Decay rates for a class of diffusive-dominated interaction equations

TL;DR

This paper investigates how diffusion dominates interaction in a mean-field particle system, providing decay rates and convergence results that show solutions behave like the heat equation over time.

Contribution

It establishes conditions under which diffusion prevails over interaction, deriving decay rates and optimal convergence rates towards the heat equation solution.

Findings

Diffusion dominates interaction under certain conditions.

Solutions decay at rates comparable to the heat equation.

Convergence to the heat kernel is optimal in the L^1 norm.

Abstract

We analyse qualitative properties of the solutions to a mean-field equation for particles interacting through a pairwise potential while diffusing by Brownian motion. Interaction and diffusion compete with each other depending on the character of the potential. We provide sufficient conditions on the relation between the interaction potential and the initial data for diffusion to be the dominant term. We give decay rates of Sobolev norms showing that asymptotically for large times the behavior is then given by the heat equation. Moreover, we show an optimal rate of convergence in the $L^1$-norm towards the fundamental solution of the heat equation.