Collaborative Research: Nonlocal Models of Aggregation and Dispersion

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In many models of physical reality, coherent system states emerge and are maintained through a balance of countervailing influences. Effects that concentrate, aggregate, or focus are generated by nonlinear mechanisms. Competing influences that diffuse, disperse or fragment are often due to linear action. This project addresses several problems which involve a dynamic balance of this kind, whose solution will require the development of new and useful mathematical tools. These problems arise in connection with investigations in animal ecology, image analysis, optimal transportation, fluid dynamics, and stochastic models of interacting agents. The new mathematical ideas to be developed promise to be of strong interest to researchers in a variety of disciplines. The project will also engage students and young researchers, and lead to results to be communicated at seminars, conferences, summer schools, and lecture series.In particular, this project's research will focus on understanding dynamic phenomena that concern four topics strongly motivated by applications and the present state of theory in partial differential equations: (1) A fundamental merging-splitting model of group-size distributions, arising in H.-S. Niwa's analysis of empirical data on fish school size in the mid-ocean; (2) Least action principles for volume-preserving shape deformations and their connection to (regularized) optimal transport metrics, motivated by modeling dynamics of constant-density crowd configurations; (3) A novel type of weakly singular shock wave dynamics in a regularized model for shallow water equations describing waves on scales large compared to the water depth; (4) How coherent waves and patterns form through gradient-like dynamics in a proposed new model of nonlocal dispersion and nonlinear focusing, related to fixed-point formulations for computing wave shapes in a variety of nonlinear systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.