Project Grant 2238219

The National Science Foundation awarded a $150,193 Project Grant to the University of South Carolina under the Mathematical and Physical Sciences program (CFDA 47.049). The grant will support research into non-local partial differential equations in collective dynamics and fluid flow from August 1, 2023 through July 31, 2028.

Specifically, the university will study three families of partial differential equations with non-local structures to model collective behavior and fluid dynamics. Researchers will investigate global phenomena and asymptotic behaviors of compressible Euler systems with nonlinear velocity alignment, aiming to understand flocking behavior in animal swarms. They will also seek to advance the long-standing question of uniqueness for weak solutions to pressureless Euler systems by approximating it with the relatively well-studied Euler-alignment system for collective dynamics. Finally, researchers will explore the embedded non-local geometric structure of Euler-Monge-Ampère systems and their relation to incompressible Euler equations, with applications for optimal transport and mean-field games. Graduate students will receive training through participation in the research projects.