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Semiclassical Pseudo-differential Calculus: properties and applications
Student-Run Analysis & PDE| Speaker: | Annalaura Stingo, UC Davis |
| Related Webpage: | https://www.math.ucdavis.edu/people/general-profile?fac_id=astingo |
| Location: | 1147 MSB |
| Start time: | Thu, Feb 27 2020, 1:10PM |
Abstract: In several examples of PDEs (wave equation, Klein-Gordon equation, water waves) some fundamental properties of the solutions are recovered by studying the equation in the phase-space (x,\xi) (the space of positions and frequencies) rather than in the physical space only. For example, solutions may satisfy better estimates in some localized regions of the phase-space than the estimates they satisfy globally, and highlighting this behavior is of great importance when proving the long-time/global existence of such solutions.
The aim of this talk is to introduce the fundamental tools used in such kind of analysis and discuss their properties. In particular, I will talk about semiclassical pseudo-differential operators and semiclassical pseudo-differential calculus, that generalize the notions already discussed by Black in a previous talk. Time permitting, I will show how I used these tools to prove the global-in-time existence of small solutions to 1D quasilinear Klein-Gordon equations.