Conference Agenda

That the signal is consumed instead of produced in consumption type variants of chemotaxis systems (as opposed to the classical Keller-Segel system, for instance) does not immediately translate into easy boundedness information for the construction of classical or weak solutions.
In this talk, we will recall some analysis of chemotaxis-consumption systems and in particular discuss recent improvements regarding their (weak) solvability.

Free boundary problems play a central role in modeling physical phenomena such as multiphase flows and the evolution of fluid patches. Examples include the motion of droplets, oil recovery processes and vortex dynamics. In this talk, we will discuss mathematical models of interface evolution in fluid dynamics and present analytical techniques for studying specific questions about the dynamics of free boundaries, such as its regularity, long-time behavior, and the formation of singularities.

We propose a new method for approximating mean curvature flow with obstacles. The scheme is an adaptation of the celebrated thresholding method of Merriman, Bence, and Osher (MBO). Despite its computational efficiency, the proposed approach is unconditionally stable, monotone, and enforces the obstacle constraints exactly.

The main focus of this talk is the convergence analysis of the scheme. We prove convergence to viscosity solutions of mean curvature flow with obstacles using a comparison-principle-based framework. In addition, we discuss suitable spatial discretizations and investigate their convergence behavior in practice.

The proposed approach naturally extends to several important variants of the MBO scheme, including volume-preserving, anisotropic, and multiphase formulations. While these extensions appear promising from a computational perspective, their rigorous convergence analysis remains open. This raises a number of interesting questions and directions for future research.

This is joint work with Tim Laux.