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

2025/10/10 by Fanch Coudreuse, Coudreuse, Fanch · 1 citation
Engineering · Environmental Science · #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics Simulations and Interactions #Geotechnical Engineering and Underground Structures #Landfill Environmental Impact Studies #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.2510.09231

openalex publication_date 2025/10/10 · openalex created_date 2025/10/13 · openalex updated_date 2026/07/28

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^∞ 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 L²_loc((0,+∞); H²(\mathbbT^d)).

Citations

• Propagation of weak log-concavity along generalised heat flows via Hamilton-Jacobi equations
• Fisher information and continuity estimates for nonlinear but 1-homogeneous diffusive PDEs (via the JKO scheme)
• On Optimal Transport Maps Between 1 /d-Concave Densities
• Lipschitz estimates on the JKO scheme for the Fokker-Plack equation on bounded convex domains
• JKO estimates in linear and non-linear Fokker-Planck equations, and Keller-Segel: L p and Sobolev bounds
• Monge-Ampère equation with bounded periodic data
• Differential Harnack inequalities for linear parabolic equations
• The Li-Yau inequality and applications under a curvature-dimension condition
• On the Jordan-Kinderlehrer-Otto scheme
• Logarithmically-concave moment measures I
• A note on Aubin-Lions-Dubinskii lemmas
• Polar factorization and monotone rearrangement of vector‐valued functions

Cited by

• Homogenization of Time-Discrete Gradient Flows

Related

• New Li--Yau--Hamilton Inequalities for the Ricci Flow via the Space-time Approach
• Sobolev estimates for the Keller-Segel system and applications to the JKO scheme
• A new Li-Yau-Hamilton estimate for Kahler-Ricci flow
• JKO schemes with general transport costs
• Strong L² H² convergence of the JKO scheme for the Fokker-Planck equation