pith. sign in

arxiv: 2604.04169 · v2 · submitted 2026-04-05 · 🧮 math.AP · math.OC

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

Fanch Coudreuse This is my paper

Pith reviewed 2026-05-14 22:08 UTC · model grok-4.3

classification 🧮 math.AP math.OC
keywords Aronson-Bénilan estimateJKO schemeporous medium equationfast diffusionBrenier potentialmaximum principleLi-Yau estimate
0
0 comments X

The pith

The JKO scheme satisfies an Aronson-Bénilan estimate for diffusion equations in dimensions 1 and 2.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives an Aronson-Bénilan / Li-Yau estimate inside the JKO time-discretization scheme for the porous-medium, heat, and fast-diffusion equations. This is done in one and two space dimensions on domains such as cubes, half-spaces, and the torus. The proof uses a maximum principle on the determinant of the Hessian of Brenier potentials that improves one step at a time along the scheme. As a result, the discrete densities obey local L^∞ bounds that stay uniform as the time step shrinks, matching the known continuous-time behavior. The work also supplies the missing optimality conditions for the fast-diffusion case.

Core claim

We derive an Aronson-Bénilan / 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^∞ 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.

What carries the argument

A maximum principle for the determinant of the Hessian of Brenier potentials, applied iteratively as a one-step improvement along the JKO scheme.

If this is right

  • Local L^∞ bounds on the density that remain uniform with respect to the time step size.
  • Consistency of the discrete estimates with the known continuous-time Aronson-Bénilan bounds.
  • Rigorous optimality conditions for the fast-diffusion equation in the JKO scheme.
  • The estimate applies uniformly on the listed simple domains in dimensions one and two.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Uniform bounds could simplify proofs of convergence for the JKO scheme to the continuous PDE as the time step vanishes.
  • Similar Hessian determinant controls might apply in related optimal transport discretizations.
  • The iterative maximum principle may extend to other variational schemes or nonlinear diffusion models on comparable domains.

Load-bearing premise

The maximum principle for the determinant of the Hessian of Brenier potentials holds and can be iterated as a one-step improvement along the scheme on the listed simple domains.

What would settle it

A numerical simulation of the JKO scheme for the heat equation in a 2D square showing that the local L^∞ bound on the density depends on the time step size and deteriorates as the step shrinks.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript derives an Aronson-Bénilan/Li-Yau estimate for the JKO scheme of the porous-medium, heat, and fast-diffusion equations in dimensions 1 and 2 on simple domains. The proof uses a maximum principle applied to the determinant of the Hessian of the Brenier potential at each proximal step, which is iterated along the scheme to obtain uniform local L^∞ bounds on the density and to derive optimality conditions in the fast-diffusion case.

Significance. If the iteration of the maximum principle holds rigorously, the result provides a discrete counterpart to classical continuous-time estimates, potentially aiding in the analysis of convergence of the JKO scheme and numerical approximations for these nonlinear diffusion equations. The byproduct on optimality conditions addresses a noted gap in the literature for fast diffusion.

major comments (2)
  1. [Main argument (abstract and §3)] The central argument rests on the maximum principle for det(D²φ) holding at each JKO step (including precise boundary conditions on cubes, quarter-spaces, half-spaces, etc.) and iterating uniformly in τ without loss of strict convexity or regularity. In dimension 2 the determinant is fully nonlinear, so any deterioration between steps would block the uniform L^∞ bound; this propagation must be verified explicitly rather than asserted as a one-step improvement.
  2. [Byproduct result (abstract and §4)] The byproduct claim that optimality conditions are rigorously derived for the fast-diffusion case requires confirmation that the argument covers the full range of m < 1 without additional regularity assumptions on the initial data or the proximal map.
minor comments (2)
  1. [Introduction and Theorem statements] Specify the precise interval for the diffusion exponent m in the statements of the main theorems and the fast-diffusion optimality conditions.
  2. [Introduction] List the exact collection of 'simple domains' (cubes, quarter-space, half-spaces, whole space, torus) at the beginning of the introduction for immediate clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major points below, providing clarifications on the rigor of the maximum principle iteration and the scope of the fast-diffusion byproduct. Revisions have been made to strengthen the presentation where the referee's observations identified opportunities for added explicitness.

read point-by-point responses

  1. Referee: [Main argument (abstract and §3)] The central argument rests on the maximum principle for det(D²φ) holding at each JKO step (including precise boundary conditions on cubes, quarter-spaces, half-spaces, etc.) and iterating uniformly in τ without loss of strict convexity or regularity. In dimension 2 the determinant is fully nonlinear, so any deterioration between steps would block the uniform L^∞ bound; this propagation must be verified explicitly rather than asserted as a one-step improvement.

    Authors: We agree that explicit verification of the iterative propagation is essential, especially in dimension 2. Section 3 establishes the maximum principle for det(D²φ) at each proximal step, with boundary conditions treated via the geometry of the domains (cubes, half-spaces, etc.) and the convexity of the Brenier potential. The bound is independent of τ, and the JKO scheme preserves strict convexity at every step by construction of the proximal map. In dimension 2 the fully nonlinear character is handled directly by the maximum principle without deterioration, as the evolution of the determinant under the proximal step does not introduce loss of regularity. To address the concern explicitly, we have added a new remark and a short inductive argument in the revised §3 confirming uniform propagation across steps. revision: partial

  2. Referee: [Byproduct result (abstract and §4)] The byproduct claim that optimality conditions are rigorously derived for the fast-diffusion case requires confirmation that the argument covers the full range of m < 1 without additional regularity assumptions on the initial data or the proximal map.

    Authors: The derivation in §4 applies to the full range m < 1. It relies solely on the local L^∞ density bounds obtained from the Aronson-Bénilan/Li-Yau estimate in dimensions 1 and 2, which hold uniformly in τ under the standard assumptions that the initial datum has finite second moment and is nonnegative (with positivity where required for the equation). No further regularity on the initial data or on the proximal map is imposed; the optimality conditions follow from the Euler-Lagrange equation for the JKO step once the density bound is available. We have expanded the discussion in the revised §4 to state the range of m explicitly and to confirm the absence of extra assumptions. revision: yes

Circularity Check

0 steps flagged

Maximum principle on det Hess of Brenier potentials iterated along JKO scheme without self-referential reduction

full rationale

The derivation applies a maximum principle to det(D²φ) for the Brenier potential φ at each proximal step in the JKO scheme, then iterates the resulting bound as a one-step improvement to obtain the Aronson-Bénilan/Li-Yau estimate and uniform-in-τ L^∞ bounds. This rests on standard optimal-transport duality and maximum-principle arguments for the Monge-Ampère equation on the listed simple domains (cubes, half-spaces, etc.), which are independent of the target estimate and do not reduce to a fitted parameter renamed as prediction or to a self-citation chain. The fast-diffusion optimality conditions are derived as a byproduct from the same construction. No load-bearing step collapses by construction to its own inputs, so the circularity score remains low.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on the existence of Brenier potentials, the validity of a maximum principle for the Hessian determinant, and standard properties of the JKO scheme on convex domains. No free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Brenier potentials exist and are sufficiently regular for the Hessian determinant to satisfy a maximum principle on the listed domains.
    Invoked to apply the maximum principle at each JKO step.
  • standard math The JKO scheme is well-defined for the porous-medium, heat, and fast-diffusion energies in dimensions 1 and 2.
    Standard background result used to set up the iteration.

pith-pipeline@v0.9.0 · 5419 in / 1363 out tokens · 28987 ms · 2026-05-14T22:08:10.036500+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages

  1. [1]

    Lectures in Mathematics

    Luigi Ambrosio, Nicola Gigli, and Giusuppe Savare.Gradient Flows: In metric spaces and in the space of probability measures. Lectures in Mathematics. ETH Z¨ urich. Birkh¨ auser Basel, 2005

    work page 2005

  2. [2]

    D. G. Aronson. Regularity Properties of Flows Through Porous Media.SIAM Journal on Applied Mathematics, 17(2):461–467, 1969

    work page 1969

  3. [3]

    D. G. Aronson and P. B´ enilan. R´ egularit´ e des solutions de l’´ equation des milieux poreux dansR⋉.CR Acad. Sci. Paris S´ er. AB, 288(2):103–105, 1979

    work page 1979

  4. [4]

    The aronson–b´ enilan estimate in lebesgue spaces

    Giulia Bevilacqua, Benoit Perthame, and Markus Schmidtchen. The aronson–b´ enilan estimate in lebesgue spaces. Annales de l’Institut Henri Poincar´ e C, Analyse non lin´ eaire, 40:259–286, 07 2022

    work page 2022

  5. [5]

    Optimal transport and Cournot-Nash equilibria.Mathematics of Operations Research, 41(1):125–145, July 2015

    Adrien Blanchet and Guillaume Carlier. Optimal transport and Cournot-Nash equilibria.Mathematics of Operations Research, 41(1):125–145, July 2015

    work page 2015

  6. [6]

    The Cauchy–Dirichlet problem for the fast diffusion equation on bounded domains.Nonlinear Analysis, 239:113394, 2024

    Matteo Bonforte and Alessio Figalli. The Cauchy–Dirichlet problem for the fast diffusion equation on bounded domains.Nonlinear Analysis, 239:113394, 2024

    work page 2024

  7. [7]

    Polar factorization and monotone rearrangement of vector-valued functions.Communications on Pure and Applied Mathematics, 44:375–417, 1991

    Yann Brenier. Polar factorization and monotone rearrangement of vector-valued functions.Communications on Pure and Applied Mathematics, 44:375–417, 1991

    work page 1991

  8. [8]

    Luis. A. Cafarelli and Avner Friedman. Regularity of the Free Boundary of a Gas Flow in an n-dimensional Porous Medium.Indiana University Mathematics Journal, 29(3):361–391, 1980

    work page 1980

  9. [9]

    Some regularity properties of solutions of Monge-Amp` ere equation.Communications on Pure and Applied Mathematics, 44:965–969, 1991

    Luis Caffarelli. Some regularity properties of solutions of Monge-Amp` ere equation.Communications on Pure and Applied Mathematics, 44:965–969, 1991

    work page 1991

  10. [10]

    Caffarelli and Avner Friedman

    Luis A. Caffarelli and Avner Friedman. Continuity of the Density of a Gas Flow in a Porous Medium. Transactions of the American Mathematical Society, 252:99–113, 1979

    work page 1979

  11. [11]

    Fisher information and continuity estimates for nonlinear but 1- homogeneous diffusive PDEs (via the JKO scheme).Bulletin of the Hellenic Mathematical Society, 2025

    Thibault Caillet and Filippo Santambrogio. Fisher information and continuity estimates for nonlinear but 1- homogeneous diffusive PDEs (via the JKO scheme).Bulletin of the Hellenic Mathematical Society, 2025. cvgmt preprint

    work page 2025

  12. [12]

    Regularity of optimal mapping between hypercubes.Advanced Nonlinear Studies, 23, 09 2023

    Shibing Chen, Jiakun Liu, and Xu-Jia Wang. Regularity of optimal mapping between hypercubes.Advanced Nonlinear Studies, 23, 09 2023

    work page 2023

  13. [13]

    Collins and Freid Tong

    Tristan C. Collins and Freid Tong. Boundary regularity of optimal transport maps on convex domains, 2025

    work page 2025

  14. [14]

    Sur le transport de mesures p´ eriodiques.Comptes Rendus de l’Acad´ emie des Sciences - Series I - Mathematics, 329(3):199–202, 1999

    Dario Cordero-Erausquin. Sur le transport de mesures p´ eriodiques.Comptes Rendus de l’Acad´ emie des Sciences - Series I - Mathematics, 329(3):199–202, 1999

    work page 1999

  15. [15]

    Li-Yau-Hamilton Inequality on the JKO Scheme for the Granular-Medium Equation, 2025

    Fanch Coudreuse. Li-Yau-Hamilton Inequality on the JKO Scheme for the Granular-Medium Equation, 2025

    work page 2025

  16. [16]

    M. G. Crandall and M. Pierre. Regularizing effects foru t = ∆ϕ(u).Trans. Amer. Math. Soc., 274(1), 1982

    work page 1982

  17. [17]

    JKO estimates in linear and non-linear Fokker–Planck equations, and Keller–Segel:L p and Sobolev bounds.Annales de l’I.H.P

    Simone Di Marino and Filippo Santambrogio. JKO estimates in linear and non-linear Fokker–Planck equations, and Keller–Segel:L p and Sobolev bounds.Annales de l’I.H.P. Analyse non lin´ eaire, 39(6):1485–1517, 2022

    work page 2022

  18. [18]

    Sobolev estimates for the Keller-Segel system and applications to the JKO scheme, 2026

    Charles Elbar. Sobolev estimates for the Keller-Segel system and applications to the JKO scheme, 2026

    work page 2026

  19. [19]

    A Li-Yau and Aronson-B´ enilan approach for the Keller-Segel system with critical exponent, 2025

    Charles Elbar, Alejandro Fern´ andez-Jim´ enez, and Filippo Santambrogio. A Li-Yau and Aronson-B´ enilan approach for the Keller-Segel system with critical exponent, 2025

    work page 2025

  20. [20]

    Lipschitz estimates on the jko scheme for the fokker–planck equation on bounded convex domains.Applied Mathematics Letters, 112:106806, 2021

    Vincent Ferrari and Filippo Santambrogio. Lipschitz estimates on the jko scheme for the fokker–planck equation on bounded convex domains.Applied Mathematics Letters, 112:106806, 2021

    work page 2021

  21. [21]

    Zurich lectures in advanced mathematics

    Alessio Figalli.The Monge-Amp` ere Equation and Its Applications. Zurich lectures in advanced mathematics. European Mathematical Society, 2017

    work page 2017

  22. [22]

    Matrix Harnack estimate for the heat equation.Communications in analysis and geometry, 1(1):113–126, 1993

    Richard S Hamilton. Matrix Harnack estimate for the heat equation.Communications in analysis and geometry, 1(1):113–126, 1993. AN ARONSON-B ´ENILAN / LI-YAU ESTIMATE IN THE JKO SCHEME IN SMALL DIMENSION 26

    work page 1993

  23. [23]

    Patacchini, and Filippo Santambrogio

    Mikaela Iacobelli, Francesco S. Patacchini, and Filippo Santambrogio. Weighted Ultrafast Diffusion Equations: From Well-Posedness to Long-Time Behaviour.Archive for Rational Mechanics and Analysis, 232:1165–1206, December 2018

    work page 2018

  24. [24]

    Kim, and Jiajun Tong

    Matt Jacobs, Inwon C. Kim, and Jiajun Tong. TheL 1-contraction principle in optimal transport.Annali Scuola Normal Superiore - Classe di Scienze, 2020

    work page 2020

  25. [25]

    On the (in)stability of the identity map in optimal transportation.Calculus of Variations and Partial Differential Equations, 58:1–25, 2017

    Yash Jhaveri. On the (in)stability of the identity map in optimal transportation.Calculus of Variations and Partial Differential Equations, 58:1–25, 2017

    work page 2017

  26. [26]

    The Variational Formulation of the Fokker–Planck Equation.SIAM Journal on Mathematical Analysis, 29(1):1–17, 1998

    Richard Jordan, David Kinderlehrer, and Felix Otto. The Variational Formulation of the Fokker–Planck Equation.SIAM Journal on Mathematical Analysis, 29(1):1–17, 1998

    work page 1998

  27. [27]

    q-moment measures and applications: a new approach via optimal transport.Journal of Convex Analysis, 28(4):1033–1052, 2021

    Huynh Khanh and Filippo Santambrogio. q-moment measures and applications: a new approach via optimal transport.Journal of Convex Analysis, 28(4):1033–1052, 2021

    work page 2021

  28. [28]

    Le.Analysis of Monge–Amp` ere Equations

    Nam Q. Le.Analysis of Monge–Amp` ere Equations. AMS, 2024

    work page 2024

  29. [29]

    Paul W. Y. Lee. On the Jordan–Kinderlehrer–Otto scheme.Journal of Mathematical Analysis and Applications, 429, 09 2015

    work page 2015

  30. [30]

    Paul W. Y. Lee. A Harnack inequality for the Jordan-Kinderlehrer-Otto scheme.Journal of Evolution Equations, 18(1):143–152, 2018

    work page 2018

  31. [31]

    On the parabolic kernel of the Schr¨ odinger operator.Acta Mathematica, 156:153–201, 07 1986

    Peter Li and Shing Yau. On the parabolic kernel of the Schr¨ odinger operator.Acta Mathematica, 156:153–201, 07 1986

    work page 1986

  32. [32]

    Monge-Amp` ere equation with bounded periodic data.Analysis in Theory and Applications, 2019

    Yanyan Li and Siyuan Lu. Monge-Amp` ere equation with bounded periodic data.Analysis in Theory and Applications, 2019

    work page 2019

  33. [33]

    Peng Lu, Lei Ni, Juan-Luis V´ azquez, and C´ edric Villani. Local Aronson–B´ enilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds.Journal de Math´ ematiques Pures et Appliqu´ ees, 91(1):1–19, 2009

    work page 2009

  34. [34]

    A synthetic approach to comparison principles for variational problems, with applications to optimal transport, 2025

    Flavien L´ eger and Maxime Sylvestre. A synthetic approach to comparison principles for variational problems, with applications to optimal transport, 2025

    work page 2025

  35. [35]

    Polar factorization of maps on Riemannian manifolds.Geometric and Functional Analysis, 11:589–608, 08 2001

    Robert McCann. Polar factorization of maps on Riemannian manifolds.Geometric and Functional Analysis, 11:589–608, 08 2001

    work page 2001

  36. [36]

    Robert J. McCann. A Convexity Principle for Interacting Gases.Advances in Mathematics, 128:153–179, 1997

    work page 1997

  37. [37]

    The Geometry of Dissipative Evolution Equations: The Porous Medium Equation.Comm Partial Differential Equations, 26, 04 2000

    Felix Otto. The Geometry of Dissipative Evolution Equations: The Porous Medium Equation.Comm Partial Differential Equations, 26, 04 2000

    work page 2000

  38. [38]

    BV Estimates in Optimal Transportation and Applications.Archive for Rational Mechanics and Analysis, 219, 03 2015

    Guido Philippis, Alp´ ar M´ esz´ aros, Filippo Santambrogio, and Bozhidar Velichkov. BV Estimates in Optimal Transportation and Applications.Archive for Rational Mechanics and Analysis, 219, 03 2015

    work page 2015

  39. [39]

    Springer, 2015

    Filippo Santambrogio.Optimal transport for applied mathematicians. Springer, 2015

    work page 2015

  40. [40]

    Dealing with moment measures via entropy and optimal transport.Journal of Functional Analysis, 271(2):418–436, 2016

    Filippo Santambrogio. Dealing with moment measures via entropy and optimal transport.Journal of Functional Analysis, 271(2):418–436, 2016

    work page 2016

  41. [41]

    StrongL 2H 2 convergence of the JKO scheme for the Fokker- Planck equation.Archive for Rational Mechanics and Analysis, 248, 10 2024

    Filippo Santambrogio and Gayrat Toshpulatov. StrongL 2H 2 convergence of the JKO scheme for the Fokker- Planck equation.Archive for Rational Mechanics and Analysis, 248, 10 2024

    work page 2024

  42. [42]

    Oxford Science Publications, 09 2007

    Juan Luis Vazquez.The Porous Medium Equation: Mathematical Theory. Oxford Science Publications, 09 2007

    work page 2007

  43. [43]

    AMS, 01 2003

    C´ edric Villani.Topics in Optimal Transportation Theory, volume 58. AMS, 01 2003

    work page 2003

  44. [44]

    Springer, 01 2008

    C´ edric Villani.Optimal transport – Old and New, volume 338. Springer, 01 2008

    work page 2008

  45. [45]

    S.-T. Yau. On the Harnack inequalities of partial differential equations.Comm. Anal. Geom., 2(4), 1994

    work page 1994