On the Behavior of Solutions of Einstein's Equations and Other Geometric Nonlinear Partial Differential Equation Systems

关于爱因斯坦方程和其他几何非线性偏微分方程组解的行为

• 批准号: 1306441
• 负责人: James Isenberg
• 金额: $ 15万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1306441&HistoricalAwards=false
• 关键词: Behavior; Solutions; Einstein; Equations; Other

A good portion of this award is devoted to the study of the formation of singularities, both in solutions of Einstein's equations and in solutions of Ricci flow. For Einstein's equations, one of the key issues is Strong Cosmic Censorship ("SCC"): Is it true that the ubiquitous geodesic incompleteness predicted by the Hawking-Penrose theorems generically involve spacetime curvature (and therefore gravitational tidal) blowup? At least in certain families of spacetimes defined by isometries, the determination that the solutions exhibit "AVTD" behavior (dominance of time derivatives of the fields over space derivatives near the singularity) is a very helpful tool for studying SCC, and is interesting in its own right. Some of our work involves the development of a framework for proving that there are large sets of smooth (non-analytic) solutions in these families which show AVTD behavior. Our studies of singularities in Ricci flow have focused on those known as "neckpinches". We have, in particular, shown that there is a discretely parametrized family of Ricci flow solutions which develop "degenerate neckpinches", with the geometry asymptotically approaching a "Bryant soliton" at a prescribed rate (consistent with Type 2 behavior). These solutions are all rotationally symmetric. Wanting to know if the detailed asymptotic behavior seen in these solutions is also characteristic of non-rotationally symmetric solutions, we have chosen to first address this issue in the case of mean curvature flow ("MCF"), since MCF neckpinches share many features with those of Ricci flow, and since it is easier with MCF to compare non-rotationally symmetric solutions with rotationally symmetric ones.In the projects discussed above, as well as in many others in my research program, the primary goal is to develop an understanding of the general behavior of large classes of solutions of complicated nonlinear systems of partial differential equations. Such equation systems, which play a major role in our modeling of physical phenomena at the terrestrial scale as well as at the astrophysical/cosmological scale, cannot be solved explicitly. However, using techniques such as those developed and used in this proposal, one can learn a tremendous amount of practical information about the behavior of solutions of these models, and therefore make very useful predictions regarding the behavior of the physical systems modeled by these equation systems.

该奖项的很大一部分致力于研究奇点的形成，无论是在爱因斯坦方程的解中，还是在利玛奇流的解中。对于爱因斯坦的方程，一个关键问题是强宇宙审查(“SCC”)：霍金-彭罗斯定理预测的普遍存在的测地线不完全性通常涉及时空曲率(因此引力潮)爆发，这是真的吗？至少在由等距线定义的某些时空族中，确定解呈现“AVTD”行为(场的时间导数相对于奇点附近的空间导数的优势)是研究SCC的一个非常有用的工具，本身就很有趣。我们的一些工作包括开发一个框架来证明在这些族中存在大量的光滑(非解析)解，它们表现出AVTD行为。我们对Ricci流中奇点的研究主要集中在被称为“颈缩”的奇点。特别地，我们已经证明了存在一个离散的参数化族的Ricci流解，它发展成“简并的颈缩”，几何以规定的速率(符合类型2的行为)渐近于一个“Bryant孤子”。这些解都是旋转对称的。为了知道在这些解中看到的详细的渐近行为是否也是非旋转对称解的特征，我们选择首先在平均曲率流的情况下解决这个问题，因为平均曲率流的颈夹与Ricci流有许多共同的特征，并且由于使用MCF更容易比较非旋转对称的解和旋转对称的解。在上面讨论的项目中，以及在我的研究计划中的许多其他项目中，主要目标是发展对复杂的非线性偏微分方程组的大类解的一般行为的理解。这些方程系统在我们对地球尺度和天体物理/宇宙尺度上的物理现象的建模中发挥着重要作用，不能明确地解决。然而，使用在本提案中开发和使用的技术，人们可以了解关于这些模型的解的行为的大量实用信息，从而对由这些方程系统建模的物理系统的行为做出非常有用的预测。