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

Grading

No final examination.

Some number of homework exercises will be assigned. Students are free to collaborate on the homework exercises. On a per question basis you must make a complete list of your collaborators. If you did not collaborate with anyone else on the homework, you must write "Collaborators: none" for the question.

Course material

Lectures will be based on my notes on dispersive PDEs, available here.

Note: my lectures may be out-of-order compared to what is included in the lecture notes.