Mathematical Sciences: Existence and Regularity of Solutionsto Nonlinear Wave Equations

这个项目将着重分析非线性严格双曲型偏微分方程的解。具体地说，工作将在构造实例中完成，其中精确描述的奇点被证明存在，并建立规则性结果，表明这些本质上是最坏的可能奇点。已知随着时间的推移，解的平滑性的自然条件会减少奇点的形成，至少在局部时间上是这样。在这项工作中要考虑的相关问题是在腐蚀性表面出现的情况下发生的奇点相互作用的问题。这与我们很好地理解的光滑特征表面上的法向奇点的简单相互作用形成对比。除此之外，还将讨论三重相互作用的问题。已知，在三重相互作用后，过去解的分段平滑性已不能保持。本研究的目标之一，将是利用给定的正法型或经典正法型数据构造解的程序。在四维空间中，简单的初始条件会产生复杂的非线性奇异集。我们将讨论这种行为的一个例子的构造。本文还将考虑凸障碍物外部的波动方程。它代表了最简单的问题，涉及奇点放牧表面附近的相互作用；它可能产生单一的非线性奇点。为了完成其中的一些构造，将需要精确的正则性定理来表明显式近似解与实际解有严格更大平滑的余数不同，并将已知的正则性结果扩展到低阶平滑。

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.