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Workgroup Nonlinear Partial Differential Equations

Secretariat
Kollegiengebäude Mathematik (20.30)
Room 3.029

Address
Karlsruher Institut für Technologie
Institut für Analysis
Englerstraße 2
76131 Karlsruhe
Germany

Office hours:
Mon-Fri 10-12, Tue+Wed 14-16

Tel.: +49 721 608 42064

Fax.: +49 721 608 46530

Dispersive equations (Winter Semester 2018/19)

Lecturer: JProf. Dr. Xian Liao
Classes: Lecture (01053400), Problem class (01053410)
Weekly hours: 3+1


Schedule
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
Lecturers
Problem classes M.Sc. Zihui He
Office hours: by appointment
Room 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.



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, 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

References

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.