Geometric Methods for Singular Solutions to Nonlinear Hyperbolic Partial Differential Equations

首席研究员（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——引力波最近的实验发现导致了诺贝尔物理学奖。关于流体和相关方程的项目将允许人们对冲击波（例如音爆）的形成和结构做出严格的数学预测。一个关键的新贡献将是解释旋涡运动的存在，这是一种在自然界中普遍存在的众所周知的复杂现象。广义相对论的项目将根据对宇宙当前状态的假设，对过去是否发生过大爆炸做出严格的预测。这些结果将严格地证实大爆炸在预期发生的所有情况下的动态稳定性，从而为一个植根于50年前的猜想提供证据。流体和重力项目的一个共同主题是，它们都涉及波状运动。在之前的工作中，PI开发了研究波的新工具，这些工具是由几何思想塑造的。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、显著的共性。这些联系使PI能够将看似独立的流体和重力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推动力。研究方向是高度跨学科的，为培养下一代跨学科研究人员提供了成熟的机会。本科生、博士生和博士后研究人员将参与该项目的工作。在第一线研究中，PI将研究三维空间中可压缩欧拉方程的稳定激波形成解，重点是给出直至边界的最大经典发展的完整描述。该研究的一个关键新特征是涡度和熵可以是非零的，并且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度和熵，因此分析需要大量新的几何技术和对流体流动的见解。在第二方面的研究中，PI将研究各种多速度拟线性双曲PDE系统的激波发展问题。这是一个描述最初光滑解越过第一个激波奇点的过渡的问题，使它们成为唯一的弱解，同时构造激波超曲面，使解跳跃。多速度系统在多空间维度上的冲击发展问题是完全开放的。这项研究需要新的技术来解释不同的溶液变量在激波超表面过渡时所表现出的独特的奇点强度。在第三条研究路线中，PI将研究爱因斯坦方程解中的稳定大爆炸形成（即沿整个类空间超表面的稳定曲率爆炸）。提出的方法是基于一种新的标准，这种标准将允许人们在整个推测发生大爆炸的地区证明稳定的大爆炸形成。由于大爆炸奇点和激波奇点的特性，这些方法与激波问题有着深刻的分析联系。相反，与激波问题相关的技术起源于广义相对论。因此，各个研究方向之间具有凝聚力。该奖项反映了美国国家科学基金会的法定使命，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 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.