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

主要研究员（PI）将研究自然界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，电磁学的麦克斯韦方程，可压缩流体力学的欧拉方程，以及新的，修改后的版本欧拉方程，该方程解释了在夸克-胶子等离子体和中子星的研究中实验发现的粘性效应。虽然这些方程已经被研究了几十年，但关于解的动力学仍有许多问题有待理解。该项目将专注于在最令人兴奋和快速发展的研究领域之一：奇点形成中得出理论结果。粗略地说，奇点是可以在解中发展的无穷大，这使得方程的研究非常具有挑战性。这样的无穷大是一些最迷人的物理现象的关键所在。突出的例子包括广义相对论中的大爆炸，时空的曲率变得无限大，以及可压缩流体中的冲击波，压力梯度变得无限大。该项目的成果将对自然规律产生深刻的新见解。PI将通过将本科生、硕士生、博士生和博士后学者纳入研究计划来整合教育、研究和科学培训。PI的目标是证明上述拟线性和双曲型PDE系统Cauchy问题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，着眼于理解最大可能经典解的全局结构，即最大全局双曲展开（MGHD）。目前还没有结果的整体结构的MGHD，这些结果是必不可少的证明的唯一性的经典解的冲击。对于粘性流体模型，目前还没有建设性的爆破结果，所以任何建设性的奇性形成的结果将是第一次。对于爱因斯坦方程（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，重点是可局部化和鲁棒的技术，从而使人们能够探索新的解决方案。在所有的问题中，基于几何和分析考虑的量规选择是分析的核心。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值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.