Regularity Properties of Partial Differential Equations

摘要马切登提出要研究两类问题。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.Klaineman以前的工作。应用于特殊双线性量的傅立叶分析方法在这项工作中具有重要意义。第二个问题是Szego核的迹的渐近行为，以及这个局域量如何与边界的整体几何关系。它是由黎曼曲面上的圆盘丛的例子所激发的。这是与J·米尔森的联合工作。第一个问题是由一些数学物理方程引起的，例如著名的、非常困难的爱因斯坦场方程。研究简化的模型，希望理解问题的数学理论，例如：在奇点(如黑洞)形成之前，良好解的哪些数学特征决定了该解保持良好的时间？第二个问题将局部量(如曲率)与全局量(如形状)相关联。

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).