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Announcement for Seminar "Free Interface Problem", Winter Semester 2024/2025 (2SWS, 3LP)

Instructor: JProf. Dr. Xian Liao (Xian.liao@kit.edu)

Assistant: M.Sc. Rebekka Zimmermann (Rebekka.Zimmermann@kit.edu)

Time & Place: 09:45-11:15, Tuesday, SR -1.017 (UG)

Preliminary meeting: 13.15pm, Tuesday July 16th, 2024, in SR 3.068 (Mathematical Building).

Time schedule: November 19, 26 & December 3, 10, 17 in 2024 (if there are more participants, there will be time slots in January 2025)

Each participant is supposed to present one topic (around 75min. + 15min. questions/discussions), and to participate actively in other presentations.
Please contact Ms. Zimmermann four weeks before your presentation.
It is recommended to prepare handouts and distribute them before your presentation.

Free interfaces between two different phases are commonly observed in everyday life. Mathematically, the free interface problem consists of one or more partial differential equations as well as an unknown, moving boundary which is to be determined.

Prüss and Simonett's book is an excellent reference for the related mathematical theory. In this seminar, we will primarily focus on the modeling and problem statements in Chapter 1 of the reference. If there are more participants, we will delve deeper into two-phase Navier-Stokes problems.

List of Topics:

(19.11.24) First principles in the bulk. (A, Section 1.1, Subsection 1.1)
(26.11.24) First principles on the interface and the dynamic problem. (A, Section 1.1, Subsections 1.2, 1.3, 1.4)
(03.12.24) Entropy and equilibria. (A, Section 1.2, Subsections 2.1, 2.2, 2.3, 2.4)
(10.12.24) The model problems. (A, Section1.3, Subsection 3.1)
(17.12.24) Transformation to a fixed domain. (A, Section1.3, Subsection 3.2)
Possible topics in January 2025: Maximal Lp-regularity, linear theory, Fixed point argument for nonlinear problem.

Analysis 1-3 & Basics of PDEs, or Höhere Mathematik 1-3. This seminar is also suitable for students from other departments, for example, ETIT, Physics.

A.   J.Prüss and G. Simonett, "Moving Interfaces and Quasilinear Parabolic Evolution Equations", 2016, Springer. https://link.springer.com/book/10.1007/978-3-319-27698-4