Mathematical Sciences: Existence and Regularity of Solutionsto Nonlinear Wave Equations

8902136 Beals这个项目将重点分析非线性严格双曲型偏微分方程解的分析。具体地说，将在构造证明存在精确描述的奇点的例子方面做工作，并建立证明这些奇点本质上是最糟糕的可能奇点的正则性结果。众所周知，溶液在时间上演化时的自然条件会减少奇点的形成，至少在时间上是局部的。在这项工作中要考虑的相关问题包括在焦散表面出现的情况下发生的奇点相互作用的问题。这与人们所熟知的光滑特征曲面上奇异点的简单相互作用形成了鲜明的对比。除此之外，还将讨论三重互动的问题。众所周知，过去解的分段光滑性在三重相互作用后是不能保持的。本研究的目标之一是利用给定的余正交型或经典余交型数据构造解的过程。在四维空间中，简单的初始条件预计会产生复杂的非线性奇点集。构建这种类型的行为的示例将被处理。文中还将考虑凸面障碍物外部的波动方程。它代表了最简单的问题，涉及奇点掠过表面附近的相互作用；它可能导致单一的非线性奇点。为了完成其中一些构造，需要精确的正则性定理来证明显式近似解与实际解的不同之处在于具有严格更大光滑性的余项，并将已知的正则性结果推广到低阶光滑性。

8902136 Beals This project will emphasize the analysis of solutions of nonlinear strictly hyperbolic partial differential equations. Specifically, work will be done in constructing examples in which precislely described singularities are shown to exist, and to establish regularity results which show that these are essentially the worst possible singularities. Natural conditions on the smmoothness of the solution as it evolves in time are known to curtail the formation of singularities, at least locally in time. Among the related problems to be considered in this work are questions concerning interactions of singularities occurring in cases where caustic surfaces appear. This contrasts with that of simple interactions of singularities conormal across smooth characteristic surfaces which are well understood. In addition to this, the question of triple interactions will be addressed. It is known that piecewise smoothness of solutions in the past is not preserved after a triple interaction. A procedure for constructing solutions with given data of conormal or classical conormal type will be one of the objectives of this research. In four dimensions, simple initial conditions are expected to give rise to complex sets of nonlinear singularities. The construction of an example of this type of behavior will be treated. The wave equation on the exterior of a convex obstacle will also be considered. It represent the simplest problem involving interaction near a grazing surface of singularities; it may give rise to single nonlinear singularity. To accomplish some of these constructions, precise regularity theorems will be required to show that explicit approximate solutions differ from actual ones by remainders of strictly greater smoothness and to extend known regularity results to lower order smoothness.