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)
      - Local & Global well-posedness in L2 / H1, 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

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.
  • 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.

Oral Exams: 31.01.2018 & 21.03.2018