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。这在动物王国--成群的哺乳动物、鱼群、鸟群--以及各种经济、工程和社会环境中经常可以观察到。通常情况下，由于系统中的代理数量庞大，这种行为的详细，微观的描述是不可能的分析和数值模拟，因此它的适用性是相当有限的。相反，在这个项目中，我们建议专注于宏观模型描述的偏微分方程（PDE）系统。这些方程使我们能够捕捉到多个代理/物种/相位之间的相互作用，在一个优雅的重新涉及少量的非局部流体动力守恒定律。它往往会导致完全未开发的系统类，但也打开了使用广泛的工具和技术的可能性，从现代理论的偏微分方程提供必要的洞察大型复杂系统的动态相互作用agents.Solving的基本问题，从这个建议的目标将提供一个深刻的数学理解偏微分方程出现在集体行为的建模。它将使我们能够验证宏观模型的定性性质，并评估它们是否适合描述自然界中可观察到的复杂模式的达成或出现。我们特别高兴能够获得正式推导出的宏观模型的知识，因为这可以为其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，开发和分析新的偏微分方程模型，沿着新的数值技术来应对数学生物学和数学物理学中的新挑战。这个环境的核心将是一个由PI，两个PDRA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，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.