论文标题
与相互作用弱的聚合扩散方程的平衡
Equilibration of aggregation-diffusion equations with weak interaction forces
论文作者
论文摘要
本文研究了聚合扩散方程的庞大时间行为。对于一个空间维度，具有相互作用电位的某些假设，扩散指数$ m $和初始数据，我们证明了随着时间的时间到达无限（平衡）的融合到唯一的稳态，并具有明确的代数速率。该证明是基于密度分布的第一瞬间的均匀限制，并结合了耗能速率估计。这是对弱限制电势$ W（r）$的一般类别的聚合扩散方程平衡的第一个结果：满足$ \ lim_ {r \ rightArrow \ rightarrow \ infty} w（r）<\ infty $的$ w（r）$。
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 $\lim_{r\rightarrow\infty}W(r)<\infty$.
