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Workgroup Applied Analysis

Secretariat
Kollegiengebäude Mathematik (20.30)
Room 2.029

Address
Englerstrasse 2
76131 Karlsruhe
Germany

Dr. Kaori Nagato-Plum
kaori.nagatou@kit.edu




Office hours:

Tel.: +49 721 608 42056

Fax.: +49 721 608 46214

Nonlinear Wave Equations (Summer Semester 2020)

Lecturer: Dr. Birgit Schörkhuber
Classes: Lecture (0156500), Problem class (0156510)
Weekly hours: 2+1


Wave equations are ubiquitous and play a fundamental role in physics and applications. In many models, nonlinearities appear naturally and introduce highly non-trivial dynamics.
The aim of this course is to give an introduction into the mathematical analysis of nonlinear wave equations. Central questions concern the existence of solutions to the initial value problem, the description of long-time behaviour, as well as the formation of singularities. Starting with a review of the linear wave equation, we will investigate nonlinear problems by means of energy estimates, tools from harmonic analysis (Strichartz type estimates) and geometric methods (vectorfield methods). The course will be held in English.

Important note: In view of the current Corona situation, the lecture starts as an online course. Please register for it in ILIAS. If you have problems with the registration, please contact me: birgit.schoerkhuber@kit.edu.

Schedule
Lecture: Friday 11:30-13:00 SR -1.013 (UG)
Problem class: Thursday 15:45-17:15 SR -1.008 (UG)
Lecturers
Lecturer, Problem classes Dr. Birgit Schörkhuber
Office hours: upon request
Room 2.023 Kollegiengebäude Mathematik (20.30)
Email: birgit.schoerkhuber@kit.edu

The course will cover the following topics:

  • The basics: The initial value problem for linear wave equation (classical solutions, weak solutions, L^2 theory, energy estimates, ...)
  • Examples of nonlinear wave equations
  • Local well-posedness theory at high regularities
  • Local well-posedness for rough initial data - Strichartz estimates
  • Formulation of singularities in finite time
  • Global well-posedness (conserved quantities, vectorfield methods)

Requirements: Functional analysis

Depending on the background of the participants, specific tools and prerequisites (such as the theory of distributions, Fourier transform, Sobolev spaces, ...) will be reviewed and discussed in the problem class on Thursdays.

Dates and course modalities:

The course starts online. Course material will be provided through the ILIAS platform.

Examination

There will be an oral examination (about 20 min) at the end of the semester.

References

  • C. Sogge, Lectures on Nonlinear Wave Equations
  • T. Tao, Nonlinear dispersive equations: local and global analysis
  • J. Shatah, M. Struwe: Geometric wave equations