An aggregation equation with degenerate diffusion: qualitative property of solutions

TL;DR

This paper investigates a nonlocal aggregation equation with degenerate diffusion on a periodic domain, establishing regularity, decay rates, and sharp results for specific parameter regimes, especially for the case m=2.

Contribution

It generalizes the McKean-Vlasov equation with porous medium diffusion, providing new regularity results and decay estimates, including sharp decay rates for the case m=2.

Findings

Regularity properties of solutions are established.

Exponential decay to uniform solutions for small interactions.

Sharp decay rate results for m=2 regime.

Abstract

We study a nonlocal aggregation equation with degenerate diffusion, set in a periodic domain. This equation represents the generalization to $m > 1$ of the McKean-Vlasov equation where here the "diffusive" portion of the dynamics are governed by Porous medium self-interactions. We focus primarily on $m\in(1,2]$ with particular emphasis on $m = 2$. In general, we establish regularity properties and, for small interaction, exponential decay to the uniform stationary solution. For $m=2$, we obtain essentially sharp results on the rate of decay for the entire regime up to the (sharp) transitional value of the interaction parameter.