Distributional solutions to mean curvature flow

TL;DR

This paper introduces distributional solutions to mean curvature flow, discussing recent advances in existence and uniqueness theories for evolving hypersurfaces, suitable for graduate students with basic PDE and measure theory knowledge.

Contribution

It provides a self-contained, accessible overview of distributional solutions and recent theoretical developments in mean curvature flow for beginners.

Findings

Recent conditional existence results

Weak-strong uniqueness principles

Simplified case of single hypersurface evolution

Abstract

These lecture notes aim to present some of the ideas behind the recent (conditional) existence and (weak-strong) uniqueness theory for mean curvature flow. Focusing on the simplest case of the evolution of a single closed hypersurface allows for a self-contained and concise presentation, which is accessible for beginning graduate students with some background in PDEs and only requires basic measure theory.