MTH943 Dispersive and Hyperbolic Equations

This syllabus applies to the Spring 2021 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); first illustrations of dispersive estimates.
• Brief introduction to the quantum phase space; basic Fourier theory.
• Model equations: Schroedinger, wave, Klein-Gordon.
• Linear dispersive estimates: pointwise decay in time, Strichartz estimates.
• Nonlinear applications.

Time-permitting, further topics of discussion include potentially

• More linear dispersive estimates: Morawetz / local smoothing / local energy decay.
• Multilinear estimates in wave-Sobolev (\(H^{s,\delta}\)) or Bourgain (\(X^{s,b}\)) type spaces and applications to nonlinear equations.

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 and material

Course will be fully asynchronous. Course lectures will be delivered via pre-recorded FlipGrid videos. The course videos themselves will be made publicly accessible and links to them will be posted on this site.

Students are required to have a (free) account on Overleaf.

Office hours

If you have any questions, mathematical or otherwise, feel free to email me or schedule an appointment to chat; the scheduling page will automatically email you a Microsoft Teams meeting link.

Grading

This course will be graded on participation only. Students are asked to prepare lecture notes to accompany the posted lecture videos. I anticipate around ≈1800 minutes of videos being prepared for this course, based on which the following grading scale is devised:

• 4.0: prepares lecture notes corresponding to 290 minutes or more of lecture material.
• 3.5: prepares lecture notes corresponding to 225 minutes or more of lecture material.
• 3.0: prepares lecture notes corresponding to 130 minutes or more of lecture material.
• 2.5: prepares lecture notes corresponding to 20 minutes or more of lecture material.

Policy and Instructions

• Grades will be computed based on what is available on the course Overleaf project on April 21, 2021.
• A short video how-to
  
  General instructions:
  • Claim a lecture to work on by editing the corresponding \lecture{<topic>} stub to read \lecture[<your name>]{<topic>} in the main.tex file in the course overleaf project.
  • Create a new file <filename>.tex with unique filename.
  • Copy the contents of template.tex into the newly created file.
  • Insert \subfile{<filename>} after the claimed \lecture... line.
  • Edit <filename>.tex and prepare the notes; the file <filename>.tex will be compilable by itself, so you can work independently of other students.
• Ground Rules:
  • You may have no more than three claimed and unfinished lectures at any given time.
  • If you claimed a lecture and did not make any progress on its notes for over a week, I reserve the right to revert your claim and open the lecture up for other students.
  • Your grades depend on the total number of minutes covered by the lecture notes you chose to prepare. Please keep track of this yourself (perhaps in a spreadsheet).

NatSci Course Info Form

For Spring 2021 the College of Natural Science required a uniform-format overview of the course structure and policy be made available to students. Please find the form here.