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异性的形成，包括爱因斯坦方程的解决方案和RICCI流动的解决方案。对于爱因斯坦的方程式来说，关键问题之一是强大的宇宙审查制度（“ SCC”）：的确，霍金 - 普通定理所预测的无处不在的大地测量不完整是否涉及时空曲率（并因此引力为引力）吗？至少在异构体定义的某些空间家族中，解决方案表现出“ AVTD”行为的决心（在奇异性附近的太空衍生物中的时间导数的优势）是研究SCC的非常有用的工具，并且本身就很有趣。我们的一些工作涉及开发一个框架，以证明这些家庭中表现出AVTD行为的平滑（非分析）解决方案。我们对RICCI流程中奇异性的研究集中在那些被称为“领带”的人。我们特别表明，有一个离散参数化的RICCI流动溶液家族，它们会形成“退化的领带”，而几何形状渐近地接近“ Bryant Soliton”，以处方率（与2型行为一致）。这些溶液在旋转上都是对称的。想要知道这些解决方案中看到的详细渐近行为是否也是非旋转对称解决方案的特征与我的研究计划中的许多其他人一样，主要目标是对偏微分方程的复杂非线性系统的大量解决方案的总体行为理解。这种方程式系统在我们在陆地尺度和天体物理/宇宙学量表上对物理现象的建模中起着重要作用，无法明确解决。但是，使用该提案中开发和使用的技术，可以学习有关这些模型解决方案行为的大量实用信息，因此，对这些方程式系统建模的物理系统的行为做出了非常有用的预测。