Nonlinear Diffusion Equations: Full characterization of Entropies

Technical Summary: Nonlinear Diffusion Equations: Full Characterization of Entropies

Problem Statement
This paper addresses the large-time behavior of quasilinear Fokker-Planck equations on the whole space $\mathbb{R}^d$ ($d \geq 1$) with a confining potential. The equation is given by:
$$\partial_t u = \Delta P(u) + \nabla \cdot (u \nabla V),$$
where $u \geq 0$ is the density, $P$ is a strictly monotone nonlinearity, and $V$ is a uniformly convex confinement potential ($\nabla^2 V \geq \lambda \text{Id}$). While the existence and uniqueness of a steady state $u_\infty$ are well-established under mild conditions, the primary objective is to characterize the complete family of relative entropy functionals $H_g(u|u_\infty)$ that guarantee exponential convergence of solutions to $u_\infty$ with the optimal rate $2\lambda$.

The authors aim to extend the celebrated Bakry-Émery method, originally developed for linear Fokker-Planck equations, to the nonlinear setting. Specifically, they seek to identify which nonlinearities $P$ admit the same rich family of entropies as the linear case, and conversely, for a fixed nonlinearity $P$, to determine all admissible entropy generators $g$ that yield exponential decay.

Methodology
The core methodology relies on a generalized Bakry-Émery approach. The authors analyze the time evolution of a general relative entropy functional $H_g$ and its dissipation $J_g$. The strategy involves:

1. Entropy Definition: Defining a broad class of relative entropies parameterized by a generating function $g$ (or equivalently $f = \log g'$). For non-degenerate diffusions, the functional takes the form $H_g(u|u_\infty) = \int G(u, u_\infty) dx$, where $G$ depends on the potential $V$ and the nonlinearity via the function $\phi(u) = \int_1^u \frac{P'(s)}{s} ds$.
2. Dissipation Analysis: Computing the second time derivative of the entropy. By integrating by parts and utilizing the convexity of $V$, the authors derive a remainder term $R$ that must be non-negative to ensure the differential inequality $-\frac{d}{dt}J_g - 2\lambda J_g \geq 0$.
3. Pointwise Conditions: The non-negativity of the remainder is reduced to a pointwise condition involving the nonlinearity curve $(\alpha(u), \beta(u))$, where $\alpha(u) = P(u)/u$ and $\beta(u) = P'(u)$, and the entropy curve $(f'(\xi), f''(\xi))$. This leads to a quadratic inequality $T(\alpha, \beta; f', f'') \leq 0$.
4. Geometric Characterization: The authors analyze the geometry of the admissible sets in the $(\alpha, \beta)$-plane defined by the entropy constraints. They utilize properties of quadrics (ellipses, parabolas, hyperbolas) to determine which nonlinearities are compatible with specific entropy generators.

Key Contributions and Results

1. Full Characterization of Admissible Entropies:
   The paper provides a necessary and sufficient condition for an entropy functional to be "admissible" (yielding exponential decay). This is formulated as a differential inequality for the entropy generator $f$ coupled with the nonlinearity curve $(\alpha, \beta)$.

   • Theorem 6.2 & Corollary 6.3: For a given nonlinearity $P$, the admissible functions $y = f'$ are characterized by a differential inequality involving the supremum of a specific function $\vartheta$ over the range of $u$.
   • Power-Law Nonlinearities ($P(u) = u^m$):
     • For fast-diffusion ($0 < m < 1$) with $m > \frac{d-1}{d}$, the only admissible entropy is the standard Ralston-Newman entropy ($g(\xi)=\xi$).
     • For porous-medium ($m > 1$), a continuous family of admissible entropies exists, interpolating between the standard entropy and a "maximal" entropy. These new entropies provide control over higher-order moments than the standard entropy.
     • For linear diffusion ($m=1$), the full family of $p$-entropies ($1 \leq p \leq 2$) is recovered.

2. Characterization of Admissible Nonlinearities:
   Conversely, for a fixed entropy generator $g$, the paper characterizes the set of nonlinearities $P$ for which the entropy method works.

   • Geometric Constraints: The nonlinearity curve $(\alpha(u), \beta(u))$ must lie within specific geometric regions (intersections of quadrics and the McCann condition line) in the $(\alpha, \beta)$-plane.
   • Stability of Linear Entropies: The paper investigates which nonlinear perturbations of the linear case ($P(u) \approx u$) still admit the full family of linear $p$-entropies. It is shown that if a nonlinearity coincides with the linear function on an interval but deviates elsewhere, it generally fails to admit all linear $p$-entropies simultaneously, unless it satisfies very specific structural constraints (e.g., lying inside a truncated ellipse or parabola in the parameter plane).

3. New Functional Inequalities:

   • Generalized Csiszár-Kullback Inequalities: The authors derive inequalities bounding the $L^1$ distance $\|u - u_\infty\|_{L^1}$ by the square root of the generalized entropy $H_g$. This extends previous results to a wider class of degenerate and non-degenerate diffusions.
   • Generalized Log-Sobolev Inequalities: New inequalities relating the entropy $H_g$ to its dissipation $J_g$ are established, expanding upon known results for the canonical entropy.
   • Moment Estimates: For degenerate diffusions, the new entropies are shown to control weighted $L^1$ norms, implying exponential convergence in moment-weighted spaces.

4. Sharpness of the Inequalities:
   The paper analyzes the conditions under which the entropy decay estimates are sharp (i.e., when the remainder term vanishes).

   • Theorem 9.5: Saturation of the convex Sobolev inequality occurs if and only if the initial data is a specific translation (or scaled translation) of the steady state, and the potential $V$ possesses specific quadratic structure in at least one direction.
   • For the standard entropy, saturation implies $V$ is quadratic in the direction of the translation. For the "maximal" entropy in the fast-diffusion case, saturation requires $V$ to be perfectly quadratic ($V(x) = \frac{\lambda}{2}|x|^2$).

Significance and Claims
The paper claims to provide a "full characterization" of the entropies that work for nonlinear Fokker-Planck equations, a task previously limited to specific cases or the linear setting.

• Extension of Bakry-Émery: It successfully extends the Bakry-Émery framework to nonlinear settings, identifying that the "McCann condition" ($d\beta \geq (d-1)\alpha$) is a fundamental prerequisite for the existence of any admissible entropy beyond the standard one.
• Discovery of New Functionals: For porous-medium equations, the work derives a large family of new entropies (beyond the Ralston-Newman entropy) that yield exponential convergence. These functionals are shown to control higher moments of the solution, offering finer information about the tail behavior of the density.
• Limitations in Fast Diffusion: A significant negative result is the characterization that fast-diffusion equations (with $m < 1$) admit only the standard entropy, contrasting with the richness of the porous-medium and linear cases.
• Optimality: The paper rigorously identifies the conditions for the sharpness of the derived inequalities, linking the optimality of the decay rate to the geometric structure of the confinement potential and the specific form of the initial data.

In summary, the work establishes a comprehensive theoretical framework linking the geometry of the nonlinearity curve, the structure of the confinement potential, and the admissibility of entropy functionals, thereby unifying and extending the understanding of long-time asymptotics for a broad class of nonlinear diffusion equations.