Singularities, Steady Structures, and Mixing in Incompressible Fluid Flows

From devising efficient agricultural practices to modelling the flow of blood in coronary arteries to investigating weather patterns on Jupiter, the study of fluid flow is essential to almost all aspects of life. This project is concerned with some of the fundamental aspects of the dynamics of incompressible fluids. The kinds of questions considered in the project have been around for millennia, even if the precise mathematical questions were only posed three centuries ago. Some of the important questions the primary investigator (PI) considers in this project relate to the degree to which the classical mathematical equations of fluid mechanics actually model physical reality, whether the equations themselves can break down, and what the equations predict on long time scales. The project contains three general directions of research: the formation of singularities in incompressible fluids, the study of steady solutions to the incompressible Euler equation in two and three dimensions, and the phenomenon of fluid mixing. The PI investigates scale-invariant and self-similar singularities in the three-dimensional Euler equation and related models and, additionally, study the possibility of non-constructive proofs of singularity formation. The study of steady solutions to the Euler equation and the phenomenon of fluid mixing represents an attempt toward understanding aspects of the long-time behavior of inviscid two-dimensional fluids in the large. 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.