Aggregation-diffusion equations on surfaces and manifolds

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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.