Fourier Analysis and its applications to PDE (Winter Semester 2021/22)
- Lecturer: JProf. Dr. Xian Liao
- Classes: Lecture (0106200), Problem class (0106210)
- Weekly hours: 3+1
In this course we will introduce the theory of Fourier analysis and then show its applications to some typical PDEs. The Fourier transform and the Littlewood-Paley decomposition techniques have shown their efficiency in the study of evolutionary equations. As examples we will study the transport-diffusion equations, Navier-Stokes equations and dispersive equations.
| Schedule | ||
|---|---|---|
| Lecture: | Wednesday 10:00-11:30 | 20.30 SR 2.66 |
| Friday 12:00-13:30 (every 2nd week) | 20.30 SR 3.61 | |
| Problem class: | Friday 12:00-13:30 (every 2nd week) | 20.30 SR 3.61 |
| Lecturers | ||
|---|---|---|
| Lecturer | JProf. Dr. Xian Liao | |
| Office hours: by appointment | ||
| Room 3.027 Kollegiengebäude Mathematik (20.30) | ||
| Email: xian.liao@kit.edu | Problem classes | M.Sc. Zihui He |
| Office hours: by appointment | ||
| Room 3.030 Kollegiengebäude Mathematik (20.30) | ||
| Email: zihui.he@kit.edu | ||
The Fourier analysis theory will include the following concepts:
-Fourier transform and Schwartz space, tempered distribution space
-Bernstein's inequality, Van der Corput's Lemma
-Littlewood-Paley decomposition
-Besov spaces and Sobolev spaces
-Paradifferential calculus
-Commutator estimates
The following types of partial differential equations will be discussed:
-Transport equations
-Navier-Stokes equations
Prerequisites:
Basic concepts from functional analysis and real analysis, e.g. Lebesgue spaces, Hölder's inequality, Young's inequality, convolution.
Lecture Notes:
Lecture Notes, version 11.02.2022
Exercise sheets:
sheet 1, to be explained on 29.10.2021
sheet 2, to be explained on 12.11.2021
sheet 3, to be explained on 26.11.2021
sheet 4, to be explained on 10.12.2021
sheet 5, to be explained on 07.01.2022
sheet 6, to be explained on 21.01.2022
sheet 7, to be explained on 04.02.2022
Examination
Oral exam on 24.02 or 08.04. Registration please send an email to zihui.he@kit.edu (until 01 February).
References
H. Bahouri, J.-Y. Chemin and R. Danchin. Fourier analysis and nonlinear partial differential equations. Springer. 2011.