Mathematical Sciences: Microlocal Analysis and Applications

在这个项目的过程中，将发展两个主题。每一个都以与偏微分方程相关的基本问题为中心。在第一个实例中，我们将研究非线性偏微分方程解的奇异性的传播。基本问题是在已知过去发生的奇点的情况下预测正时间的奇点。一个类似的问题是根据柯西数据在初始时刻的知识来描述奇点。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域现象。这一原理的现代阐述是由波前集给出的，根据索博列夫空间包含的解，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。第二步研究主型伪微分算子的局部可解性。一个是对相关非齐次方程的解感兴趣。Nirenberg和Treves的早期工作证明了微分算子解析情况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，在二维情况下证明了该条件的充分性。工作将集中于解除这个尺寸限制。

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.