Nonlinear aggregation-diffusion equations with Riesz potentials

TL;DR

This paper studies a nonlinear aggregation-diffusion model with Riesz potentials, proving existence, uniqueness, and asymptotic behavior of stationary states, and constructing solutions via the JKO scheme.

Contribution

It introduces a new analysis of the competition between nonlinear diffusion and Riesz potential aggregation, including asymptotic and existence results.

Findings

Existence and uniqueness of stationary states.

Asymptotic characterization as s approaches zero.

Construction of gradient flow solutions using the JKO scheme.

Abstract

We consider an aggregation-diffusion model, where the diffusion is nonlinear of porous medium type and the aggregation is governed by the Riesz potential of order s. The addition of a quadratic diffusion term produces a more precise competition with the aggregation term for small s, as they have the same scaling if s=0. We prove existence and uniqueness of stationary states and we characterize their asymptotic behavior as s goes to zero. Moreover, we prove existence of gradient flow solutions to the evolution problem by applying the JKO scheme.