V5B8 - Selected Topics in Analysis - The vector field method and quasilinear wave equations
Summer Semester 2022
- Dr. Dongxiao Yu
- 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.
Tue 14-15, Room 2.005, Endenicher Allee 60. Or by appointment.
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)
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.
There will be an oral exam at the end of the semester. Further details will be forthcoming.