Microlocal Analysis and Nonlinear Waves

9706489威廉姆斯拟议的研究旨在应用和进一步发展微局部方法(相空间局部化；傅立叶分析；拟微分、傅立叶积分和拟微分算子)来研究几种类型的非线性波动问题：拟线性边值问题的连续和间断(激波)解的严格非线性几何光学，爆破的新机制，以及光滑障碍对一般奇点的半线性绕射。关于冲击的工作与现实世界的应用程序联系最紧密。冲击发生在许多情况下，涉及到某种类型的“流体”的流动，通常指的是像空气一样的气体或像水一样的液体。例如，它们与喷气式飞机的飞行、航天飞机的再入以及燃烧或爆炸有关。几何光学是在构造和分析“近似激波”的基础上，用来获得定性信息和预测激波的一种数学方法。在过去，对于近似冲击是否真的像物理冲击一样，一直存在不确定性。我们的工作旨在准确地了解近似激波何时是精确激波，何时不是精确激波。

9706489 Williams The proposed research aims to apply and further develop microlocal methods (localization in phase space; Fourier analysis; pseudodifferential, Fourier integral, and paradifferential operators) for the study of several types of nonlinear wave problems: rigorous nonlinear geometric optics for continuous and discontinuous (shock) solutions of quasilinear boundary problems, new mechanisms for blowup, and semilinear diffraction of generic singularities by smooth obstacles. The work on shocks has the closest connection to real-world applications. Shocks arise in many contexts involving the flow of some type of "fluid", which usually means a gas like air or a liquid like water. They form in connection with flights of jets, reentry of space shuttles, and as a result of combustion or explosion, for example. Geometric optics is a mathematical method that can be used to obtain qualitative information and make predictions about shocks, which is based on constructing and analyzing "approximate shocks". In the past there has been uncertainty about whether the approximate shocks really behave like the exact physical shocks. Our work aims to understand precisely when the approximate shocks are and are not close to exact shocks.