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Fall 2022, Math 597, Graduate Dispersive PDEs:
Course Description: Nonlinear dispersive models are commonly used to model wave propagation in physics and engineering such as water waves, waves in plasmas, nonlinear optics, Bose-Einstein condensation, gravitational waves in general relativity, among many others. Standard models are nonlinear wave equations and Schr\"odinger equations. There have been a lot of interest in the last few decades, yet the large time dynamics of general solutions remains a great challenge for most of dispersive pdes (e.g., whether general solutions will scatter into a sum of solitons, plus the free dynamics, known as the soliton conjecture). In this course, I plan to cover the basic theory for the dispersive equations such as dispersive estimates, local smoothing estimates, Morawets estimates, Strichartz estimates, local vs global well-posedness theory, and scattering theory. There are no prerequisite background in PDEs, though the familiarity of Sobolev embeddings and Fourier transform is helpful.
Suggested Lecture Notes: I plan to follow the following lecture notes
A few suggested papers for presentation: