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Partial Differential Equations II (Summer 2022)

Phan Thành Nam, Arnaud Triay, Uni2work, Homework

General Information

Goal: We will discuss nonlinear Schrödinger equations. In the first part, we focus on the existence of stationary solutions by variational techniques. In the second part, we focus on the time-dependent dispersive equations where the well-posedness, blowup, and scattering theory will be analyzed.

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 Uni2work.

Time and place: All of the meetings will be held in-person at Theresienstr. 39, unless otherwise notified.
  • Lectures: Tuesday 14:15-16:00 (room B 041) and Wednesday 10:15-12:00 (room B 134).
  • Exercises: Monday 14:15-16:00 (room B 251).
The first lecture takes place on Tuesday 26.04.2022, from 14:15 at B 041. The first exercise section takes place on Monday 02.05.2022.

References:
  • Lawrence C. Evans: Partial Differential Equations (Second Edition). AMS Graduate Studies in Mathematics, Volume 19, 2010.
  • 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
  • Elliott Lieb and Michael Loss: Analysis (Second Edition). AMS Graduate Studies in Mathematics, Volume 14, 2001.
Grade: Final grade is determined by your total performance:
  • You can get up to 100 points in the final exam.
  • You can get up to 10 bonus points in the midterm exam.
  • You can get up to 1 bonus point per week for solving exercises.
You need 50 points to pass the course and 85 points to get the final grade 1.0.

Exams: We will have exams on June 15 (midterm) and July 25 (final). You can use your notes (lecture notes, homework sheets and tutorial materials) in the exams.

Contents of the lectures

26 & 27.4. Introduction. Chapter 1: Sobolev inequalities. Lp spaces. Distributions. Sobolev spaces. Uncertainty principles.

3 & 4.5. Sobolev inequalities. Linear Schroedinger equation.

11 & 17. 5. Chapter 2: Nonlinear Schroedinger equation (stationary problem). Concentration compactness method.

18 & 24. 5. Binding vs. strict binding inequality. Existence of minimizers. Translation-invariant case.

25 & 31.5. Chapter 3: Optimizer of Sobolev inequalities. Sub-critical case. Critical case. Concentration compactness method for measures. Lions' proof of the existence. Another proof of the existence via Gérard-Meyer-Oru inequality.

1 & 8.6. Computation of the Sobolev optimizers. Notes. Zoom video. Lecture Notes of Rupert Frank.

21 & 22.6. Chapter 4: Rearrangement techniques. The simplest (Hardy–Littlewood) rearrangment inequality. Riesz rearrangement inequality. Pólya–Szegő inequality. Radial symmetry of the Sobolev optimizers. Lecture Notes of Almut Burchard.

28 & 29. 6. The method of moving planes. Our discussion is based on Chen-Li-Ou's paper. For another approach, see Lecture Notes of Rupert Frank.

5.7. Uniqueness of radial solutions in the energy sub-critical case.

6.7. Chapter 5: Time-dependent nonlinear Schrödinger equations. Abstract global wellposedness in the locally Lipschitz case.

12 & 13. 6. Local existence of the NLS in the energy space. Uniqueness in 1D. Strichartz estimates. Uniqueness in higher dimensions.

19 & 20. 6. Virial identity. Finite time blow-up solutions in the focusing case. Global wellposedness, dispersive estimate and scattering theory in the defocusing case.