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Seminar (Water Waves Problem) (Summer Semester 2024)

Preliminary meeting: '''13.15pm, Thursday February 8th''', 2024, in SR 3.068 (Mathematical Building).

Time plan
May 8 & 15, June 5, 12, 19, 26 (if there are more participants, there will be time slots in July)

Seminar: Wednesday 9:45-11:15 20.30 -01.17 Begin: 8.5.2024
Lecturer JProf. Dr. Xian Liao
Office hours: by appointment
Room 3.027 Kollegiengebäude Mathematik (20.30)
Email: xian.liao@kit.edu
Lecturer M. Sc. Rebekka Zimmermann
Office hours: by appointment
Room 3.030 Kollegiengebäude Mathematik (20.30)
Email: rebekka.zimmermann@kit.edu

Water waves can be observed almost everywhere in our daily life, and water waves problem has been extensively studied by scientists, engineers and also mathematicians. However, the mathematical theory is far from satisfactory due to the complexity of the models, e.g. the presence of the free boundary.

In this seminar we focus instead on the mathematical formulation and modelling of relevant models in water waves problem. While very elementary analysis techniques are required, we can derive (at least formally) very interesting prototypical models such as Korteweg-de Vries equation, Camassa-Holm equations, Davey-Stewartson model, etc. from free surface Euler equations. We will follow David Lannes' book (namely Chapters 1, 5, 7-8) for the derivation of the approximation models, and skip the rigorous mathematical results (as presented in e.g. Chapters 2-4 in the book).

This seminar is also suitable for students from other departments, for example, ETIT, Physics. Some lecture notes for mechanical engineers can be found below.

Analysis 1-3 & Basics of PDEs, or Höhere Mathematik 1-3.

A. Techet (Dept. Mechanical Engineering), Handout Free-Surface Waves
B. MIT-OpenCourseWare: Marine Hydrodynamics: Water Waves.
C. Lannes, The Water Waves Problem - Mathematical Analysis and Asymptotics, 2013, AMS. (Please let us know if you need the .pdf file.)

Each participant is supposed to present one topic (around 75min. + 15min. questions/discussions), and to participate actively in other presentations. Please contact your "Betreuer-in" four weeks before your presentation.

List of Topics (Topics 1-2 are overviews.)

  1. (08.05.24, Betreuerin: Rebekka Zimmermann, Reference A: Pages 1-27) Modelling of the free surface water waves problems.
  2. (15.05.24, Betreuer: Robert Wegner, Reference B) Shallow water waves, intermediate depth and deep water waves.
  3. (05.06.24, Betreuerin: Rebekka Zimmermann, Reference C: Pages 1-5) Formulations for free surface Euler equations.
  4. (12.06.24, Betreuerin: Rebekka Zimmermann, Reference C: Pages 5-8 & Page 12) Lagrangian formulation for free surface problems.
  5. (19.06.24, Betreuerin: Rebekka Zimmermann, Reference C, Section 1.3: Pages 13-20) Nondimensionalization of the equations.
  6. (26.06.24, Betreuer: Robert Wegner, Reference C, Section 5.1: Pages 121-130) Derivation of shallow water system.
  7. Contents in July 2024 (Reference C, without rigorous proofs) Section 5.2: Boussinesq equations; Section 7.1: KdV equation; Section 7.3: Camassa-Holm equation; Section 8.1-8.2: Deep-water models