Dispersive equations (Winter Semester 2018/19)
- Lecturer: JProf. Dr. Xian Liao
- Classes: Lecture (01053400), Problem class (01053410)
- Weekly hours: 3+1
|Lecture:||Friday 11:30-13:00||SR 2.066|
|Wednesday 11:30-13:00 (every 2nd week)||SR 2.059 Beginn 17.10.2018|
|Problem class:||Wednesday 11:30-13:00 (every 2nd week)||SR 2.059 Beginn 24.10.2018|
|Problem classes||M.Sc. Zihui He|
|Office hours: by appointment|
|Room 3.030 Kollegiengebäude Mathematik (20.30)|
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
Conserved energies for one dimensional cubic (NLS)
- Conserved energies, by use of invariant transmission coefficient
In 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.
Basic concepts from functional analysis, e.g. Lebesgue spaces, Sobolev spaces, Fourier transform, Hölder's inequality, Young's inequality, convolution.
Lecture Notes, February 01, 2019
Exercise sheet 1, to be explained on October 24, 2018
Exercise sheet 2, to be explained on November 07, 2018
Exercise sheet 3, to be explained on November 21, 2018
Exercise sheet 4, to be explained on December 05, 2018 & January 16, 2019
Exercise sheet 5, to be explained on December 19, 2018 & January 16, 2019
Exercise sheet 6, to be explained on January 30, 2019
Exercise sheet 7, to be explained on February 06, 2019
Exercise sheet 8, to be explained on February 08, 2019
T. Cazenave: Semilinear Schrödinger equations.
F. Linares, G. Ponce: Introduction to nonlinear dispersive equations.
T. Tao: Nonlinear dispersive equations - local and global analysis.
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T.Tao: The theory of nonlinear Schödinger equations.
H. Koch, D. Tataru: Conserved energies for the cubic NLS in 1-d.