Mathematical Sciences: Research in Partial Differential Equations

本项目继续对线性和非线性一阶偏微分方程过定系统解的存在性和光滑性进行数学研究。令人吃惊的是，与常微分方程相反，有一些简单的线性偏微分方程没有解。对这些问题的分析，以及相关的基本问题，是本研究的核心。这项工作，用被称为伪微分算子的变换方程来表述，试图获得关于方程组的局部可解性的信息。通过研究一阶非线性方程的奇异性，即解析波前集，对一阶非线性方程的微局部可解性进行了相关研究。目前已经发现的是奇点的传播和线性情况完全一样。一项新的调查也已经开始。本文考虑流形上复向量场系统的研究。它将经典的微分算子思想与流形上同调理论相结合，可以用来定义广义特征类。第一个目标是研究这些阶级不消失的影响。同样要考虑的是由于这种不消失而没有的几何性质。偏微分方程是物理科学中数学建模的基础。涉及连续变化的现象，例如在运动、材料和能量中看到的现象，都遵循某些一般规律，这些规律可以用偏导数之间的相互作用和关系来表示。数学的关键作用不是说明关系，而是从中提取定性和定量的意义。

This project continues mathematical research on the existence and smoothness of solutions of overdetermined systems of first-order partial differential equations, both linear and nonlinear. It is striking that, in contrast to ordinary differential equations, there are simple linear partial differential equations possessing no solution. The analysis of this, and related basic questions, lies at the heart of this research. The work, phrased in terms of transformed equations known as pseudodifferential operators, seeks to obtain information about local solvability of systems of equations. Related work is going forward on the microlocal solvability of first order nonlinear equations through study of the singularities of the equations known as the analytic wave-front set. What has been discovered so far is that the singularities propagate exactly as in the linear case. A new line of investigation has also begun. This considers studies of systems of complex vector fields on a manifold. It mixes classical ideas of differential operators with cohomology theory on the manifold which can be used to define generalized characteristic classes. The first objective will be to examine the implications of the nonvanishing of those classes. Also to be considered are the geometrical properties absence of due to this nonvanishing. Partial differential equations form the backbone of mathematical modeling in the physical sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not to state the relationships, but rather, to extract qualitative and quantitative meaning from them.