Equilibration of aggregation-diffusion equations with weak interaction forces

TL;DR

This paper proves that solutions to certain one-dimensional aggregation-diffusion equations with weak interaction forces converge to a unique steady state over time, providing explicit rates and covering a broad class of potentials.

Contribution

It establishes the first equilibration results for aggregation-diffusion equations with weakly confining potentials, using novel energy and moment bounds.

Findings

Solutions converge to steady state with explicit algebraic rate.

First proof of equilibration for weakly confining potentials.

Uniform bounds on the first moment are achieved.

Abstract

This paper studies the large time behavior of aggregation-diffusion equations. For one spatial dimension with certain assumptions on the interaction potential, the diffusion index $m$, and the initial data, we prove the convergence to the unique steady state as time goes to infinity (equilibration), with an explicit algebraic rate. The proof is based on a uniform-in-time bound on the first moment of the density distribution, combined with an energy dissipation rate estimate. This is the first result on the equilibration of aggregation-diffusion equations for a general class of weakly confining potentials $W(r)$: those satisfying $\lim_{r\rightarrow\infty}W(r)<\infty$.