Workshop on Geometry and Analysis of Fluid Flows

Vanderbilt University, Nashville TN

Investigators

Abstract

This award supports participation in the "Workshop on the Geometry and Analysis of Fluid Flows" hosted at Stony Brook University during the week of January 16-20, 2023. The fundamental equations of fluid mechanics describe many natural phenomena, including the motion of air in weather modeling, the lift of an airplane, mixing of fluids with applications in industry, flow of liquids through pipes, generation of electricity through wind and water, and the flow of blood through the body. The full equations are too difficult to solve explicitly or even by computer, hence mathematical attention focuses on properties of the equations. Such properties include whether solutions for given initial conditions exist for a long time or break down in finite time; whether solutions are stable and can be predicted with small errors, or fundamentally unstable and unpredictable; and the validity of simplifying approximations used to make the equations more manageable. These questions are difficult and longstanding. For instance, the question of long-time existence for the idealized three-dimensional fluid equations has remained open for well over a century. Both analysis and geometry have been used to study such questions, and the primary goal of this conference is to bring together senior and junior researchers with expertise in these two areas to share perspectives, techniques, and results, and to train the new generation of mathematicians. The workshop will feature a mathematically diverse group of speakers whose expertise spans multiple relevant areas. Topics to be discussed at the workshop include free boundary problems in fluid dynamics, the geometry of infinite-dimensional groups, singular limits in fluid flows, well-posedness and regularity of fluid equations, and differential geometric methods in mathematics and physics. Some of the talks will focus on issues of global existence for the three-dimensional Euler and Navier-Stokes equations; the limiting behavior as viscosity, compressibility, or surface tension approaches zero; the infinite-dimensional geometry describing fluid flows as geodesics; and well-posedness results for free boundary problems. The organizers are currently writing a textbook on geometric and analytic methods in fluid mechanics; a draft of this book will be distributed during the workshop for the benefit of participants. https://my.vanderbilt.edu/geoanalysisff 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.