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

1. Problem Statement and Context

The JKO Scheme:
The paper investigates the Jordan-Kinderlehrer-Otto (JKO) scheme, a variational time-discretization method used to approximate gradient flows in the Wasserstein space ($W_2$). Specifically, it addresses the approximation of the porous-medium, heat, and fast-diffusion equations:
$$\partial_t \rho = \Delta \rho^m$$
where $m > 0$. The scheme iterates the minimization problem:
$$\rho_{k+1}^\tau \in \arg\min_{\eta \in \mathcal{P}_2(\Omega)} \left( E_m[\eta] + \frac{1}{2\tau} W_2^2(\eta, \rho_k^\tau) \right)$$
where $E_m$ is the entropy functional associated with the exponent $m$, and $\tau$ is the time step.

The Core Challenge:
A fundamental property of the continuous-time solutions to these equations is the Aronson-Bénilan / Li-Yau estimate. For the pressure variable $p$ (defined as $p = \frac{m}{m-1}\rho^{m-1}$ for $m \neq 1$ and $p = \log \rho$ for $m=1$), the continuous flow satisfies a sub-harmonic lower bound:
$$\Delta p \geq -\frac{\alpha_{d,m}}{t}, \quad \text{where } \alpha_{d,m} = \frac{d}{d(m-1)+2}$$
This estimate is crucial for deriving $L^1-L^\infty$ regularization effects and studying free boundaries.

The Gap:
While qualitative properties of the JKO scheme are well-understood, recovering quantitative estimates like the Aronson-Bénilan bound uniformly in the time step $\tau$ has been difficult. Previous works (e.g., Lee [30]) established Li-Yau-Hamilton type estimates for the heat equation ($m=1$) on the torus, but a general proof for the porous-medium and fast-diffusion regimes ($m \neq 1$), particularly for $m < 1$, was missing. Furthermore, the fast-diffusion case ($m < 1$) presents analytical difficulties regarding the existence of minimizers and optimality conditions due to the non-super-linear behavior of the entropy.

2. Methodology

The author employs a strategy based on maximum principles applied to the determinant of the Hessian of the Brenier potentials. The methodology proceeds in several stages:

A. Analysis of the One-Step Problem (Regime $m < 1$)

The paper first rigorously fills a gap in the literature regarding the fast-diffusion regime ($m < 1$):

• Existence and Uniqueness: Proves existence and uniqueness of minimizers for the one-step JKO problem, even on unbounded domains, provided the domain is "volume-regular."
• Optimality Conditions: Derives the Euler-Lagrange equations for minimizers. Crucially, it proves that for $m < 1$, minimizers are absolutely continuous (no singular part) and strictly positive almost everywhere. This allows for the derivation of the condition:
  $$\tau f_m'(\rho) + \frac{1}{2}|x|^2 = \text{convex function}$$
  where $f_m'$ is the derivative of the entropy density.

B. One-Step Improvement of Monge-Ampère Bounds

The core technical innovation is a one-step improvement lemma (Theorem 4.1).

• Setup: Let $u = \tau f_m'(\rho) + \frac{1}{2}|x|^2$. The author assumes a lower bound on the Monge-Ampère measure: $\det(D^2 u)^{1/d} \geq \lambda$.
• Mechanism: By analyzing the Monge-Ampère equation at a minimum point of the determinant, the author derives an algebraic inequality relating the determinant at step $k+1$ to step $k$.
• Dimensional Restriction: This algebraic inequality relies on the relationship between the trace and the determinant of the Hessian. Specifically, the proof utilizes the identity $\text{Tr}(A^{-1}) = \frac{\text{Tr}(A)}{\det(A)}$, which holds specifically for dimensions $d=1$ and $d=2$.
• Boundary Handling: For simple domains (cubes, torus), the author analyzes the behavior of the optimal transport map on the boundary. Using reflection arguments and the specific geometry of cubes, they show that the gradient of the determinant vanishes at boundary minima, allowing the interior maximum principle arguments to extend to the boundary.

C. Iterative Scheme and Asymptotics

• Sequence Construction: The one-step improvement is iterated to define a sequence $(X_k)_{k \geq 1}$ in $[0, 1]$ such that:
  $$\det(D^2 u_k)^{1/d} \geq 1 - X_k$$
• Asymptotic Analysis: The sequence $X_k$ is shown to decay as $X_k \sim \frac{1}{\alpha_{d,m} k}$.
• Recovery of Continuous Estimate: By relating the discrete step $k$ to continuous time $t \approx k\tau$, the discrete bound converges to the continuous Aronson-Bénilan estimate as $\tau \to 0$.

3. Key Contributions

1. First Discrete Aronson-Bénilan Estimate: The paper provides the first proof of an Aronson-Bénilan / Li-Yau type estimate for the JKO scheme in the porous-medium ($m>1$) and fast-diffusion ($0<m<1$) regimes.
2. Resolution of Fast-Diffusion Regularity: It rigorously establishes the existence, uniqueness, and absolute continuity of minimizers for the JKO scheme when $m < 1$, a case previously not fully treated due to the lack of strict convexity in the entropy.
3. Dimensional Constraint Strategy: The proof introduces a novel algebraic approach using the determinant of the Hessian, which is currently restricted to dimensions $d=1$ and $d=2$. This highlights a specific algebraic barrier in extending these estimates to higher dimensions without new ideas.
4. Uniform $L^\infty$ Bounds: As a direct consequence, the paper derives local $L^\infty$ bounds on the density $\rho_k^\tau$ that are uniform in the time step $\tau$, consistent with the continuous-time regularization effects.

4. Main Results

Theorem 1.1 (Aronson-Bénilan in JKO Scheme):
For dimensions $d=1, 2$ and domains $\Omega \in \{\text{Torus, Cube, Quarter-space, Half-space, Whole space}\}$, let $\rho_k^\tau$ be the JKO iterates. Define the discrete pressure potential $u_k^\tau = \tau f_m'(\rho_k^\tau) + \frac{1}{2}|x|^2$. Then:

1. $u_k^\tau$ is convex.
2. There exists a sequence $(X_k) \subset [0, 1]$ depending only on $m$ and $d$ such that:
   $$\det(D^2 u_k^\tau)^{1/d} \geq 1 - X_k$$
   in the sense of Monge-Ampère measures.
3. The sequence behaves asymptotically as $X_k \sim \frac{1}{(d(m-1)+2)k}$.

Corollary 1.2 (Local Uniform $L^\infty$ Bounds):
For any small ball $B_r \subset \Omega$, there exists a constant $M$ (independent of $\tau$) such that for $t \geq t_0$:
$$\|\rho_k^\tau\|_{L^\infty(B_r)} \leq M$$
This confirms that the JKO scheme preserves the instantaneous smoothing properties of the continuous equations.

Theorem 5.1 (Asymptotic Convergence):
As $\tau \to 0$ and $k\tau \to t$, the discrete estimate converges to the classical Aronson-Bénilan estimate:
$$\Delta f_m'(\rho_t) \geq -\frac{d}{d(m-1)+2} \frac{1}{t}$$

5. Significance and Limitations

Significance:

• Robustness of JKO: The result demonstrates that the JKO scheme is not just a convergence tool but preserves deep qualitative and quantitative regularity properties of the underlying PDEs.
• Regularity Theory: It provides a new pathway to derive $L^\infty$ bounds and regularity for discrete approximations without relying on continuous-time PDE techniques.
• Literature Gap: It settles the theoretical foundations for the fast-diffusion case in the JKO framework, which is essential for numerical analysis and theoretical studies of these equations.

Limitations:

• Dimensionality: The proof is strictly limited to dimensions 1 and 2. The algebraic inequality used to control the determinant via the trace fails in higher dimensions (where one can construct matrices with large trace but small determinant). Extending this to $d \geq 3$ remains an open problem.
• Domain Geometry: The proof relies on specific boundary behaviors (corners and faces) of simple domains like cubes and tori. Extending this to general convex domains with curved boundaries requires new techniques to handle boundary maximum points.

In summary, this paper represents a significant step forward in the rigorous analysis of the JKO scheme, successfully bridging the gap between discrete variational schemes and continuous regularity estimates for a broad class of diffusion equations in low dimensions.