Dispersive and Wave Equations in the Presence of Background Geometry

本项目旨在进一步理解几何背景下波动方程的解。例如，为了达到必要的精度，GPS依赖于广义相对论，而广义相对论又假设重力是波在弯曲的时空上传播的结果，而不是外力。在很多涉及非线性方程的应用中，控制系统的几何形状依赖于解而解又依赖于几何形状。爱因斯坦的方程是广义相对论的核心，它决定了宇宙从给定的起始状态开始的演化，它可以在某些坐标系中被实现为这样一个波动方程系统。在背景几何上，波沿着被称为测地线的特殊曲线流动，而不是我们更熟悉的射线。当这些测地线中的一些始终保持在一个有界集合中时，就会出现一种称为捕获的现象。例如，在已知的黑洞时空中，确实存在光绕黑洞运行的区域，而不是向无限延伸的区域。捕获是典型色散测量的已知障碍，本项目的主要重点是精确量化其在多种情况下的影响。该项目为本科生和研究生提供研究训练机会。要研究的问题主要集中在综合局部能量估计上，这是Morawetz原始估计的推广，以及它们在非线性方程中的应用。主要举措包括提高我们对存在捕获和在非平稳背景下的这种估计的理解。具有简并俘获的时空构造提供了第一个例子，其中正则性的代数损失对于恢复局部能量估计既是必要的也是充分的。与这些例子相关的许多问题仍未被探索，包括在高度对称的翘曲积设置之外发现具有简并俘获的时空。局部能量估计可以用来建立非线性方程的长期存在，并且在存在背景几何的情况下具有特别的好处。与此相关的计划工作包括检查半空间上的波动方程，以及将Dafermos和Rodnianski的相关加权估计应用于与Strauss猜想相关的临界阻尼方程。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。

This project seeks to further the understanding of solutions to the wave equation on geometric backgrounds. To achieve the requisite accuracy, GPS, for example, relies on general relativity, which in turn postulates that gravity is the result of a curved space-time on which waves travel rather than an external force. And in many applications involving nonlinear equations, the governing systems have geometry that depends on the solution but the solution in turn depends on the geometry. Einstein’s equations, which are at the heart of general relativity and determine the evolution of a universe from a given starting state, can be realized as such a system of wave equations in certain coordinate systems. On background geometries, waves flow along special curves called geodesics rather than rays as is more familiar. A phenomenon called trapping occurs when some of these geodesics remain in a bounded set for all time. This occurs, for example, on known black hole space-times where there are indeed regions where light orbits the black hole rather than tending toward infinity. Trapping is a known obstruction to typical measures of dispersion, and a major focus of this project is to precisely quantify its effect in numerous scenarios. The project provides research training opportunities for both undergraduate and graduate students.The problems to be examined largely focus on integrated local energy estimates, which are generalizations of the original estimates of Morawetz, and their application to nonlinear equations. Major initiatives include improving our understanding of such estimates in the presence of trapping and on non-stationary backgrounds. The construction of space-times with degenerate trapping provided the first examples where an algebraic loss of regularity is both necessary and sufficient for recovering local energy estimates. Numerous questions related to these examples remain unexplored, including the discovery of space-times with degenerate trapping outside of the highly symmetric warped product setting. Local energy estimates can be used to establish long-time existence for nonlinear equations and have particular benefits in the presence of background geometry. Planned work related to this include the examination of wave equations on half-spaces and applications of a related weighted estimate of Dafermos and Rodnianski to critically damped equations related to the Strauss conjecture.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.