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An Aronson-Bénilan / Li-Yau estimate in the JKO scheme in small dimension

Coudreuse Fanch

Abstract

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 $L^\infty$ 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.

An Aronson-Bénilan / Li-Yau estimate in the JKO scheme in small dimension

Abstract

We derive an Aronson-Bénilan / Li-Yau estimate in the JKO scheme associated to the porous-medium, heat, and fast-diffusion equations, in dimensions and , 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 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.

Paper Structure

This paper contains 17 sections, 22 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Main Result
  3. Structure of the paper
  4. Acknowledgment
  5. Preliminaries
  6. The Wasserstein distance
  7. Brenier's theorem and Caffarelli's regularity
  8. Entropy functional
  9. Monge-Ampère measure
  10. A $L^1-L^\infty$-regularization Lemma
  11. The JKO Scheme
  12. Existence and Qualitative properties
  13. Optimality conditions
  14. Stability of the JKO scheme
  15. One-Step Improvement of Monge-Ampère lower bound
  16. ...and 2 more sections

Key Result

Theorem 1.1

Suppose that $\Omega$ is either: the torus, a cube, a quarter space, a half-space, or the whole space, in dimension $d=1$ or $2$. Then for all $k \geq 1$, $u_k^\tau := \tau p_k^\tau + \frac{1}{2} |\cdot|^2$ is convex finite on $\Omega$, and there exists a universal sequence $(X_k)_{k \geq 1}$ valued Furthermore, as $k \to +\infty$ we have $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (44)

  • Theorem 1.1: Aronson-Bénilan in JKO Scheme
  • Corollary 1.2: Local uniform $L^\infty$-bounds on the JKO
  • Definition 2.1: Wasserstein distance of order $2$
  • Theorem 2.2: Kantorovich duality
  • Theorem 2.3: Brenier - Cordero - McCann
  • Theorem 2.4: Caffarelli's regularity in torus and cubes
  • Definition 2.5: $m$-entropy
  • Proposition 2.6: Lower semi-continuity of entropy
  • proof
  • Proposition 2.7: Lower bound on entropy
  • ...and 34 more