Mathematical Sciences: Microlocal Analysis and Applications

在此过程中将发展两个主题 项目 每一个中心的基本问题， 偏微分方程 首先，工作将 研究奇点的传播， 非线性偏微分方程的解 基本 问题是预测正时间内的奇点 知道过去发生的奇异点 一个 类似的问题是描述的奇点， 在初始时刻的柯西数据的知识。 的 奇点的传播是一种微局部现象， 惠更斯在几个世纪前就指出了这一点 现代博览会 该原理由波前集给出，并且取决于 其中Sobolev空间包含一个解决方案，奇点旅行 沿着双特征，如在线性情况下，否则需要 复杂的描述-这代表了这个目标之一， 工作 第二条研究路线将处理当地的 主型伪微分算子的可解性 一个是感兴趣的解决方案，相关的非齐次 方程 尼伦伯格和特里维斯的早期工作证明了 解析情形下局部可解的充分条件 微分算子。 大致相同的条件下， 拟微分方程局部可解必要条件 运营商 最近，该条件的充分性被证明为 二维案例 工作将集中在解除这个 尺寸限制。

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.