Regularity Properties of Partial Differential Equations

摘要：本文主要研究两类问题。第一个问题是非线性波动方程的最优正则性和爆破问题，延续了S. Klainerman先前的工作。应用于特殊双线性量的傅里叶分析方法在这项工作中是至关重要的。第二个问题是Szego核的迹的渐近性，以及这个局部量与边界的全局几何的关系。它是由黎曼曲面上的圆盘束的例子激发的。这是与J. Millson的合作。第一个问题是由一些数学物理方程引起的，比如著名的、非常难解的爱因斯坦场方程。简化的模型被研究，希望能理解一些问题的数学理论，如：一个好的解的什么数学特征决定了这个解在奇点（如黑洞）形成之前能保持多久？第二个问题将局部量（如曲率）与全局量（如形状）联系起来。

Abstract Machedon It is proposed to study two kinds of questions. The first one is the Question of optimal regularity and blow-up of non-linear wave equations, continuing previous work with S. Klainerman. Fourier analysis methods applied to special bilinear quantities are of cental importance in this work. The second one is the question of asymptotic behaviour of the trace of the Szego kernel, and how this local quantity relates to the global geometry of the boundary. It is motivated by the example of a disc bundle over a Riemann surface. It is joint work with J. Millson. The first problem is motivated by some equations of Mathematical Physics, such as the famous and very difficult Einstein Field equation. Simplified models are studied, hoping to understand the mathematical theory of questions such as: What mathematical features of a nice solution determine for how long that solution stays nice, before a singularity (such as a black hole) forms? The second question relates local quantities (such as curvature) to global quantities (such as shape).