Studies on Dispersive and Wave Equations

这个项目的目的是提高我们对复杂背景几何对某些波动或色散方程的解的影响的理解。例如，几何可能是由于引入边界、非平坦度规的影响或考虑非均匀材料而产生的。这个项目的一个主要方面是研究变系数波动方程的全局时间弥散度量，例如局部能量估计和Strichartz估计。与平面情况相比，人们遇到了额外的困难，例如，由于哈密顿射线、囚禁射线的聚焦，以及本征函数或共振的可能性。一个特别有趣的例子是研究关于爱因斯坦方程的非平坦解的波动方程，例如施瓦茨柴尔德时空。这个项目的第二个主要组成部分是研究某些非线性波方程或弹性波方程解的长期存在性。这里特别感兴趣的是各向异性弹性问题或外部区域中的非线性波动方程的存在性问题，其中某些通常用于证明长期存在的不变性是不可用的。理论物理中的一类基本开放问题围绕着爱因斯坦方程解的稳定性。也就是说，如果宇宙开始时接近某个已知的解，人们可能会试图证明它在以后的所有时间都保持在那个解的附近。唯一已知的稳定性的严格证明是关于所谓的平坦解的。人们认为，对波动方程色散性质的良好理解将在任何稳定性的证明中起到至关重要的作用，上面描述的第一组问题试图提供一些关于这一点的见解。了解偏微分方程解的寿命是很有意义的。我们的目标是估计一个解在变得无限大(爆炸现象)之前可能存在的时间量。在这个项目中特别感兴趣的是非线性弹性中某些问题的解的寿命。这些问题类似于非线性波动方程中的某些问题，其中长期存在的证明在很大程度上依赖于线性波动方程的许多不变性。然而，在非均匀材料中，这些不变性可能会丢失，因此需要不同的技术。作为平面上各向同性保持不变的问题的模型问题，如六方晶体，我们将首先研究某些二维各向同性问题。在这里，人们必须更巧妙地利用方程的非线性结构，才能证明这种长期存在。

The purpose of this project is to improve our understanding of the effect of complicated background geometry on solutions to certain wave or dispersive equations. The geometry may result, for example, from the introduction of a boundary, from the influence of a nonflat metric, or from consideration of nonhomogeneous materials. One major aspect of this project is to study global-in-time measures of dispersion, such as localized energy estimates and Strichartz estimates, for variable coefficient wave equations. As compared to the flat cases, one encounters additional difficulties that arise, for instance, because of the focusing of Hamiltonian rays, trapped rays, and the possibility of eigenfunctions or resonances. A particular example of interest is to study the wave equation on nonflat solutions to Einstein's equations, such as the Schwarzschild space-time. A second major component of this project is to study the long-time existence of solutions to certain nonlinear wave or elastic wave equations. Of special interest here are problems in anisotropic elasticity or existence questions for nonlinear wave equations in exterior domains where certain invariances, which are typically used to prove long-time existence, are not available.A class of fundamental open questions in theoretical physics revolves around the stability of solutions to Einstein's equations. Namely, if the universe starts close to a certain known solution, one might seek to prove that it remains close to that solution for all later times. The only known rigorous proofs of stability are for the so-called flat solution. It is believed that a good understanding of the dispersive nature of wave equations will play an essential role in any proof of stability, and the first set of questions described above seeks to provide some insights into this. Understanding the lifespan of solutions to partial differential equations is of basic interest. The objective is to estimate the amount of time that a solution may exist prior to, say, becoming infinitely large (blow-up phenomena). Of particular interest in this project is the lifespan of solutions to certain problems in nonlinear elasticity. These problems resemble certain questions in nonlinear wave equations where the proofs of long-time existence rely heavily on the many invariances of the linear wave equation. In nonhomogeneous materials, however, these invariances may be lost, so different techniques are required. As a model problem for problems where isotropy is maintained in planes, such as hexagonal crystals, certain two-dimensional isotropic problems will first be studied. Here one must make more delicate use of the nonlinear structure of the equations in order to show such long-time existence.