Vorlesung: Partielle Differentialgleichungen II (PDG2) (SoSe 2019)

Lecture (Vorlesung):
Tue 14--16 & Wed 14--16 (in A 027).

Exercises (Ãœbungen):
See separate webpage.

Tutorials (Tutorien):
See separate webpage.

Synopsis (Kurzbeschreibung):
In this course we will study time-dependent dispersive Partial Differential Equations (PDEs), more explicitly the linear and nonlinear Schroedinger equation. This equation has applications in Physics (nonlinear optics, laser physics, Bose-Einstein condensation), but is also mathematically intrinsically interesting.
The course will start by developing certain tools in Analysis, needed to study the PDE, and interesting in their own right. (Students having seen some of these in recent courses should be aware that the presentation might be sligtly different in this course)
Keywords: Fourier transform, tempered distributions, oscillatory integrals, interpolation theorems (Riesz-Thorin, Marcinkiewicz, Stein), functional inequalities (Young, Hausdorff-Young, Hardy-Littlewood etc), Hardy-Littlewood Maximal Function, Sobolev spaces and pseudodifferential operators (crash course!).
For the PDE-part, some keywords are: Global and local smoothing effect, local well-posedness of the initial value problem (IVP) (L^2, H^1, and H^2 - theory).

Audience (Hörerkreis):
Master students of Mathematics (WP 40), Master students of `Finanz- und Versicherungsmathematik' (WP 27), TMP-Master.

Credits:
9 (6+3) ECTS.

Prerequisites (Vorkenntnisse):
Analysis I-III, Functional Analysis (in particular, Integration Theory and L^p-spaces), some Complex Analysis.
Note: PDE 1 is not a prerequisite: This course does not build on PDE 1 from last semester.
Here you find a handout with the needed facts (without proofs): Version 0.5 (06 May).

Language (Sprache):
English. (Die mündliche Prüfung kan auch auf Deutsch gemacht werden).

Exam (Prüfung):
There will be an oral exam (Es wird eine mündliche Prüfung geben). See separate webpage.

Content (Inhalt):

0. Introduction and motivation

1. Fourier transform

   1.1 Fourier transform on L^1(R^d)
   1.2 Fourier transform on L^2(R^d)
   1.3 Tempered distributions
   

2. Interpolation of operators

   2.1 Riesz-Thorin Convexity Theorem
   2.2 Marcinkiewicz Interpolation Theorem (Diagonal Case)
   2.3 The Stein Interpolation Theorem
   2.4 The Mikhlin-Hörmander Multiplier Theorem
   

3. Sobolev Spaces and Pseudo-Differential Operators

   3.1 Sobolev Spaces
   3.2 Pseudodifferential Operators
   

4. The Linear Schrödinger Equation

   4.1 Basic Results
   4.2 Global Smoothing Effects
   4.3 Local Smoothing Effects
   

5. The Nonlinear Schrödinger Equation: Local theory

   5.0 Preliminaries
   5.1 L^2 theory
   5.2 H^1 theory
   5.3 H^2 theory
   

6. Asymptotic behaviour of solutions (outlook)

   6.1 Global results
   6.2 Formation of singularities
   

Literature (Literatur):
There will be no lecture notes. Above you will find a short description of the content of the lecture (to be updated as we go along). The lecture will mainly follow the book by Linares and Ponce mentioned below.
(Es wird kein Skript geben. Hier wird laufend eine Kurzübersicht der Vorlesung erstellt. Die Vorlesung wird größtenteils auf folgendes Buch basieren:)

[LP] F. Linares, G. Ponce, Introduction to Nonlinear Dispersive Equations, 2nd edition, Springer (Universitext), 2015. (Login with your Campus-account.)

Supplementary literatur (Ergänzende Literatur):

• J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, AMS (1999).
• T. Cazenave, Semilinear Schrödinger Equations, AMS (2003).
• C. E. Kenig, Lectures on the Energy Critical Nonlinear Wave Equation, AMS (2015).
• W. A. Strauss, Nonlinear Wave Equations, AMS (1989).
• C. Sulem, P.-L. Sulem, The nonlinear Schrödinger equation, Springer (1999).
• T. Tao, Nonlinear Dispersive Equations, AMS (2006).

Office hours (Sprechstunde):
Wednesday 10:15-11:00 (Room B 408) or by appointment via email.


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Letzte Änderung: 23 September 2019 (No more updates)

Thomas Østergaard Sørensen