Nonlinear kinetic Fokker-Planck equations: existence and diffusion limits

Technical Summary: Nonlinear Kinetic Fokker-Planck Equations: Existence and Diffusion Limits

Problem Statement
This paper investigates a class of nonlinear kinetic Fokker-Planck equations where the nonlinearity arises from a nonlinear diffusion term in the velocity variable. The primary equation under study is:
$$\partial_t f + v \cdot \nabla_x f = \Delta_v f^m + \nabla_v \cdot (v f),$$
defined for $t > 0$, $x \in \Omega$ (where $\Omega$ is either the torus $\mathbb{T}^d$ or the whole space $\mathbb{R}^d$), and $v \in \mathbb{R}^d$. The parameter $m$ governs the nature of the diffusion: $m > 1$ corresponds to the porous medium regime (degenerate diffusion), while $0 < m < 1$ corresponds to the fast diffusion regime (singular diffusion).

The central objectives are twofold:

1. To establish the existence and uniqueness of global-in-time weak solutions for this equation in suitable Lebesgue spaces.
2. To rigorously derive the macroscopic diffusion limit of this kinetic equation as the scaling parameter $\varepsilon \to 0$. The authors aim to show that the macroscopic density $\rho(t, x) = \int f \, dv$ converges to a solution of a nonlinear diffusion equation of the form $\partial_t \rho = \Delta_x (\nu_1 \rho^k)$, where the exponent $k$ is determined by $m$ and the dimension $d$.

Methodology
The paper employs a combination of functional analysis, entropy methods, and compactness arguments.

• Existence of Solutions: The authors utilize a regularization strategy. They first solve a regularized, uniformly elliptic version of the equation using a fixed-point argument (based on Lions' approach involving averaging lemmas and $H^1_v$ regularity). They then pass to the limit in the regularization parameters.

  • For $m > 1$, the proof relies on deriving a priori estimates for the entropy and moments, utilizing the convexity of the entropy functional.
  • For $m < 1$, the analysis is more delicate due to the heavy tails of the equilibrium states and the potential singularity of the diffusion. The authors establish bounds on specific moments of the solution (in both $x$ and $v$) and utilize generalized Carlson-Levin inequalities to control the nonlinear terms.
  • Uniqueness is proven via a relative entropy argument applied to the difference of two solutions, leveraging the monotonicity of the nonlinear operator.

• Diffusion Limits: Two distinct approaches are developed to prove the convergence of the rescaled solution $f_\varepsilon$ to the local equilibrium $G[\rho]$:

  1. Compactness Method: This approach relies on the entropy inequality to derive uniform bounds on the family of solutions. By establishing tightness in $L^1$ and using velocity averaging lemmas, the authors prove the strong convergence of the macroscopic density $\rho_\varepsilon$ and the distribution function $f_\varepsilon$. This method requires fewer assumptions on the initial data but imposes stricter constraints on the exponent $m$ (particularly for $m < 1$).
  2. Relative Entropy Method: This approach compares the kinetic solution $f_\varepsilon$ to a local equilibrium $G[\rho_0, \varepsilon w_0]$ constructed from the solution $\rho_0$ of the limiting macroscopic equation. The authors define a specific relative entropy functional that accounts for the macroscopic flux. By deriving a differential inequality for this functional (Proposition 1.8), they establish a convergence rate. This method is more robust regarding the range of $m$ (covering the full range $m > \frac{d}{d+2}$) but requires the existence of smooth solutions to the limiting macroscopic equation.

Key Contributions and Results

• Well-posedness: The paper proves the existence and uniqueness of global weak solutions for the nonlinear kinetic Fokker-Planck equation for both $m > 1$ and $m < 1$ (with specific lower bounds on $m$ depending on the domain $\Omega$).

  • For $m > 1$, solutions exist for initial data in $L^1 \cap L^\infty$ with finite second moments.
  • For $m < 1$, existence is shown for $m \in (\frac{d-2}{d}, 1)$ on the torus and $m \in (\frac{d-1}{d}, 1)$ in $\mathbb{R}^d$, with additional moment conditions required for the whole space.
  • The solutions satisfy a crucial entropy inequality, which is central to the subsequent analysis.

• Macroscopic Limits: The authors rigorously justify the diffusion limit. They demonstrate that as $\varepsilon \to 0$, the distribution function $f_\varepsilon$ converges strongly to the local equilibrium $G[\rho](v)$, where $\rho$ solves the nonlinear diffusion equation:
  $$\partial_t \rho = \Delta_x (\nu_1 \rho^k).$$
  The exponent $k$ is linked to $m$ via the relation $\frac{1}{k-1} = \frac{1}{m-1} + \frac{d}{2}$.

  • When $m > 1$, the limit is a porous medium equation ($k > 1$).
  • When $\frac{d}{d+2} < m < 1$, the limit is a fast diffusion equation ($0 < k < 1$).

• Convergence Rates: Through the relative entropy method, the paper provides a quantitative convergence rate for the kinetic solution to the macroscopic limit, provided the macroscopic solution is sufficiently smooth.

Significance
The paper claims to bridge the gap between microscopic stochastic particle dynamics and macroscopic continuum models for systems with nonlinear velocity diffusion. Its significance lies in:

1. Extending Classical Theory: It generalizes the well-known diffusion limits of linear Fokker-Planck equations to the nonlinear regime, covering both porous medium and fast diffusion types.
2. Rigorous Justification: It provides a rigorous mathematical derivation of the macroscopic equations, moving beyond formal asymptotic expansions.
3. Dual Methodological Approach: By presenting both a compactness-based proof (robust for rough data) and a relative entropy-based proof (providing rates and valid for a wider range of $m$), the paper offers a comprehensive toolkit for analyzing such kinetic limits.
4. Entropy Estimates: The work highlights the utility of relative entropy methods in the context of diffusion limits for nonlinear kinetic equations, a technique traditionally more common in the study of the Boltzmann equation or purely macroscopic nonlinear diffusion.

The authors note that the intermediate regime where the equilibrium lacks a variance (specifically $m \in (\frac{d-2}{d}, \frac{d}{d+2}]$) is excluded from the diffusion limit analysis, as it may lead to fractional regimes outside the scope of the current work.