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Derivation of nonlinear aggregation-diffusion equation from a kinetic BGK-type equation

Young-Pil Choi, Jeongho Kim, Oliver Tse

TL;DR

This work rigorously derives a nonlinear aggregation-diffusion PDE with porous-medium-type diffusion from a scaled BGK-type kinetic model. It introduces a weak entropy framework and leverages a relative entropy structure together with velocity averaging to obtain uniform estimates and strong compactness, enabling a diffusion limit without bounded initial data. Under milder assumptions on the potentials $V$ and $K$, the authors establish global existence of weak entropy solutions to the kinetic model and prove convergence to the limit equation, including convergence of the kinetic density to the local equilibrium $M_\gamma[\rho]$ and convergence of $f^\varepsilon$ to $M_\gamma[\rho]$. The results extend prior works by removing boundedness requirements on the initial data and providing a robust convergence analysis that handles nonlocal interactions and porous-medium diffusion. Overall, the paper advances understanding of diffusion limits in kinetic-to-continuum transitions and strengthens the mathematical foundation for nonlinear drift-diffusion models arising in swarming, granular media, and pattern formation.

Abstract

This paper investigates the diffusion limit of a kinetic BGK-type equation, focusing on its relaxation to a nonlinear aggregation-diffusion equation, where the diffusion exhibits a porous-medium-type nonlinearity. Unlike previous studies by Dolbeault et al. [Arch. Ration. Mech. Anal., 186, (2007), 133-158] and Addala and Tayeb [J. Hyperbolic Differ. Equ., 16, (2019), 131-156], which required bounded initial data, our work considers initial data that need not be bounded. We develop new techniques for handling weak entropy solutions that satisfy the natural bounds associated with the kinetic entropy inequality. Our proof employs the relative entropy method and various compactness arguments to establish the convergence and properties of these solutions.

Derivation of nonlinear aggregation-diffusion equation from a kinetic BGK-type equation

TL;DR

This work rigorously derives a nonlinear aggregation-diffusion PDE with porous-medium-type diffusion from a scaled BGK-type kinetic model. It introduces a weak entropy framework and leverages a relative entropy structure together with velocity averaging to obtain uniform estimates and strong compactness, enabling a diffusion limit without bounded initial data. Under milder assumptions on the potentials and , the authors establish global existence of weak entropy solutions to the kinetic model and prove convergence to the limit equation, including convergence of the kinetic density to the local equilibrium and convergence of to . The results extend prior works by removing boundedness requirements on the initial data and providing a robust convergence analysis that handles nonlocal interactions and porous-medium diffusion. Overall, the paper advances understanding of diffusion limits in kinetic-to-continuum transitions and strengthens the mathematical foundation for nonlinear drift-diffusion models arising in swarming, granular media, and pattern formation.

Abstract

This paper investigates the diffusion limit of a kinetic BGK-type equation, focusing on its relaxation to a nonlinear aggregation-diffusion equation, where the diffusion exhibits a porous-medium-type nonlinearity. Unlike previous studies by Dolbeault et al. [Arch. Ration. Mech. Anal., 186, (2007), 133-158] and Addala and Tayeb [J. Hyperbolic Differ. Equ., 16, (2019), 131-156], which required bounded initial data, our work considers initial data that need not be bounded. We develop new techniques for handling weak entropy solutions that satisfy the natural bounds associated with the kinetic entropy inequality. Our proof employs the relative entropy method and various compactness arguments to establish the convergence and properties of these solutions.

Paper Structure

This paper contains 29 sections, 19 theorems, 247 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Other derivations of \ref{['main_eq']}
  3. Main results
  4. (HV)
  5. (HK)
  6. Global existence of weak entropy solutions
  7. Diffusion limit of \ref{['main_kin']}: Derivation of \ref{['main_eq']}
  8. Properties of the local equilibrium
  9. Uniform-in-$\varepsilon$ estimates
  10. Second spatial moment bound and tightness
  11. Implication of control on dissipation
  12. Strong compactness
  13. Passing to the limit $\varepsilon\to 0$: Proof of Theorem \ref{['thm_main']}
  14. Weak formulations
  15. Passing to the limit $\varepsilon \to 0$
  16. ...and 14 more sections

Key Result

Theorem 1.4

Let $T>0$. Further, let $V$ and $K$ satisfy (HV) and (HK), respectively. Then, for any initial datum $f_0\in L^1_2 \cap L^{1+2/n}(\mathbb R^d \times \mathbb R^d)$ with finite energy $\mathcal{E}[f_0]<+\infty$, there exists a weak entropy solution $f$ to the equation main_kin in the sense of Defini

Figures (1)

  • Figure 1: Conditions for the exponents $q$ and $r$. The left figure indicates the region of $(\frac{1}{r},\frac{1}{q})$ that satisfying the condition \ref{['condi_hk2']}, while the right figure shows the region of exponents for the 3D Coulomb potential case with $\gamma=7/5$.

Theorems & Definitions (37)

  • Remark 1.1
  • Remark 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • ...and 27 more