第二十五讲暨数学学科创建110周年系列报告
数学学科创建110周年系列报告
报告题目(Title)：Aggregation-Diffusion Equations for Collective Behaviour in the Sciences
报告人(Speaker)： José Antonio Carrillo 院士（牛津大学）
地点(Place)：后主楼1124
时间(Time)：2025年4月22日（周二） 14:00-15:00
报告摘要
Many phenomena in the life sciences, ranging from the microscopic to macroscopic level, exhibit surprisingly similar structures. Behaviour at the microscopic level, including ion channel transport, chemotaxis, and angiogenesis, and behaviour at the macroscopic level, including herding of animal populations, motion of human crowds, and bacteria orientation, are both largely driven by long-range attractive forces, due to electrical, chemical or social interactions, and short-range repulsion, due to dissipation or finite size effects. Various modelling approaches at the agent-based level, from cellular automata to Brownian particles, have been used to describe these phenomena. An alternative way to pass from microscopic models to continuum descriptions requires the analysis of the mean-field limit, as the number of agents becomes large. All these approaches lead to a continuum kinematic equation for the evolution of the density of individuals known as the aggregation-diffusion equation. This equation models the evolution of the density of individuals of a population, that move driven by the balances of forces: on one hand, the diffusive term models diffusion of the population, where individuals escape high concentration of individuals, and on the other hand, the aggregation forces due to the drifts modelling attraction/repulsion at a distance. The aggregation-diffusion equation can also be understood as the steepest-descent curve (gradient flow) of free energies coming from statistical physics. Significant effort has been devoted to the subtle mechanism of balance between aggregation and diffusion. In some extreme cases, the minimisation of the free energy leads to partial concentration of the mass. Aggregation-diffusion equations are present in a wealth of applications across science and engineering. Of particular relevance is mathematical biology, with an emphasis on cell population models. The aggregation terms, either in scalar or in system form, is often used to model the motion of cells as they concentrate or separate from a target or interact through chemical cues. The diffusion effects described above are consistent with population pressure effects, whereby groups of cells naturally spread away from areas of high concentration. This talk will give an overview of the state of the art in the understanding of aggregation-diffusion equations, and their applications in mathematical biology.
主讲人简介
José A. Carrillo，英国牛津大学数学系教授，欧洲人文与自然科学院院士、国际工业与应用数学学会会士，1996年获西班牙格拉纳达大学博士学位。他长期从事偏微分方程的数学理论与数值分析研究，在动力学方程、非线性与非局部扩散模型等方向取得了重要成果。他运用最优传输与熵方法深入研究偏微分方程的梯度流结构与奇异性，其研究广泛应用于颗粒介质、半导体、集体行为、生物系统等多个领域，同时发展了保持自由能耗散特性的非线性扩散数值格式。迄今已在 Inventiones Mathematicae、Duke Math J、Comm. Pure Appl. Math. 等国际顶尖数学期刊发表论文200余篇。现任国际工业与应用数学理事会执行委员、欧洲科学院数学部主任，并担任多个国际高水平数学期刊编委。因其卓越的学术贡献，曾获西班牙皇家科学院最高科学奖 Echegaray 奖章（2022）和意大利林琴国家科学院“Luigi Tartufari”国际数学奖（2024）。
