CAREER: Nonlocal partial differential equations in collective dynamics and fluid flow

The collective behaviors of large groups of similar animals, e.g., birds, insects, or fishes, are ubiquitous in nature. In recent times, the mathematical study of collective dynamics has become an active and fast-growing field of research. Many mathematical models of collective behavior rely on partial differential equations with nonlocal interactions to describe the resulting emergent behavior. It turns out that these models are intimately connected to other models traditionally used in fluid dynamics. The goal of this project is to study several models of nonlocal partial differential equations to model collective behavior or fluid flows, and to develop novel and robust analytical techniques to understand the collective behaviors driven by nonlocal structures. The training and professional development of graduate students and young researchers is an integral part of the project. This project studies three families of partial differential equations with shared nonlocal structures that can affect the solutions of the equations: existence, uniqueness, regularity, and long-time asymptotic behaviors. The first problem is on the compressible Euler system with nonlinear velocity alignment, which describes the remarkable flocking phenomenon in animal swarms. Global phenomena and asymptotic behaviors of the system will be investigated, with a focus on the nonlinearity in the velocity alignment. The second problem is on the pressureless Euler system, aiming at the long-standing question of the uniqueness of weak solutions. The plan is to approximate the system by the relatively well-studied Euler-alignment system in collective dynamics. The third problem is on the Euler-Monge-Ampère system which is closely related to the incompressible Euler equations in fluid dynamics. The embedded nonlocal geometric structure of the system will be explored, with interesting applications in optimal transport and mean-field games. 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.