Mathematical Sciences: Microlocal Analysis and Applications

数学科学：微局域分析与应用

• 批准号: 8802821
• 负责人: Nicolas Lerner
• 金额: $ 3.82万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-06-01 至 1990-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8802821&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Microlocal; Analysis; Applications

Two themes will be developed during the course of this project. Each centers on fundamental questions related to partial differential equations. In the first instance work will be done investigating the propagation of singularities for solutions of nonlinear partial differential equations. The basic question is one of predicting singularities in positive time knowing the singularities which occurred in the past. An analogous problem is the description of the singularities from knowledge of the Cauchy data at the initial time. The propagation of singularities is a microlocal phenomenon, as pointed out by Huygens centuries ago. The modern exposition of this principle is given by the wave front set and, depending on which Sobolev space contains a solution, the singularities travel along bicharacteristics, as in the linear case, or else require intricate description - which represents one of the goals of this work. A second line of research will deal with the local solvability for pseudo-differential operators of principal type. One is interested in solutions of the associated nonhomogeneous equation. Earlier work of Nirenberg and Treves proved a sufficient condition for local solvability in the analytic case for differential operators. Roughly the same condition gives a necessary condition for local solvability for pseudo-differential operators. Recently, sufficiency of the condition was shown for the two-dimensional case. Work will concentrate on lifting this dimension restriction.

在这个项目的过程中，将制定两个主题。每本书都围绕着与偏微分方程有关的基本问题。首先，我们将研究非线性偏微分方程解的奇性传播问题。基本问题是在已知过去发生的奇点的情况下，在正时间预测奇点的问题。一个类似的问题是根据初始时间的柯西数据的知识来描述奇点。正如惠更斯几个世纪前指出的那样，奇点的传播是一种微局部现象。这一原理的现代阐述是由波前集合给出的，取决于哪个索波列夫空间包含解，奇点沿着双特征传播，就像在线性情况下一样，或者需要复杂的描述--这代表了这项工作的目标之一。第二个研究将讨论主型拟微分算子的局部可解性。人们对相关的非齐次方程的解很感兴趣。Nirenberg和Treves的早期工作证明了微分算子在解析情形下局部可解的一个充分条件。大致相同的条件给出了伪微分算子局部可解的必要条件。最近，对于二维情形，证明了该条件的充分性。工作将集中在取消这一尺寸限制上。