Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow

TL;DR

This paper introduces a new Brakke inequality tailored for volume preserving mean curvature flow, demonstrating the global existence of solutions via phase field methods and extending prior results in the field.

Contribution

It proposes a novel Brakke inequality for volume preserving mean curvature flow and establishes the existence of solutions using phase field techniques, extending previous work.

Findings

Existence of integral varifolds solving the flow globally-in-time

Varifolds are solutions in the $L^2$-flow sense

Extension of previous results by one of the authors

Abstract

In this paper, we propose a new notion of Brakke inequality for volume preserving mean curvature flow. We show the existence of integral varifolds solving the flow globally-in-time in the corresponding Brakke sense using the phase field method. Morever, such varifolds are solutions to volume preserving mean curvature flow in the $L^2$-flow sense as well. We thus extend a previous result by one of the authors [25].