Mathematical Questions in Classical and Relativistic Fluids and Applications

Vanderbilt University, Nashville TN

Investigators

Abstract

Mathematical fluid dynamics investigates the mathematical properties of the equations employed to describe the motion of fluids. This subject has a long history but many important questions remain open. This project focuses on free-boundary problems for compressible flows and relativistic fluids. The atmosphere, stars, and gaseous planets can be described by compressible flows with a free surface. The equations of free-boundary fluids present many challenges and only recently we have begun to understand their mathematical properties. By relativistic fluids, one means fluids in a regime where Einstein's theory of relativity cannot be neglected. Neutron stars and quark-gluon plasma (an exotic state of matter that forms in heavy-ion collisions) can be described by viscous relativistic fluids. The equations of relativistic fluid dynamics present a rich mathematical structure that has been the source of intensive investigation in recent years. This project investigates a variety of problems in classical and relativistic fluids that are part of very active research areas. This project will investigate the free-boundary Euler equations, with focus on the case of compressible fluids, a subject much less developed than its incompressible counterpart. This project will investigate questions of existence, uniqueness, long-time behavior, the incompressible limit, and the role of surface tension and rotation in the flow. For relativistic fluids, the focus will be primarily in the case of fluids with viscosity. This is an area that has witnessed a great deal of activity in physics in recent years but whose mathematical foundations are underdeveloped. This project will contribute to put the theory of relativistic viscous fluids on a firm mathematical basis and use this knowledge to solve problems of direct relevance to physics. 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.