Mathematical Sciences: Existence and Regularity of Solutionsto Nonlinear Wave Equations

8902136 比尔斯 本项目将重点分析解决方案， 非线性严格双曲型偏微分方程 具体而言，工作将在构建示例中完成， 这些精确描述的奇点被证明是存在的， 并建立规律性的结果表明，这些是 本质上是最糟糕的奇点 自然条件下的smsmsmoothness的解决方案， 随着时间的推移， 奇点，至少在时间上是局部的。 在相关的 在这项工作中要考虑的问题是关于 在焦散线 表面出现。 这与简单的 光滑域上余法向奇点的相互作用 这些特征表面是很好理解的。 此外 为此，将讨论三重相互作用的问题。 已知解的分段光滑性 在三重相互作用后不被保留。 的程序 用给定的余正规或经典数据构造解 余正常类型是本研究的目标之一。 在四维空间中， 从而产生复杂的非线性奇点。 的 这种行为的一个例子的建设将是 治疗。 凸曲面外的波动方程 障碍也将被考虑。 它代表了最简单的 涉及到的相互作用问题附近的放牧表面的 奇点;它可能会引起单一的非线性 奇点 为了完成其中的一些构造， 需要精确的正则性定理来证明， 显式近似解与实际解的差异如下： 严格的更大的光滑度和扩展已知 规则性导致较低阶的平滑性。

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.