报告时间：2025年6月12日 上午9:00开始
报 告 人：闫绪恺 （副教授，俄克拉荷马州立大学）
报告地点：腾讯会议：675-144-052
报告题目：On interaction energy and steady states of aggregation-diffusion equations
报告摘要: In this talk, I will talk about two results related to the interaction energy $E[f]= \int f(x)f(y)W(x-y) dxdy$ for a nonnegative density f and radially decreasing interaction potential W. The celebrated Riesz rearrangement shows that $E[f] \le E[f^*]$, where $f^*$ is the radially decreasing rearrangement of $f$. It is a natural question to make a quantitative stability estimate of $E[f^*]-E[f]$ in terms of some distance between $f$ and $f^*$. I will describe some previous results about the stability estimate for characteristic functions. I will then present a joint work with Yao Yao, where we establish the stability estimate for general densities.
I will also talk about another work with Matias Delgadino and Yao Yao about the uniqueness and non-uniqueness of steady states of aggregation equations with degenerate diffusion, where the convexity of the interaction energy plays an important role. For the diffusion power $m\ge 2$, we constructed a novel interpolation curve between any two radially decreasing densities with the same mass, and showed that the interaction energy is convex along this interpolation, which leads to the uniqueness of the steady state, and the threshold is sharp. In the case $1<m<2$, we construct examples of smooth attractive potentials, such that there are infinitely many radially decreasing stationary solutions of the same mass.
报告人简介：闫绪恺，美国俄克拉荷马州立大学数学系副教授。其在中国科学技术大学取得学士学位，随后于美国罗格斯大学取得博士学位，并在佐治亚理工大学从事博士后研究。2020年至今就职于俄克拉荷马州立大学。主要从事非线性偏微分方程的研究。在CPAM, ARMA, JFA, JDE, CVPDE, DCDS等杂志发表多篇论文。
