Agent-Based Dynamics, Nonlinear Transport, and Social Hydrodynamics

自然和人类社会提供了许多集体动力学的例子，这些动力学往往会自我组织成大规模的模式。 该项目的目标是研究非线性传输方程，该方程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对化学刺激（趋化性）的反应。 本研究的主要重点是在相应的输运方程的解决方案，包括其平滑行为，自组织，并出现的模式，如大规模集群到共识，群集，并出现领导者的全局特征的持久性。本研究项目包括五个部分：（i）研究基于主体的动力学系综，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时间出现。(ii)一个集体动力学的新范式的研究，其中通信发生沿着那些代理人的预测是“向前移动”。“（三）众所周知，由非线性运输控制的集体社会流体动力学的光滑解必须群集。问题来了，这些方程的光滑性什么时候能保持？社会流体动力学中的非线性输运方程的规律性，以及它们对初始配置空间中临界阈值的依赖性将被研究。(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。当一个细菌种群过于拥挤时，会出现一个排斥峰;这就需要一个新的不可压缩的趋化性模型。将研究这种“峰值”排斥的消失粘性极限。(v)将进行一项与退化的当地拉普拉斯算子驱动的社会流体动力学相关的非线性运输研究。计划实施用于研究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.