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Partial Differential Equations 2 (SoSe 25)

Lectures: A. Triay, Tutorials and exercises: L. Read, L. Meng
Registration via LSF (if you cannot register on LSF, you can do it on Moodle, key: "pde2")

Description:

In this course we will mainly focus on the study of the non-linear Schrödinger equation \[ i\partial_t u = (-\Delta \pm \lambda |u|^{\alpha -1 }) u. \] for \(u : (-T,T) \times \mathbb{R}^d \to \mathbb{C}, \alpha >0, \lambda \in \mathbb{R} \) and initial value \(u(t) = u_0 : \mathbb{R}^d\to \mathbb{C}\) given.
This equation models the propagation of waves in different areas of physics, and in particular in quantum mechanics where it models the time evolution of Bose-Einstein condensates. We will mainly follow the book of Linares and Ponce (see reference below).


Credits:
9 (6+3) ECTS.

Audience:

Master students of Mathematics (WP34), Master students of `Finanz- und Versicherungsmathematik' (WP12), TMP-Master, motivated bachelor students (they will get "Schein" when they pass the course).
The course is (to some extent) self-contained and does not require to have followed PDE1. However a good training in analysis is expected.

Schedule:

- Lectures: Tuesday 10:00-12:00 and Thursday 14:00-16:00, Room: B046
- Tutorium: Monday 16:00-18:00 Room B134, and Exercises: Thursday 16:00-18:00, Room B046
- First lecture Thursday 24 April, execise session and and tutorial begin the next week.

References:
  • Felipe Linares and Gustavo Ponce: Introduction to Nonlinear Dispersive Equations, Universitext, Springer, Second Edition 2014.
  • Lawrence C. Evans: Partial Differential Equations (Second Edition). AMS Graduate Studies in Mathematics, Volume 19, 2010.
  • Brezis: Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext 2011
  • Lieb-Loss: Analysis, Amer. Math. Soc. 2001.