Collaborative Research: Kinetic Models of Aggregation and Dispersion

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In many mathematical models of physical reality, coherent structures are formed and maintained by a balance of competing influences. On the one hand, focusing, concentration, or aggregation effects are typically produced by nonlinear mechanisms. These effects are often counterbalanced by other processes that disperse, defocus, fragment, or spread things out in some way. This proposal aims to develop several novel and useful mathematical tools for analyzing how such competing effects achieve dynamic balance. The particular models of aggregation and dispersion to be studied arise specifically in studies of: animal ecology, crowd dynamics, shape matching, hydrodynamics, and mass transportation. The mathematical lessons learned are expected to be fundamental, and contribute to a body of understanding that promises to be useful to researchers across a range of disciplines. Further, the investigators plan to be substantially engaged in training and interacting with students and young researchers, at summer schools, lecture series, and disseminating results at conferences, workshops, and seminars.The proposed research focuses on the study of dynamic behavior in four areas strongly motivated by applications and the theory of partial differential equations. The first area considers a fundamental coagulation-fragmentation model without detailed balance, coming from Niwa's scaling analysis of a large body of empirical data on fish school size in the mid-ocean. The second area will develop metrics and geodesics for crowd-configuration paths and related hydrodynamic problems for shape distances proposed by image analysts. The third area focuses on the long-time dynamics and gradient structure in a new model of nonlocal dispersion and nonlinear concentration, related to fixed-point equations for solitary wave profiles. The final area considers random sticky particle dynamics, seeking to build on recent advances in PDE theory that tie sticky particle dynamics to singular solutions of conservation laws, and on related progress for random shock clustering. Real-world applications include the fields of animal ecology, image analysis, fluid dynamics, and stochastic interacting particle systems.