报告人：姚珧 （美国乔治亚理工学院）
时 间：2019年6月17日下午15：30-16：30
地点：华南数学应用与交叉研究中心学术报告厅111
题目： Uniqueness and non-uniqueness of stationary solutions of aggregation-diffusion equation
摘要：
In this talk, I will discuss a nonlocal aggregation equation with degenerate diffusion, which describes the mean-field limit of interacting particles driven by nonlocal interactions and localized repulsion. When the interaction potential is attractive, it is previously known that all stationary solutions must be radially decreasing up to a translation, but uniqueness (for a given mass) within this class was open, except for some special interaction potentials. For general attractive potentials, we show that the uniqueness/non-uniqueness criteria are determined by the power of the degenerate diffusion, with the critical power being m=2. Namely, for m>=2, we show the stationary solution for any given mass is unique for any attractive potential, by tracking the associated energy functional along a novel interpolation curve. And for 1<m<2, we construct examples of smooth attractive potentials, such that there are infinitely many radially decreasing stationary solutions of the same mass. This is a joint work with Matias Delgadino and Xukai Yan.