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A unified relative entropy framework for macroscopic limits of Vlasov--Fokker--Planck equations

Young-Pil Choi, Jinwook Jung

TL;DR

The paper develops a unified relative entropy framework to derive quantitative macroscopic limits for Vlasov--Fokker--Planck equations with Riesz interactions and Fokker--Planck relaxation. By combining entropy dissipation, Fisher information control, and modulated interaction energies with the bounded Lipschitz distance, it yields both strong and weak convergence toward three prototypical limits: the diffusive drift--diffusion equation, the high-field drift-driven aggregation equation, and the gSQG-type equation. The results hold for well-prepared as well as mildly prepared initial data, providing explicit stability bounds and, in favorable cases, improved convergence rates such as $O(\varepsilon^2)$ in the diffusive regime. The work thus unifies disparate singular limits under a single, quantitative entropy-based methodology and highlights the role of relative entropy in linking microscopic dissipation to macroscopic dynamics in nonlocal, highly structured kinetic systems.

Abstract

We develop a unified relative entropy framework for macroscopic limits of kinetic equations with Riesz-type interactions and Fokker-Planck relaxation. The method combines entropy dissipation, Fisher-information control, and modulated interaction energies into a robust stability theory that yields both strong and weak convergence results. For the strong convergence, we establish quantitative relative entropy estimates toward macroscopic limits under well-prepared data, extending the scope of the method to settings where nonlocal forces and singular scalings play a decisive role. For the weak convergence, we prove that quantitative convergence propagates in bounded Lipschitz topologies, even when the initial relative entropy diverges with respect to the singular scaling parameter. This dual perspective shows that relative entropy provides not only a tool for strong convergence, but also a new mechanism to handle mildly prepared initial states. We establish quantitative convergence toward three prototypical limits: the diffusive limit leading to a drift-diffusion equation, the high-field limit yielding the aggregation equation in the repulsive regime, and the strong magnetic field limit producing a generalized surface quasi-geostrophic equation. The analysis highlights the unifying role of relative entropy in connecting microscopic dissipation with both strong and weak macroscopic convergence.

A unified relative entropy framework for macroscopic limits of Vlasov--Fokker--Planck equations

TL;DR

The paper develops a unified relative entropy framework to derive quantitative macroscopic limits for Vlasov--Fokker--Planck equations with Riesz interactions and Fokker--Planck relaxation. By combining entropy dissipation, Fisher information control, and modulated interaction energies with the bounded Lipschitz distance, it yields both strong and weak convergence toward three prototypical limits: the diffusive drift--diffusion equation, the high-field drift-driven aggregation equation, and the gSQG-type equation. The results hold for well-prepared as well as mildly prepared initial data, providing explicit stability bounds and, in favorable cases, improved convergence rates such as in the diffusive regime. The work thus unifies disparate singular limits under a single, quantitative entropy-based methodology and highlights the role of relative entropy in linking microscopic dissipation to macroscopic dynamics in nonlocal, highly structured kinetic systems.

Abstract

We develop a unified relative entropy framework for macroscopic limits of kinetic equations with Riesz-type interactions and Fokker-Planck relaxation. The method combines entropy dissipation, Fisher-information control, and modulated interaction energies into a robust stability theory that yields both strong and weak convergence results. For the strong convergence, we establish quantitative relative entropy estimates toward macroscopic limits under well-prepared data, extending the scope of the method to settings where nonlocal forces and singular scalings play a decisive role. For the weak convergence, we prove that quantitative convergence propagates in bounded Lipschitz topologies, even when the initial relative entropy diverges with respect to the singular scaling parameter. This dual perspective shows that relative entropy provides not only a tool for strong convergence, but also a new mechanism to handle mildly prepared initial states. We establish quantitative convergence toward three prototypical limits: the diffusive limit leading to a drift-diffusion equation, the high-field limit yielding the aggregation equation in the repulsive regime, and the strong magnetic field limit producing a generalized surface quasi-geostrophic equation. The analysis highlights the unifying role of relative entropy in connecting microscopic dissipation with both strong and weak macroscopic convergence.
Paper Structure (30 sections, 16 theorems, 288 equations)

This paper contains 30 sections, 16 theorems, 288 equations.

Table of Contents

  1. Introduction
  2. Functional frameworks
  3. Relative entropy
  4. Energy and modulated functionals
  5. Bounded Lipschitz distance
  6. Main results
  7. Diffusive limit
  8. High-field limit
  9. gSQG limit
  10. Novelty and contributions
  11. Diffusive limit
  12. High-field limit
  13. gSQG limit
  14. Organization of paper
  15. Preliminaries
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $T>0$, and let $\rho \in L^\infty([0,T];\mathcal{P}(\mathbb R^d))$ be the unique classical solution to eq_agdi satisfying Let $\{f^\varepsilon\}_{\varepsilon > 0}$ be a family of weak entropy solutions to the equation eq_diff_sca on the time interval $[0,T]$, with initial data $\{f^\varepsilon_0\}_{\varepsilon > 0}$ satisfying If the initial data is well-prepared in the sense that then we h

Theorems & Definitions (41)

  • Remark 1.1
  • Definition 1.1
  • Theorem 1.1: Diffusive limit
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.2: High-field limit
  • Remark 1.6
  • Remark 1.7
  • ...and 31 more