Uniform in Time Interacting Particle Approximations for Nonlinear Equations of Patlak-Keller-Segel type

TL;DR

This paper analyzes a modified Patlak-Keller-Segel model using interacting particle systems, proving uniform in time convergence, exponential concentration bounds, and introducing an Euler scheme with error bounds for long-term simulation accuracy.

Contribution

It establishes uniform in time convergence of particle approximations for a modified PKS equation, along with exponential concentration bounds and a convergent Euler discretization scheme.

Findings

Proved uniform in time convergence of empirical measures to PDE solutions.

Established exponential concentration bounds under integrability conditions.

Developed an Euler scheme with proven uniform in time error bounds.

Abstract

We study a system of interacting diffusions that models chemotaxis of biological cells or microorganisms (referred to as particles) in a chemical field that is dynamically modified through the collective contributions from the particles. Such systems of reinforced diffusions have been widely studied and their hydrodynamic limits that are nonlinear non-local partial differential equations are usually referred to as Patlak-Keller-Segel (PKS) equations. Under the so-called "quasi-stationary hypothesis" on the chemical field the limit PDE is closely related to granular media equations that have been extensively studied probabilistically in recent years. Solutions of classical PKS solutions may blow up in finite time and much of the PDE literature has been focused on understanding this blow-up phenomenon. In this work we study a modified form of the PKS equation for which global existence and uniqueness of solutions holds. Our goal is to analyze the long-time behavior of the particle system approximating the equation. We establish, under suitable conditions, uniform in time convergence of the empirical measure of particle states to the solution of the PDE. We also provide uniform in time exponential concentration bounds for rate of the above convergence under additional integrability conditions. Finally, we introduce an Euler discretization scheme for the simulation of the interacting particle system and give error bounds that show that the scheme converges uniformly in time and in the size of the particle system as the discretization parameter approaches zero.