Asymptotic simplification of Aggregation-Diffusion equations towards the heat kernel

TL;DR

This paper establishes conditions under which aggregation-diffusion equations simplify to heat kernel behavior at large times, using novel estimates and techniques for these equations.

Contribution

It provides sharp conditions for asymptotic behavior and introduces a new approach with modulus of continuity techniques for analyzing these equations.

Findings

Solutions behave like the heat kernel under certain decay conditions.

The potential $W(x) \,\sim\, \log |x|$ marks a transition in asymptotic behavior.

Uniform-in-time estimates are obtained for entropy, moments, and regularity.

Abstract

We give sharp conditions for the large time asymptotic simplification of aggregation-diffusion equations with linear diffusion. As soon as the interaction potential is bounded and its first and second derivatives decay fast enough at infinity, then the linear diffusion overcomes its effect, either attractive or repulsive, for large times independently of the initial data, and solutions behave like the fundamental solution of the heat equation with some rate. The potential $W(x) \sim \log |x|$ for $|x| \gg 1$ appears as the natural limiting case when the intermediate asymptotics change. In order to obtain such a result, we produce uniform-in-time estimates in a suitable rescaled change of variables for the entropy, the second moment, Sobolev norms and the $C^\alpha$ regularity with a novel approach for this family of equations using modulus of continuity techniques.