Nonlocal Hydrodynamic Models of Interacting Agents

交互系统中的代理通常会根据邻居的行为来组织自己的动态。这经常出现在动物界--哺乳动物群、鱼群、鸟群--但也存在于各种经济、工程和社会环境中。通常情况下，由于系统中存在大量的智能体，对这种行为的详细的微观描述是不可能进行数值分析和模拟的，因此其适用性是相当有限的。这些方程使我们能够捕捉多个试剂/物种/相之间的相互作用，在一个优雅的重新公式中，涉及少量非局部流体动力学守恒定律。它经常导致完全未探索的系统类别，但也打开了使用现代偏微分方程理论的广泛工具和技术的可能性，以提供对相互作用的主体的大型复杂系统的动力学的基本洞察。从本提案的目标解决基本问题将提供对偏微分方程在集体行为建模中出现的深刻的数学理解。它将使我们能够描述宏观模型的定性属性，并评估它们是否适合描述达成共识或在自然界中观察到的复杂模式的出现。该项目的长期目标是利用伦敦大学学院的多学科环境，建立一个新的研究小组，开发和分析新的PDE模型，以及新的数值技术，以应对数学生物学和数学物理中新出现的挑战。这个环境的核心将是一个由PI、两个PDRA和伦敦大学学院资助的博士生组成的团队。结果将在不同的环境中广泛传播，包括数学流体力学和动力学理论社区，但也在其他学科，如计算科学，或土木工程。

Agents in an interacting system typically organise their dynamics based on the behaviour of their neighbours. This is often observed in the animal kingdom - herds of mammals, schools of fish, flocks of birds - but also in a variety of economic, engineering, and social settings. Usually, due to huge number of agents in the system, detailed, microscopic description of such behaviour is impossible to analyse and to simulate numerically, and thus its applicability is rather limited.Instead, in this project we propose to focus on the macroscopic models described by systems of Partial Differential Equations (PDEs). These equations allow us to capture interactions between multiple agents/species/phases in an elegant reformulation involving a small number of nonlocal hydrodynamic conservation laws. It often leads to completely unexplored classes of systems, but also opens the possibility of using a wide range of tools and techniques from the modern theory of PDEs to provide essential insight into the dynamics of large complex systems of interacting agents.Solving the fundamental problems from the objectives of this proposal will provide a profound mathematical understanding of PDEs emerging in the modelling of collective behaviour. It will allow us to characterise the qualitative properties of the macroscopic models, and to asses whether they are fit to describe the achieving of consensus or emergence of complex patterns observable in nature. We are especially excited to gain this knowledge for the formally derived macroscopic models, as this could provide the arguments for their validity and applicability.The long-term goal of the project is to use the multidisciplinary environment of the University College London to establish a new research group developing and analysing new PDE models, along with novel numerical techniques for emerging challenges in Mathematical Biology and Mathematical Physics. The core of this environment will be a team composed of the PI, two PDRAs and the UCL funded PhD student. Results will be widely disseminated in diverse environments, including the mathematical fluid mechanics and kinetic theory communities, but also within other disciplines such as computational science, or civil engineering.