V5B4 Selected Topics in PDE and Mathematical Models - Dispersive Equations

Wintersemester 2017/2018



Instructor: Dr. Xian Liao

Lectures: Friday, 10 (c.t.) - 12, SR 1.007

Topics: We will focus on the mathematical theory of nonlinear Schrödinger equations (NLS)

  • Well-posedness issue of (NLS)
  • Asymptotic behaviour of the solutions of (NLS)
  • Conserved energies for one dimensional (NLS)
  • Well-posedness issue and stability of solitons of Korteweg-de Vries equation (KdV)

Prerequisites: Basic concepts from functional analysis and PDEs, e.g. Lebesgue spaces, Sobolev spaces, Fourier analysis, Hölder's inequality, Young's inequality, convolution.

Notes

References:

  • T. Cazenave: Semilinear Schrödinger equations.
  • F. Linares, G. Ponce: Introduction to nonlinear dispersive equations.
  • T. Tao: Nonlinear dispersive equations - local and global analysis. First three chapters
  • J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T.Tao: The theory of nonlinear Schödinger equations. Lecture notes
  • H. Koch, D. Tataru: Conserved energies for the cubic NLS in 1-d. Article

Oral Exams:

    Feb. 5~6 & March 13~14 (expected)