Course Information:
Lectures: TuTh 2:00-3:15pm
Classroom: Phillips 301
Instructor: Jason Metcalfe
E-mail: metcalfe@email.unc.edu
Office: Phillips 324K
Office Hours: TuTh 1:00-2:00pm and by appointment.
Course Webpage: http://www.unc.edu/~metcalfe/teaching/math891s09/
Textbook: The lectures will be drawn from a number of resources. Here are some recommended reads:
- S. Selberg's Lecture notes on nonlinear wave equations
- C. Sogge, Lectures on nonlinear wave equations.
- L. Hormander, Lectures on nonliner hyperbolic equations
- J. Shatah and M. Struwe, Geometric wave equations
- W. Strauss, Nonlinear wave equations
- M. Ikawa, Hyperbolic partial differential equations and wave equations
- T. Tao, Nonlinear dispersive equations
Final Project: There will be no collected assignments or exams for this course. The final grade for the course will be based on class participation and a final project. The final project will be a 5-7 page essay on a more advanced topic related to the course material. The project is due on the last day of lecture, Thursday April 23.
Some ideas for final projects:
- Improved local existence results using, e.g., Strichartz estimates
- The counterexamples of John and Sideris that show finite time blow up in n=3
- Localized energy estimates on black hole backgrounds
- Morawetz's conformal local energy estimate
- Exponential decay of local energy in nontrapping exterior domains
- Quasilinear wave equations in exterior domains
- Quasilinear wave equations in waveguides
- The weak null condition
- A more detailed exposition on long-time existence when the nonlinearity depends on the solution not just its derivatives
- Christodoulou's method for proving global existence with the null condition
- Long time existence with a null condition in n=2