Mathematical Institute of the University of Bonn

V5B8 - Selected Topics in Analysis - The vector field method and quasilinear wave equations

Summer Semester 2022

Dr. Dongxiao Yu

Instructor

Lectures

• Tue 12-14, Room 1.007, Endenicher Allee 60. See here for the current COVID policy for the Mathematics Center.
• Lectures may be held on Zoom occasionally.

Office hours

Tue 14-15, Room 2.005, Endenicher Allee 60. Or by appointment.

Topics

In this course we will study the lifespans of solutions to some special types of quasilinear wave equations with small, smooth and localized initial data. Using the so-called ''vector field method'', we will prove several global or almost global existence results for quasilinear wave equations.

The prerequisites include basic real analysis and basic knowledge about partial differential equations.

Here is a tentative schedule.

• A review on the linear wave equation. (1.5 weeks)
• Energy estimates. (1.5 weeks)
• Local wellposedness and blowup criteria. (1 week)
• Commuting vector fields and some related inequalities. (2 weeks)
• An almost global existence result. (2 weeks)
• The null condition and a global existence result. (4 weeks)

Literature

Here are the lecture notes.

The following four books are recommended but not required:

• Christopher Sogge, Lectures on Non-Linear Wave Equations (Second Edition).
• Lars Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations.
• Serge Alinhac, Geometric Analysis of Hyperbolic Differential Equations: An Introduction.
• Serge Alinhac, Hyperbolic Partial Differential Equations.

My plan is to follow Chapter I and II in Sogge’s book and use the other three books as complements.

Exam

There will be an oral exam at the end of the semester. Further details will be forthcoming.

Important links

Basis page

Ecampus page