Equilibration of aggregation-diffusion equations with weak interaction forces

Ruiwen Shu

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arXiv · 2003.04230

Equilibration of aggregation-diffusion equations with weak interaction forces

Ruiwen Shu

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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 $\lim_{r\rightarrow\infty}W(r)<\infty$.

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Ruiwen Shu. 2020-03-09. Equilibration of aggregation-diffusion equations with weak interaction forces. https://arxiv.org/abs/2003.04230

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