# MTH943 Dispersive Equations

This syllabus applies to the Spring 2019 edition of the course.

## General Information

This is an introductory course on dispersive partial differential equations and their a priori estimates. At the end of the course, students will understand the physical origins of dispersion and how it manifests in qualitative and quantitative behaviors of solutions to dispersive PDEs. A rough list of topics that we will aim to cover include

• A first look at dispersion via the Vlasov equation (linear transport in classical phase space); both the representation formula and vector-field method approaches to dispersive estimates will be introduced.
• Brief introduction to the quantum phase space; just enough Fourier theory will be introduced.
• Model equations: Schroedinger, Airy, Wave, Klein-Gordon.
• Dispersive estimates via representation formula (oscillatory integration).
• Dispersive estimates via vector field method.
• Basic interpolation theory; $$L^p$$ decay.
• Introduction to Littlewood-Paley decompositions, the $$TT^\star$$ method and Strichartz estimates.
• Nonlinear applications.

Time-permitting, we will also connect the theory developed to Fourier restriction phenomenon and discuss multilinear estimates in wave-Sobolev ($$H^{s,\delta}$$) or Bourgain ($$X^{s,b}$$) type spaces.

## Prerequisites

• (Required) Solid background in basic real analysis ("advanced calculus"); e.g. MTH828.
• (Recommended) Background in PDE; e.g. MTH847, 849.
• (Optional) prior knowledge of basic Fourier theory.

## Course meeting

Lectures: M W F 9:10am - 10:00am -- A306 Wells Hall

Office hours: by appointment only