Geometric Techniques for Studying Singular Solutions to Hyperbolic Partial Differential Equations in Physics

Vanderbilt University, Nashville TN

Investigators

Abstract

The Principal Investigator (PI) will study evolution equations that arise in several physical models of nature, including Einstein’s equations of general relativity, Maxwell’s equations of electromagnetism, Euler’s equations of compressible fluid mechanics, and new, modified versions of Euler’s equations that account for viscous effects that were experimentally discovered in the study of the Quark-Gluon Plasma and neutron stars. While these equations have been studied for many decades, much remains to be understood about the dynamics of solutions. This project will focus on deriving theoretical results in one of the most exciting and rapidly advancing areas of study: singularity formation. Roughly, singularities are infinities that can develop in solutions, making the equations exceptionally challenging to study. Such infinities lie at the crux of some of the most fascinating physical phenomena. Outstanding examples include Big Bangs in general relativity, where the curvature of spacetime becomes infinite, and shock waves in compressible fluids, where pressure gradients become infinitely large. The results of the project will shed deep new insights into the laws of nature. The PI will integrate education, research, and scientific training by incorporating undergraduates, Master’s degree students, PhD students, and postdoctoral scholars into the research program. The PI aims to prove novel stable blowup-results in multidimensions for solutions to the Cauchy problem for the PDE systems mentioned above, which are quasilinear and hyperbolic in character. For compressible Euler flow, the goal is to prove shock-formation, with an eye towards understanding the global structure of the largest possible classical solution, i.e, the Maximal Globally Hyperbolic Development (MGHD). There are currently no results on the global structure of the MGHD, and such results are essential for proving the uniqueness of classical solutions with shocks. For the viscous fluid models, there are currently no constructive blowup-results, so any constructive singularity-formation result would be the first of its kind. For Einstein’s equations (coupled to various matter models), the goal is to understand the structure and stability of spacetime singularities, with a focus on techniques that are localizable and robust, thus allowing one to probe new solution regimes. In all of the problems, gauge choices motivated by geometric and analytical considerations lie at the heart of the analysis. 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.