Weak-strong uniqueness for volume-preserving mean curvature flow

TL;DR

This paper establishes a stability and weak-strong uniqueness principle for volume-preserving mean curvature flow, introducing a new calibration concept and providing stability estimates for distributional solutions.

Contribution

It introduces volume-preserving gradient flow calibrations and proves weak-strong uniqueness and stability results for the flow.

Findings

Strong solutions are calibrated under certain regularity.

A stability estimate in terms of relative entropy is established.

The results apply to distributional solutions of the flow.

Abstract

In this note, we derive a stability and weak-strong uniqueness principle for volume-preserving mean curvature flow. The proof is based on a new notion of volume-preserving gradient flow calibrations, which is a natural extension of the concept in the case without volume preservation recently introduced by Fischer et al. [arXiv:2003.05478]. The first main result shows that any strong solution with certain regularity is calibrated. The second main result consists of a stability estimate in terms of a relative entropy, which is valid in the class of distributional solutions to volume-preserving mean curvature flow.