非线性偏微分方程的微局部分析

Microlocal Analysis of Nonlinear Partial Differential Equations

这是一篇介绍当前方程界十分重要的课题——非线性微局部分析——的综述文章。作为这一研究领域的开拓者,我们在不太长的篇幅里,从相当的理论高度简洁地介绍该领域近十年来一些最引人注目的工作。本文首先阐述了一般微局部分析的基本思想,然后介绍近十年来对非线性偏微分方程起重要推动作用的仿微分计算(如仿乘积,仿微分算子,仿复合等),以及有着更深刻内容的高次微局部的思想。同时,也大量介绍这些思想在非线性偏微分方程弱奇性分析中的应用,如奇性的传播,反射与绕射,余法型奇性的相互作用,非线性亚椭圆性,以及三个奇性波的相互作用等,这些均是当前方程界的热门课题。

This is a survey on a very active topic in recent decades, i.e.nonlinear microlocal analysis.As one of the pioneers in this area, it gives a very concise introduction to the most remarkable results in recent years. The survey starts from an explanation of the basic ideas and technique of nonlinear microlocal analysis. Then a systematic exposition on 'para' technique also the (paraproduct, para-differential operators, and para-composition) is given,deeper idea of microlocalization of higher order is introduced.The remaining part of the survey consists of some most active problems,such as the analysis of weak singularities of non-linear P.D.E.,their propagation, reflections and diffraction, conormal singularities and their interaction, non-linear hypoellipticity, interaction of three waves.