Dispersive equations (Summer Semester 2022)
- Lecturer: JProf. Dr. Xian Liao
- Classes: Lecture (0170100), Problem class (0170110)
- Weekly hours: 3+1
At the beginning of the lecture course there will be an introduction part, where the basic concepts (such as dispersion, symmetries, solitons) and the motivations will be clarified.
| Schedule | ||
|---|---|---|
| Lecture: | Monday 9:45-11:15 | 20.30 SR 3.61 |
| Friday 14:00-15:30 (every 2nd week) | 20.30 SR 2.66 | |
| Problem class: | Friday 14:00-15:30 (every 2nd week) | 20.30 SR 2.66 |
| Lecturers | ||
|---|---|---|
| Lecturer | JProf. Dr. Xian Liao | |
| Office hours: by appointment | ||
| Room 3.027 Kollegiengebäude Mathematik (20.30) | ||
| Email: xian.liao@kit.edu | Problem classes | M. Sc. Julia Henninger |
| Office hours: by appointment | ||
| Room 3.038 Kollegiengebäude Mathematik (20.30) | ||
| Email: julia.henninger@kit.edu | ||
We are going to study the mathematical theory of the nonlinear Schrödinger equations (NLS) as follows.
Well-posedness issue of (NLS)
- Local & Global well-posedness, by use of Strichartz estimates & Sobolev embedding & conservation laws
Long time behaviour of the solutions of (NLS)
- Blowup & Scattering, by use of Virial & Morawetz idenities
Solitary waves of (NLS)
- Orbital stability, by use of variational description & concentration-compactness
Prerequisites:
Basic concepts from functional analysis, e.g. Lebesgue spaces, Sobolev spaces, Fourier transform, Hölder's inequality, Young's inequality, convolution.
Lecture Notes:
Lecture Notes, version 18.07.2022
Exercise sheets:
sheet 1, to be explained on 29.04.2022
sheet 2, to be explained on 13.05.2022
sheet 3, to be explained on 27.05.2022
sheet 4, to be explained on Monday, 20.06.22 at 11:30 in seminar room -1.008
sheet 5, to be explained on Monday, 04.07.22 at 11:30 in seminar room -1.008
sheet 6, to be explained on 15.07.2022
sheet 7, to be explained on Monday, 25.07.22 at 11:30 in seminar room -1.008
Examination
Oral exam.
References
T. Cazenave: Semilinear Schrödinger equations.
F. Linares, G. Ponce: Introduction to nonlinear dispersive equations.