Tim Laux | Teaching
The gradient flow structure of mean curvature flow
Abstract: Mean curvature flow is the most basic geometric evolution equation for embedded surfaces. Folklore says that it can be viewed as a gradient flow. This course aims at making this statement more precise and harnessing this structure for rigorous analysis.
After a brief introduction to the field, I will present some of the ideas behind the (conditional) existence and (weak-strong) uniqueness theory for solutions to mean curvature flow. Focusing on the simple two-phase case, i.e., the evolution of a closed hypersurface, allows for a self-contained and concise presentation, which is accessible for graduate students (master or PhD) with some background in PDEs and functional analysis.
The course is structured into four lectures as follows. The first lecture provides an overview, basic examples, and some motivation from numerics and data science. In the second lecture, I'll discuss different weak solution concepts, present a (conditional) closure theorem and relate different solution concepts. The third lecture is devoted to the weak-strong uniqueness principle via a concept of gradient flow calibrations. In the last lecture, I'll show how to use this structure for proving optimal convergence rates for numerical schemes.
Simulation: A solution to the Allen-Cahn equation (with periodic boundary conditions) starting from random initial data. Very quickly, diffuse interfaces form, which then move approximately by mean curvature flow.
I'll post the lecture notes and additional material here for the summer course at Heidelberg.
Here's a link to some typed notes: arxiv.org/abs/2108.08347 There will be some overlap but some parts (for example the solution concept) will be generalized in this course.
