Nonlocal Hydrodynamic Models of Interacting Agents

交互系统中的代理通常根据其邻居的行为来组织其动态。这在动物界——哺乳动物群、鱼群、鸟群——以及各种经济、工程和社会环境中都经常观察到。通常，由于系统中主体数量庞大，对这种行为进行详细、微观的描述是不可能进行分析和数值模拟的，因此其适用性相当有限。相反，在这个项目中，我们建议将重点放在偏微分方程（PDEs）系统描述的宏观模型上。这些方程使我们能够在涉及少量非局部流体动力学守恒定律的优雅重新表述中捕捉多个介质/物种/相之间的相互作用。它经常导致完全未探索的系统类别，但也打开了使用来自pde现代理论的广泛工具和技术的可能性，以提供对相互作用代理的大型复杂系统动力学的基本见解。从本提案的目标中解决基本问题将为在集体行为建模中出现的偏微分方程提供深刻的数学理解。它将使我们能够描述宏观模型的定性特性，并评估它们是否适合描述在自然界中可观察到的共识的达成或复杂模式的出现。我们特别兴奋地为正式推导的宏观模型获得这些知识，因为这可以为它们的有效性和适用性提供论据。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，以及针对数学生物学和数学物理学中新出现的挑战的新颖数值技术。这个环境的核心将是一个由PI，两个PDRAs和UCL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，以及其他学科，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.