Aggregation-diffusion equations on surfaces and manifolds

本研究的重点是对表面和流形上自集体行为的数学模型进行理论和数值研究。一类重要的模型由非线性微分方程描述，其中包括非局部相互作用和扩散。除了极少数例外，聚集模型都是在欧几里得空间中建立的。然而，在生物学或工程（机器人）中有许多应用需要一定的地形或环境/移动限制。提出的研究解决了这些模型在非线性空间中的重要扩展。我们建议采用内在的方法，只考虑流形的内在几何形状来建模个体的两两相互作用。这与外在方法相反，在外在方法中，个体在流形的欧几里得空间中相互作用。我们建议与牛津大学的Carrillo教授合作。提议的合作研究将调查解决方案的适定性和长期行为，以及从离散（微观）到连续（宏观）描述的严格通道。我们还将开发鲁棒和有效的数值方法来解释这类模型的特殊几何结构。聚集模型在种群生物学、颗粒介质、趋化性、机器人和舆论形成等领域有广泛的应用。本研究的一个关键应用是证明在没有任何领导者或外部协调的情况下，在表面和流形上出现自我集体行为。我们将解决机器人领域的覆盖和共识问题；这种配置在监视和跟踪应用中是必不可少的。拟议的项目为具有不同技能水平和研究兴趣的HQP提供培训机会。

The focus of this research is a theoretical and numerical investigation of mathematical models for self-collective behaviour on surfaces and manifolds. An important class of such models is described by nonlinear differential equations that include nonlocal interactions and diffusion. With very few exceptions, aggregation models are set up in the Euclidean space. Nevertheless, there are many applications in biology or engineering (robotics) that require a certain topography or environment/mobility constraints. The proposed research addresses the important extension of these models to nonlinear spaces. We propose to take the intrinsic approach, and only consider the intrinsic geometry of the manifold in modelling individuals' pairwise interaction. This is in contrast to the extrinsic approach, where individuals interact in an ambient Euclidean space of the manifold. We propose to initiate a collaboration with Prof. Carrillo from University of Oxford. The proposed collaborative research will investigate the well-posedness and long-time behaviour of solutions, and the rigorous passage from the discrete (microscopic) to the continuum (macroscopic) description. We will also develop robust and efficient numerical methods that account for the special geometrical structure of this class of models. Aggregation models have a wide range of applications, in areas such as population biology, granular media, chemotaxis, robotics and opinion formation. A key application of this research is to demonstrate emergence of self-collective behaviour on surfaces and manifolds, in the absence of any leader or external coordination. We will address coverage and consensus problems in robotics; such configurations are essential in surveillance and tracking applications. The proposed projects offer training opportunities for HQP with a variety of skill levels and research interests.