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Rigorous derivation of the mean-field limit for the signal-dependent Keller-Segel system

Jinhuan Wang, Keyu Li, Hui Huang

TL;DR

This work rigorously derives a two-dimensional Keller–Segel-type system with signal-dependent sensitivity from a stochastic moderately interacting particle model, described by $\partial_t u = \Delta(e^{-v}u+u)$ and $-\Delta v + v = \chi u$. It introduces stopping times to prove convergence in probability under algebraic scaling $ε \sim N^{-γ}$, improving upon previously established logarithmic rates. The authors then apply a relative-entropy framework to obtain strong $L^1$ propagation of chaos with an explicit algebraic rate $\|u_{N,r}^{ε} - u^{ε\otimes r}\|_{L^∞(0,T^*;L^1)} \le C ε^{β}$, providing a quantitative link between microscopic particle dynamics and the macroscopic nonlocal PDE. These results advance the rigorous justification of cross-diffusion chemotaxis limits and yield sharper convergence rates that are useful for multiscale modeling and simulations of chemotactic systems.

Abstract

We rigorously derive a two-dimensional Keller-Segel type system with signal-dependent sensitivity from a stochastic interacting particle model. By employing suitably defined stopping times, we prove that the convergence of the interacting particle system towards the corresponding mean-field limit equations in probability under an algebraic scaling regime which improves upon existing results with logarithmic scaling. Building on this, we apply the relative-entropy method to obtain strong $L^1$ propagation of chaos, and establish an algebraic convergence rate.

Rigorous derivation of the mean-field limit for the signal-dependent Keller-Segel system

TL;DR

This work rigorously derives a two-dimensional Keller–Segel-type system with signal-dependent sensitivity from a stochastic moderately interacting particle model, described by and . It introduces stopping times to prove convergence in probability under algebraic scaling , improving upon previously established logarithmic rates. The authors then apply a relative-entropy framework to obtain strong propagation of chaos with an explicit algebraic rate , providing a quantitative link between microscopic particle dynamics and the macroscopic nonlocal PDE. These results advance the rigorous justification of cross-diffusion chemotaxis limits and yield sharper convergence rates that are useful for multiscale modeling and simulations of chemotactic systems.

Abstract

We rigorously derive a two-dimensional Keller-Segel type system with signal-dependent sensitivity from a stochastic interacting particle model. By employing suitably defined stopping times, we prove that the convergence of the interacting particle system towards the corresponding mean-field limit equations in probability under an algebraic scaling regime which improves upon existing results with logarithmic scaling. Building on this, we apply the relative-entropy method to obtain strong propagation of chaos, and establish an algebraic convergence rate.
Paper Structure (4 sections, 5 theorems, 109 equations)

This paper contains 4 sections, 5 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Convergence in probability
  3. $L^1$ convergence for the propagation of chaos
  4. Interest statement

Key Result

Lemma 1

(Existence of weak solution, BOL2026113712). Let $u_0\ge0$ be a initial probability density that satisfies Furthermore, assume $\chi<4/c_*$, where $c_*$ is the optimal constant in the Gagliardo-Nirenberg inequality: $\|\omega\|_{L^4(\mathbb{R}^2)}^4\le c_*\|\omega\|_{L^2(\mathbb{R}^2)}^2\|\nabla\omega\|_{L^2(\mathbb{R}^2)}^2$. Then for any $T>0$ and $t\in[0,T]$, the problem rpde possesses weak so

Theorems & Definitions (10)

  • Lemma 1
  • Theorem 1
  • Remark 1
  • Theorem 2
  • Remark 2
  • Lemma 2
  • proof : Proof of Theorem \ref{['propagation of chaos']}
  • Theorem 2
  • Remark 3
  • proof : Proof of Theorem \ref{['Propagation of chaos in the strong sense']}