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

研究物理学中双曲偏微分方程奇异解的几何技术

• 批准号: 2349575
• 负责人: Jared Speck
• 金额: $ 33.09万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-06-15 至 2027-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2349575&HistoricalAwards=false
• 关键词: Geometric; Techniques; Studying; Singular; Solutions; Geometric; Techniques; Studying; Singular; Solutions

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.

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 electronmagnetism, 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.尽管这些方程已经进行了数十年的研究，但有关解决方案的动态仍有许多待理解。该项目将着重于在最令人兴奋和迅速发展的研究领域之一中得出理论结果：奇异性形成。粗略地，奇异性是可以在解决方案中发展的无限态度，从而使方程式变得异常挑战。这样的无限性在于一些最迷人的物理现象的关键。出色的例子包括一般相对论中的大爆炸，时空的曲率变得无限，以及可压缩的笛子中的冲击波，其中压力梯度变得无限大。该项目的结果将为自然定律提供深刻的新见解。 PI将通过转换本科生，硕士学位学生，博士生和博士后学者将教育，研究和科学培训整合到研究计划中。该PI旨在证明在上述PDE系统中解决cauchy问题的多维稳定性重新质量的新颖稳定性爆炸性，并且在上述PDE系统中提到了Quasilearear和超级bolicolocolic and quasilocolic of quasilocolic inthere and themal in themal intemal in Charnation。对于可压缩的Euler流，目标是证明冲击形成，以了解最大可能的经典解决方案的全球结构，即最大的全球双曲线发育（MGHD）。目前尚无关于MGHD的全球结构的结果，此类结果对于证明具有冲击的经典解决方案的独特性至关重要。对于粘性流体模型，目前尚无建设性的爆炸结果，因此任何建设性的奇异性形成结果都将是同类产品中的第一个。对于爱因斯坦的方程式（与各种物质模型相结合），目标是了解时空奇点的结构和稳定性，重点关注的是可本地化且可靠的技术，从而允许人们探测新的解决方案方案。在所有问题中，由几何和分析考虑的仪表选择位于分析的核心。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准，通过评估被认为是宝贵的支持。