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Geometric Methods for Singular Solutions to Nonlinear Hyperbolic Partial Differential Equations

非线性双曲偏微分方程奇异解的几何方法

基本信息

批准号：
2054184
负责人：
Jared Speck
金额：
$ 33.61万
依托单位：
Vanderbilt University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-06-01 至 2024-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054184&HistoricalAwards=false
关键词：
Geometric Methods Singular Solutions Nonlinear

项目摘要

The principal investigator (PI) will study equations with physical and geometric origins, including Euler’s equations, which describe the motion of fluids such as air and water, and Einstein’s equations of General Relativity, which describe the propagation of gravitational waves – whose recent experimental detection led to the Nobel Prize in Physics. The projects on fluids and related equations will allow one to make rigorous mathematical predictions about the formation and structure of shock waves (e.g. sonic booms). A key new contribution will be accounting for the presence of swirling motion, a notoriously complex phenomenon that is ubiquitous in nature. The projects in General Relativity will make rigorous predictions about whether a Big Bang occurred in the past, based on assumptions about the present state of the universe. The results will rigorously confirm the dynamic stability of the Big Bang for the full range of situations where it has been expected to occur, thus providing a proof of a conjecture that has its roots in ideas stretching back 50 years. A common theme unifying the projects in fluids and gravity is that they involve wave-like motion. In previous work, the PI developed new tools for the study of waves, shaped by ideas from geometry. In particular, his recent work has shown that the equations of fluid motion have some unexpected, remarkable commonalities with Einstein’s equations. These connections allow the PI to blend insights and techniques from the seemingly separate fields of fluids and gravity, which in turn serves as a driving force behind the projects. The research directions are highly interdisciplinary and are ripe with opportunities for training the next generation of researchers across disciplines. Undergraduates, Ph.D. students, and postdoctoral researchers will be involved in the work of the project. In a first line of research, the PI will study stable shock-forming solutions to the compressible Euler equations in three spatial dimensions, with a focus on giving a complete description of the maximal classical development up to the boundary. A key new feature of the research is that the vorticity and entropy can be non-zero, and the behavior of these quantities must be tracked all the way up to the boundary. Because the shape of the boundary is unknown in advance, and because elliptic estimates are needed to control the vorticity and entropy, the analysis requires a multitude of new geometric techniques and insights about fluid flow. In a second line of research, the PI will study the shock development problem for various multiple speed quasilinear hyperbolic PDE systems. This is the problem of describing the transition of initially smooth solutions past their first shock singularity in a manner such that they become unique weak solutions, while simultaneously constructing the shock hypersurface, across which the solution jumps. The shock development problem for multiple speed systems in multiple spatial dimensions is completely open. This research requires new techniques that account for the distinct singularity strengths exhibited by the different solution variables as they transition across the shock hypersurface. In a third line of research, the PI will study stable Big Bang formation (i.e., stable curvature blowup along an entire spacelike hypersurface) in solutions to Einstein's equations. The proposed approach is based on a new gauge that will allow one to prove stable Big Bang formation in the entire regime where it has been conjectured to occur. Due to the character of Big Bang singularities and shock singularities, the methods have deep analytical connections to the problems on shocks. Conversely, the techniques relevant for the problems on shocks have their origins in General Relativity. Thus, there is cohesiveness between all the research directions.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.

主要研究员（PI）将研究具有物理和几何起源的方程，包括描述空气和水等流体运动的欧拉方程，以及描述引力波传播的爱因斯坦广义相对论方程-最近的实验检测导致诺贝尔物理学奖。关于流体和相关方程的项目将使人们能够对冲击波（例如音爆）的形成和结构进行严格的数学预测。一个关键的新贡献将是解释漩涡运动的存在，这是一种众所周知的复杂现象，在自然界中无处不在。广义相对论中的项目将根据对宇宙目前状态的假设，对过去是否发生过大爆炸进行严格的预测。这些结果将严格证实大爆炸在预期发生的所有情况下的动态稳定性，从而证明了一个起源于50年前的猜想。一个共同的主题统一的项目，在流体和重力是，他们涉及波一样的运动。在以前的工作中，PI开发了新的工具来研究波，这些波是由几何学的思想塑造的。特别是，他最近的工作表明，流体运动方程与爱因斯坦方程有一些意想不到的，显着的共同点。这些联系使PI能够将流体和重力这两个看似独立的领域的见解和技术融合在一起，这反过来又成为项目背后的驱动力。研究方向是高度跨学科的，并且有机会培养跨学科的下一代研究人员。本科生，博士学生和博士后研究人员将参与该项目的工作。在研究的第一线，PI将研究稳定的激波形成解决方案，可压缩欧拉方程在三维空间，重点是给出一个完整的描述最大的经典发展到边界。这项研究的一个关键新特征是涡量和熵可以是非零的，这些量的行为必须一直跟踪到边界。由于边界的形状事先是未知的，并且由于需要椭圆估计来控制涡量和熵，因此分析需要大量新的几何技术和对流体流动的见解。在第二条研究线中，PI将研究各种多速准线性双曲PDE系统的激波发展问题。这是一个描述初始光滑解经过第一个激波奇点的过渡的问题，使得它们成为唯一的弱解，同时构造激波超曲面，解跳过该超曲面。多维空间中多速度系统的激波发展问题是完全开放的。这项研究需要新的技术，占不同的解决方案的变量所表现出的独特的奇异性强度，因为他们跨越冲击超曲面过渡。在第三条研究路线中，PI将研究稳定的大爆炸形成（即，稳定曲率沿着整个类空超曲面爆破）。所提出的方法是基于一个新的规范，这将允许一个证明稳定的大爆炸形成在整个制度，它已被证实发生。由于大爆炸奇异性和激波奇异性的特点，这些方法与激波问题有着深刻的分析联系。相反，与冲击问题相关的技术则起源于广义相对论。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。