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Nonlinear Maxwell Equations (Winter Semester 2024/25)

  • Lecturer: Prof. Dr. Roland Schnaubelt
  • Classes: Lecture (0105360), Problem class (0105370)
  • Weekly hours: 4+2
  • Audience: Mathematics (from 7. semester)

All information and materials for this lecture are provided in the course "Nonlinear Maxwell Equations" within Ilias, including communication via email and forum. Please join this course if you want to participate.

Schedule
Lecture: Monday 9:45-11:15 SR 2.067 Begin: 21.10.2024
Tuesday 9:45-11:15 SR 2.067
Problem class: Thursday 15:45-17:15 SR 3.069 Begin: 24.10.2024
Lecturers
Lecturer Prof. Dr. Roland Schnaubelt
Office hours: Tuesday at 12:00 - 13:00, and by appointment.
Room 2-047 (Englerstr. 2) Kollegiengebäude Mathematik (20.30)
Email: schnaubelt@kit.edu
Problem classes Richard Nutt M.Sc.
Office hours: Just drop by whenever I am in the office.
Room 2.043 Kollegiengebäude Mathematik (20.30)
Email: richard.nutt@kit.edu

The Maxwell equations are the fundamental laws governing electro-magnetic phenomena, and they are important building blocks for various coupled systems. Constitutive relations describe the interaction between the fields and the material by determining the polarization or magnetization. We study nonlinear relations focusing on an instantaneous material response and the full space case. The Maxwell system then becomes a quasilinear hyperbolic system. We first establish local wellposedness for initial values in H^3 by linearization and energy methods. Further blow-up phenomena despite energy preservation will be treated. In the second main part we investigate dispersive behavior in terms of Strichartz inequalities and improve the wellposedness theory. These theorems were shown only very recently. We also intend to give an outlook to further recent results on the longtime behavior and problems on domains. The methods can also be applied to the simpler, scalar (quasilinear) wave equation.

The lectures requires knowledge in functional analysis and basics of Sobolev spaces. Further tools (mainly from harmonic analysis) will be discussed in the lectures, partly without giving complete proofs.

Examination

There is an oral exam of about 30 min (for 8 ECTS points).

References

  • S. Benzoni-Gavage and D. Serre: Multidimensional Hyperbolic Partial Differential Equations.