Phase space analysis and nonlinear dispersive equations
speaker: Daniel Tataru (University of Berkeley California)abstract: The aim of the course is to introduce a phase space approach to the study of linear and nonlinear dispersive equations. We first introduce the Bargman transform and a corresponding approach to the study of pseudodifferential and Fourier integral operators. Then we consider Schrödinger like equations and obtain phase space representations for the Fourier integral operators governing the corresponding evolution. In other words, this amounts to obtaining representations of solutions as superpositions of wave packets, i.e. special solutions which are highly localized in the phase space and move essentially along the Hamilton flow. A more delicate application of this is to study long time outgoing parametrices for Schrödinger equations with variable coefficient principal part.
A second part of the course is devoted to a related approach for the wave equation. This is similar in spirit but the geometry is different. The mainn application that is discussed is to the study of quasilinear wave equations.
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