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Dispersive equations (Wintersemester 2018/19)

  • Dozent*in: JProf. Dr. Xian Liao
  • Veranstaltungen: Vorlesung (01053400), Übung (01053410)
  • Semesterwochenstunden: 3+1
Vorlesung: Freitag 11:30-13:00 SR 2.066
Mittwoch 11:30-13:00 (14-tägig) SR 2.059 Beginn 17.10.2018
Übung: Mittwoch 11:30-13:00 (14-tägig) SR 2.059 Beginn 24.10.2018
Übungsleiterin M.Sc. Zihui He
Sprechstunde: by appointment
Zimmer 3.030 Kollegiengebäude Mathematik (20.30)
Email: zihui.he@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

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:
Lecture Notes, February 01, 2019

Exercise Sheets:
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.