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Fourier Analysis and its applications to PDE (Winter Semester 2021/22)

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
-Littlewood-Paley decomposition
-Besov spaces and Sobolev spaces
-Paradifferential calculus

The following types of partial differential equations will be discussed:
-Transport equations
-Navier-Stokes equations
-Dispersive 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 08.12.2021

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

Examination

Oral exam.

References

H. Bahouri, J.-Y. Chemin and R. Danchin. Fourier analysis and nonlinear partial differential equations. Springer. 2011.