function
  • ...and 81 more