学术报告报告题目: Nonlinear aggregation-diffusion equations:(reverse) HLS inequalities and equilibration
报 告 人: Jose Antonio Carrillo 教授（英国帝国理工学院）
报告时间: 2019年 8 月 27 日（星期 二）下午 16:00-17:00
报告地点: 4号楼 4318 室
邀 请 人: 金海洋 副教授
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数学学院
2019年 8 月 21 日
报告摘要：
We analyse under which conditions equilibration between two competing effects, repulsion modelled by nonlinear diffusion and attraction modelled by nonlocal interaction, occurs. I will discuss the regimes that appear in aggregation diffusion problems with homogeneous kernels. We will discuss the main qualitative properties in terms of stationary states and minimizers of the free energies. In particular, all the porous medium cases are critical while the fast diffusion are not in this regime. In the second part, I will discuss the diffusion dominated case for porous medium cases in which this balance leads to continuous compactly supported radially decreasing equilibrium configurations for all masses. All stationary states with suitable regularity are shown to be radially symmetric by means of continuous Steiner symmetrisation techniques. Calculus of variations tools allow us to show the existence of global minimizers among these equilibria. Finally, in the fast diffusion regime we are able to find conditions for the existence of global minimizers in a different range by means of reverse HLS inequalities. Concentration at the origin in part of this range is not ruled out. This talk is based on works in collaboration with S. Hittmeir, B. Volzone, and Y. Yao (to appear in Inventiones Mathematicae), with V. Calvez and F. Hoffmann (Nonlinear Analisys TMA), and with J. Dolbeault, M. Delgadino, F. Hoffmann and R. Frank (to appear in J. Math. Pure Appl.).
报告人简介：
Jose Antonio Carrillo，英国帝国理工学院数学系数值分析与应用数学讲席教授, 1996年在University of Granada获得博士学位；主要从事偏微分方程、生物数学等方面的理论与数值分析研究；已在Invent Math、 Duke Math J、 CPAM、 JEMS、CMP、ARMA、SIAM、PNAS等国际一流数学期刊上发表学术论文200余篇，被SCI引用4500余次（H指数：33）；现任SIAM J Math Anal、Proc London Math Soc、J London Math Soc、Bull London Math Soc、Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal等20余个高水平SCI学术期刊编委。