250142 VO Topics in Analysis (2022W)

The aim of this course is to provide an introduction to the rigorous mathematical analysis of solutions to partial differential equations with dispersive or wave features. Classical such examples are the (nonlinear) Schrodinger, wave or KdV equations, which arise in various physical contexts (such as quantum mechanics, electrodynamics, fluid motion and relativity theory) and will guide the exposition. The necessary technical tools, including in particular basics of distribution theory and harmonic/Fourier analysis, will be developed along the way.Plan of the course:- Introduction- Some background in distributions and Harmonic / Fourier analysis- Solutions to linear equations and their behavior- What do we mean by solving a nonlinear equation?- Semilinear Equations: Local Solutions- Semilinear Equations: Global Solutions and (long time) dynamics?- Quasilinear Equations: Spacetime resonance approachPrerequisites:Solid foundations in analysis, including measure theory and some basics of functional analysis. No prior knowledge of Sobolev spaces / PDE necessary.