Analysis of Continuum PDE's in Collective Behavior and Related Models

This project concerns the fundamental mechanisms underpinning collective behavior of large groups of agents, such as flocks of birds, schools of fish, or swarms of bacteria. Mathematical models for these phenomena offer insights into how large-scale structures emerge from small-scale interactions in physical systems, with potential applications in technology, including in computer graphics. In order to efficiently study systems with an otherwise intractable number of agents, this project will focus on the "effective" large-scale dynamics rather than on individual trajectories. Taking this perspective brings the problems of interest into the realm of partial differential equations. The models that arise in these problems bear substantial resemblance to equations found in fluid dynamics and continuum mechanics, a connection that will be leveraged extensively in the research to be carried out. The mentorship, training, and professional development of students and junior researchers will also be a key goal of the project. The proposed analysis will center on the effects of a nonlocal velocity alignment mechanism in isolation, as manifested in the class of hydrodynamic equations known as Euler Alignment systems. The PI will investigate the consequences of imposing different communication rules, especially as they relate to the large-time structure and regularity of the density profile. Emphasis will be placed on the as-of-yet poorly understood transition between qualitatively different regimes of interactions. In particular, the PI will leverage the additional structure available in settings with simple geometries to draw connections between models that incorporate strongly localized alignment and those that feature sticky particles. The PDEs governing alignment dynamics serve as a paradigm for more general nonlocal equations, and the proposed research has the potential to advance the understanding of classes of nonlocal models far beyond those explicitly studied in the project. 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.