Phan Thành Nam

Selected Topics in Nonlinear Schrödinger Equations (Winter 2026-2027)

General Information

Goal: We will discuss some selected advanced topics on nonlinear Schrödinger equations. In the first part, we focus on the existence of stationary solutions by variational techniques. In the second part, we consider the local and global well-posedness of the dynamical problem with initial data in the energy space. In the last part, we study the dynamical problem for irregular initial data using Gibbs measures. If time permits, we will also discuss the construction of Gibbs measures using an SPDE approach.

Audience : Master students of Mathematics and Physics, Master students of ‘Finanz und Versicherungsmathematik’, TMP Master. Other participants will get "Schein" if pass the course.

Please register via LSF.

Time and place:

• Lectures: Wednesday 10:15-12:00 (B251) and Friday 10:15-12:00 (B252).
• Exercises: Tuesday 10:15-12:00 (B251).

The first lecture takes place on Wednesday 16.10.2026.

References:

• P.-L. Lions: The concentration-compactness principle in the calculus of variations. The locally compact case, Part II. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 223–283.
• Thierry Cazenave: Semilinear Schrödinger Equations. AMS Courant Lecture Notes, Volume 10, 2003.
• Felipe Linares and Gustavo Ponce: Introduction to Nonlinear Dispersive Equations, Universitext, Springer, Second Edition 2014.
• Terence Tao: Nonlinear dispersive equations - local and global analysis. CBMS Regional Conference Series in Mathematics Volume 106, 2006. https://www.math.ucla.edu/~tao/preprints/chapter.pdf
• J. Bourgain. Periodic nonlinear Schrödinger equation and invariant measures. Comm. Math. Phys. 166 (1994), pp. 1-26.
• J. Bourgain. Invariant measures for the 2D-defocusing nonlinear Schrödinger equation. Comm. Math. Phys. 176 (1996), pp. 421–445.
• G. Da Prato and A. Debussche: Strong solutions to the stochastic quantization equations. Ann. Probab. 31 (2003), no. 4, 1900–1916.
• M. Gubinelli, Stochastic quantization of the Euclidean $\Phi^4_3$ quantum field theory. https://www.iam.uni-bonn.de/abteilung-gubinelli/sq-lectures-milan-ws2021