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

Andrea Chiesa, Keisuke Takasao

TL;DR

The paper addresses the challenge of formulating and constructing a weak evolution for volume-preserving mean curvature flow by introducing a Brakke inequality adapted to the volume constraint. Using phase-field (Allen–Cahn) approximations, it proves the global existence of integral varifold solutions that satisfy a modified Brakke inequality and that converge to an $L^2$-flow with a velocity decomposed as $\vec{v}=\vec{h}-\lambda\frac{d\|\nabla\psi\|}{d\mu_t}\vec{\nu}$. This builds on prior results by Takasao, showing that the limit is a volume-preserving Brakke-flow under periodic boundary conditions, and it bridges phase-field methods with a Brakke-flow framework for constrained MCF. The key contribution is the establishment of a Brakke-flow for VPMCF with a controlled error term, enabling a robust weak theory for constrained multi-phase evolution with volume preservation.

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].

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

TL;DR

The paper addresses the challenge of formulating and constructing a weak evolution for volume-preserving mean curvature flow by introducing a Brakke inequality adapted to the volume constraint. Using phase-field (Allen–Cahn) approximations, it proves the global existence of integral varifold solutions that satisfy a modified Brakke inequality and that converge to an -flow with a velocity decomposed as . This builds on prior results by Takasao, showing that the limit is a volume-preserving Brakke-flow under periodic boundary conditions, and it bridges phase-field methods with a Brakke-flow framework for constrained MCF. The key contribution is the establishment of a Brakke-flow for VPMCF with a controlled error term, enabling a robust weak theory for constrained multi-phase evolution with volume preservation.

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 -flow sense as well. We thus extend a previous result by one of the authors [25].

Paper Structure

This paper contains 7 sections, 4 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Definitions and assumptions
  3. Notions of solution: -flow and Brakke-flow
  4. Assumptions on the initial data
  5. Main results
  6. Previous results: Existence of an -flow
  7. Main result: Existence of a volume-preserving Brakke-flow

Key Result

Proposition 2.1

Let $D \in (0,\infty), \ \Omega=\mathbb{R}^d$ or $\mathbb{T}^d$ and $(\{\mu_t\}_{t\in[0,T)}, \psi)$ be a volume-preserving Brakke-flow with $\mu_t(\Omega)\leq \mu_0(\Omega)<\infty$ for almost every $t\in [0,T]$. Suppose that $U\subset \Omega$ be a bounded open set. Then $\{\mu_t\}_{t\in[0,T)}$ satis for every $\eta\in C^1_c({U}{\times} (0,T))$, for some constant $C_T>0$. In particular, $\{\mu_t\}_

Theorems & Definitions (10)

  • Definition 2.1: $L^2$-flow TakasaoVPMCF
  • Remark 2.1
  • Definition 2.2: Volume-preserving Brakke-flow
  • Proposition 2.1
  • proof
  • Proposition 3.1: TakasaoVPMCF
  • Theorem 3.1: TakasaoVPMCF
  • Remark 3.1
  • Theorem 3.2: Existence of a weak Brakke-flow
  • proof