Hydrodynamics of Collective Phenomena and Applications

Among the various collective phenomena observed in nature there are many that resemble the motion of a fluid. Such examples are abundant in biology, for example, a large flock of birds swinging constantly changing shape or a school of fish swirling in a milling pattern. Other examples include crowd dynamics and social networking, for example the formation of friend-clusters on social media, or dynamics of opinions among a large group of individuals.

Examples also arise in technology, such as the coordinated fight of an escort of unmanned aerial vehicles or satellite navigation. All of these large systems are governed by models similar to those we use to study motion of a liquid, like water or gas. A set of new ideas on how to study this analogy recently developed into a new mathematical subject, called Hydrodynamics of Collective Behavior.

This project will analyze hydrodynamic collective models both from the point of view of their mathematical properties and with a view towards their applications to self-organized dynamics and emergent phenomena.

Central to the project will be analysis of the so-called Euler Alignment Systems (EAS for short). Particular focus will be placed on justification of a class of isentropic EASs via the hydrodynamic limit from a noisy kinetic Fokker-Plank model. In the framework of systems with singular communication introduced in PI’s earlier works, the project sets forth a program of research on understanding topological interactions prevalent in many biological systems.

Such interactions enrich the models with a possibility of more diverse collective outcomes and, from theoretical perspective, endow them with a fractional parabolic structure. The regularity theory exploiting this parabolic structure will be developed in the context of unidirectional flocks. Applications of this research will be made to several fields of mathematics including the so-called opinion mean-field games and finding their Nash equilibria, alignment of interfacial profile in two-fluid porous media problem, and modeling turbulent energy cascade in 2D inviscid fluid flows.

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.