Teaching

Kind:	Place:
Lecture 3 hrs.	online

This is a joint course with Ulisse Stefanelli, who will lecture about Gradient flows, curves of maximal slope, rate-independent flows and Luca Scarpa, who will lecture about Semilinear parabolic equations. I will cover some topics about the Nonlinear Schrödinger equation.

As announced I would like to present some results about local/global existence for the Nonlinear Schrödinger (NLS) equation i u_t + Δ u = ± |u|^α-1 u,   u(0)=g My course is divided into three parts. The proper setting for such type of semilinear evolutions equations are operator semigroups. With these tools local existence of solutions with sufficiently smooth initial conditions can be established easily. More advanced results for L² and H¹ initial data require more advanced tools from Harmonic Analysis, namely Strichartz estimates. The proper formulation requires the use of Bochner-Lebesgue spaces.

Since neither the Bochner integral nor operator semigroups are part of the standard courses, these will constitute the first two parts, while the last part will be the applications to the NLS equation. For the Bochner integral and Lebesgue-Bochner spaces I ask you to read Sections 5.5 and 5.6 from my notes "Topics in Real Analysis". While far from complete, this should still give you a good working knowledge. For operator semigroups I ask you to read Chapter 8 from my notes "Topics in Linear and Nonlinear Functional Analysis". Again this only covers some of the most basic results (and in fact much more than we will need, since the linear Schrödinger equation is easily understood with the help of the Fourier transform), but again it should get you started in this area and set the stage for more fur further study. The applications to the NLS equation will be in Chapter 9.

Of course your feedback on my notes (from typographical errors to mathematical mistakes, suggestions for simplifications or requests for clarifications if some parts are unclear or hard to follow) is more than welcome! I am very much looking forward to comments!

Take home exam.

My lecture notes:

1. G. Teschl, Topics in Linear and Nonlinear Functional Analysis, Lecture Notes.
2. G. Teschl, Topics in Real Analysis, Lecture Notes.