具有弱相互作用力的聚集-扩散方程的平衡性
Equilibration of aggregation-diffusion equations with weak interaction forces
摘要：
本文研究了聚集-扩散方程的大时间行为。对于一个空间维度的相互作用势，扩散指数$m$，和初始数据的某些假设，我们证明了收敛到唯一的稳定状态，随着时间的推移到无穷大（平衡），与一个明确的代数率。证明是基于一个统一的时间约束的第一时刻的密度分布，结合能量耗散率估计。这是第一个结果的平衡的聚集-扩散方程的一般类弱限制势$W(r)$：那些满足$\lim_{r\rightarrow\infty}W(r)<\infty$。
英文摘要：
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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