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Quantitative approximation of a Keller--Segel PDE by a branching moderately interacting particle system and suppression of blow-up

Thomas Cavallazzi, Alexandre Richard, Milica Tomasevic

TL;DR

The paper analyzes the Keller–Segel chemotaxis model with logistic damping on $\mathbb{R}^d$ and proves global well-posedness under a damping threshold, with special suppression of blow-up in $d=2$. It then constructs a microscopic, moderately interacting particle system with a singular Coulomb-type kernel and a branching mechanism that encodes logistic damping, establishing a quantitative propagation of chaos-type result. A main contribution is a rigorous rate of convergence for the mollified empirical measure $u^N$ to the PDE solution $u$, with $\sup_{t\in[0,T]} \|u^N_t-u_t\|_{L^1\cap L^r} \lesssim N^{-\varrho}$ where $\varrho$ depends on the regularization and integrability parameters, and a corollary giving convergence in Kantorovich–Rubinstein distance for the un-mollified empirical measure. The results bridge macroscopic PDE dynamics and microscopic stochastic particle models in a setting with singular interactions and branching, providing insight into blow-up suppression through demographic effects and offering explicit convergence rates useful for numerical schemes and simulations.

Abstract

The Keller--Segel PDE is a model for chemotaxis known to exhibit possible finite-time blow-up. Following a seminal work by Tello and Winkler, a logistic damping term is added in this PDE and local well-posedness of mild solutions is proven. When the space dimension is $2$ or when the damping is strong enough, the solution is global in time. In the second part of this work, a microscopic description of this model is introduced in terms of a system of stochastic moderately interacting particles. This system features two main characteristics: the interaction between particles happens through a singular (Coulomb-type) kernel which is attractive; and the particles are subject to demographic events, birth and death due to local competition with other particles. The latter induces a branching structure of the particle system. Then the main result of this work is the convergence of the empirical measure of the particle system towards the Keller--Segel PDE with logistic damping, with a rate of order $N^{-\frac{1}{2(d+1)}}$.

Quantitative approximation of a Keller--Segel PDE by a branching moderately interacting particle system and suppression of blow-up

TL;DR

The paper analyzes the Keller–Segel chemotaxis model with logistic damping on and proves global well-posedness under a damping threshold, with special suppression of blow-up in . It then constructs a microscopic, moderately interacting particle system with a singular Coulomb-type kernel and a branching mechanism that encodes logistic damping, establishing a quantitative propagation of chaos-type result. A main contribution is a rigorous rate of convergence for the mollified empirical measure to the PDE solution , with where depends on the regularization and integrability parameters, and a corollary giving convergence in Kantorovich–Rubinstein distance for the un-mollified empirical measure. The results bridge macroscopic PDE dynamics and microscopic stochastic particle models in a setting with singular interactions and branching, providing insight into blow-up suppression through demographic effects and offering explicit convergence rates useful for numerical schemes and simulations.

Abstract

The Keller--Segel PDE is a model for chemotaxis known to exhibit possible finite-time blow-up. Following a seminal work by Tello and Winkler, a logistic damping term is added in this PDE and local well-posedness of mild solutions is proven. When the space dimension is or when the damping is strong enough, the solution is global in time. In the second part of this work, a microscopic description of this model is introduced in terms of a system of stochastic moderately interacting particles. This system features two main characteristics: the interaction between particles happens through a singular (Coulomb-type) kernel which is attractive; and the particles are subject to demographic events, birth and death due to local competition with other particles. The latter induces a branching structure of the particle system. Then the main result of this work is the convergence of the empirical measure of the particle system towards the Keller--Segel PDE with logistic damping, with a rate of order .
Paper Structure (17 sections, 15 theorems, 171 equations)

This paper contains 17 sections, 15 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Organisation of the paper.
  3. Preliminaries and main results
  4. Notations and standard properties
  5. Well-posedness of the logistic Keller--Segel PDE
  6. Definition of the branching interacting particle system
  7. Quantitative approximation of the PDE by the particle system
  8. A priori estimates and well-posedness of the PDE
  9. Properties of the Keller--Segel interaction kernel
  10. Global well-posedness of the PDE: Proof of Theorem \ref{['Thm_WP_KS']}
  11. Convergence of the moderately interacting particle system to the Keller--Segel PDE
  12. Equation satisfied by the empirical measure
  13. Control of the number of particles in the system
  14. Estimates on the stochastic convolution integrals
  15. A priori bounds on the mollified empirical measure
  16. ...and 2 more sections

Key Result

Theorem 1

Let $d\geq2$ and $\mu,\nu \geq 0$. Assume that $u_{0}\in L^1\cap L^\infty$.

Theorems & Definitions (33)

  • Definition 2.1: Local mild solution
  • Definition 2.2: Global mild solution
  • Theorem 1
  • Remark 2.3
  • Theorem 2
  • Corollary 2.4
  • Remark 2.5: I.i.d. initial conditions
  • Remark 2.6: Heuristics on the rate $\varrho$
  • Remark 2.7: Optimising $\alpha$ or $\varrho$
  • Remark 2.8: Removing the cutoff in \ref{['eq:IPS']}
  • ...and 23 more