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解的影响的理解。例如，几何形状可以由边界的引入、由非平坦度量的影响或由非均匀材料的考虑而产生。这个项目的一个主要方面是研究全球的时间分散措施，如局部能量估计和Escherichartz估计，变系数波动方程。与平面情况相比，人们遇到了额外的困难，例如，由于哈密尔顿射线的聚焦，捕获的射线，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爱因斯坦方程的非平坦解的波动方程，例如史瓦西时空。本项目的第二个主要组成部分是研究某些非线性波动或弹性波动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非线性波动方程在外部区域的存在性问题，其中某些通常用于证明长期存在性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里都保持接近那个解。唯一已知的严格的稳定性证明是所谓的平坦解。人们相信，对波动方程色散性质的良好理解将在任何稳定性证明中发挥重要作用，上述第一组问题旨在对此提供一些见解。理解偏微分方程解的寿命是基本的兴趣。 我们的目标是估计一个解决方案可能存在的时间量之前，说，成为无限大（爆破现象）。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决方案的寿命。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严重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。作为一个模型问题的问题，各向同性是保持在平面，如六方晶体，某些二维各向同性问题将首先进行研究。在这里，人们必须更精细地利用方程的非线性结构，以表明这种长期存在。

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.