Microlocal Analysis and Nonlinear Waves

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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.