Agent-Based Dynamics, Nonlinear Transport, and Social Hydrodynamics

自然界和人类社会提供了许多集体动力倾向于自我组织成大规模模式的例子。这个项目的目标是研究非线性输运方程，它给出了集体动力学的各种表现形式的数学描述，例如鸟类的群集或微生物群体响应化学刺激（趋化性）的运动。本研究的主要重点是相应输运方程解的全局特征的持久性，包括它们的平滑行为、自组织和模式的出现，如大规模聚类达成共识、群集和领导者的出现。该项目涉及研究生和博士后研究人员。本研究项目包括五个部分：(i)研究基于主体的动力学集合，以预测不同的局部参与规则如何影响一个或多个集群的大规模出现。研究集体动力的新范例，其中交流是沿着那些“前进”的行动者的预测进行的。（三）已知由非线性输运控制的集体社会流体动力学的光滑解必须聚集。问题来了，这些方程的平滑性在什么时候会持续？本文将研究社会流体力学中非线性输运方程的规律性及其与初始构型空间中临界阈值的关系。（iv）趋化性是由（化学）吸引和（体积填充）排斥驱动的群体动力学的典型例子。当一个细菌种群过度拥挤时，会有一个排斥峰；这就需要一种新的不可压缩趋化模型。将研究这种“峰值”斥力的消失粘度极限。将对退化的当地拉普拉斯人推动的与社会流体动力学有关的非线性输运进行研究。计划实现用于研究具有退化扩散的非线性守恒律正则化的速度平均技术。

Nature and human societies offer many examples of collective dynamics that tend to self-organize into large-scale patterns. The goal of this project is to study nonlinear transport equations, which give a mathematical description of diverse manifestations of collective dynamics, such as flocking of birds or movement of the population of microorganisms in response to a chemical stimulus (chemotaxis). The principal focus of this research is on the persistence of global features of solutions of the corresponding transport equations, including their smoothing behavior, self-organization, and the emergence of patterns, such as large-scale clustering into consensus, flocking, and the emergence of leaders. The project involves graduate students and postdoctoral fellows in the research.This research project comprises five parts: (i) A study of the ensemble of agent-based dynamics in order to predict how different rules of local engagement affect the large-time emergence of one or more clusters. (ii) A study of a new paradigm of collective dynamics, for which communication takes place along the projections of those agents that are "moving ahead." (iii) It is known that smooth solutions of collective social hydrodynamics governed by nonlinear transport must flock. The question arises, when does the smoothness of these equations persist? The regularity of nonlinear transport equations in social hydrodynamics, and their dependence on critical thresholds in the space of initial configurations will be investigated. (iv) Chemotaxis is a canonical example for group dynamics driven by (chemo-) attraction and (volume-filling) repulsion. When a population of bacteria is overcrowded, there is a repulsion peak; this calls for a new incompressible model for chemotaxis. The vanishing viscosity limit of such a "peaked" repulsion will be studied. (v) A study of nonlinear transport associated with social hydrodynamics driven by degenerate local Laplacians will be undertaken. It is planned to implement the velocity-averaging techniques used to study the regularization of nonlinear conservation laws with degenerate diffusion.