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

首席研究员(PI)将研究自然的几个物理模型中出现的演化方程，包括爱因斯坦的广义相对论方程、麦克斯韦的电磁方程、可压缩流体力学的欧拉方程，以及在研究夸克胶子等离子体和中子星时实验发现的粘性效应的新的、修正的欧拉方程。虽然这些方程已经研究了几十年，但关于解的动力学仍有许多需要了解的地方。这个项目将专注于在最令人兴奋和快速发展的研究领域之一：奇点形成方面得出理论结果。粗略地说，奇点是可以在解中形成的无穷大，这使得方程的研究格外具有挑战性。这种无限存在于某些最吸引人的物理现象的症结所在。突出的例子包括广义相对论中的大爆炸，其中时空的曲率变得无限大，以及可压缩流体中的冲击波，其中压力梯度变得无限大。该项目的结果将为自然规律带来深刻的新见解。PI将把本科生、硕士生、博士后和博士后纳入研究计划，将教育、研究和科学培训融为一体。PI旨在证明上述具有拟线性和双曲性特征的偏微分方程组柯西问题的解的多维稳定爆破结果。对于可压缩欧拉流，目标是证明激波的形成，着眼于了解最大可能的经典解的全局结构，即最大全局双曲发展(MGHD)。目前还没有关于MGHD整体结构的结果，而这些结果对于证明带有激波的经典解的唯一性是必不可少的。对于粘性流体模型，目前还没有建设性的井喷结果，所以任何建设性的奇点形成结果都将是此类结果中的第一个。对于爱因斯坦的方程(耦合到各种物质模型)，目标是了解时空奇点的结构和稳定性，重点放在可局部化和健壮的技术上，从而允许人们探索新的解决方案。在所有的问题中，出于几何和分析考虑的量规选择是分析的核心。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.