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Time global existence of generalized BV flow via the Allen--Cahn equation

Kiichi Tashiro

TL;DR

This work analyzes the mean curvature flow realized as a singular limit of the Allen–Cahn equation and proves it lies not only in the Brakke flow class but also in the generalized BV flow class of Stuvard and Tonegawa. By combining Brakke flow theory with BV-type evolution through measure-function pairs, the authors derive a BV-type volume-change formula that characterizes the phase-region dynamics in time. The main result shows the AC limit produces a pair consisting of a Brakke flow and a family of finite-perimeter sets that satisfy the BV relation, with no extra convergence assumptions beyond those already used to obtain the Brakke limit. This strengthens the link between phase-field models and geometric flows and provides a framework for BV-flow-type uniqueness up to topological changes in this setting.

Abstract

We show that a mean curvature flow obtained as the limit of the Allen--Cahn equation is not only a Brakke flow but also a generalized BV flow proposed by Stuvard and Tonegawa.

Time global existence of generalized BV flow via the Allen--Cahn equation

TL;DR

This work analyzes the mean curvature flow realized as a singular limit of the Allen–Cahn equation and proves it lies not only in the Brakke flow class but also in the generalized BV flow class of Stuvard and Tonegawa. By combining Brakke flow theory with BV-type evolution through measure-function pairs, the authors derive a BV-type volume-change formula that characterizes the phase-region dynamics in time. The main result shows the AC limit produces a pair consisting of a Brakke flow and a family of finite-perimeter sets that satisfy the BV relation, with no extra convergence assumptions beyond those already used to obtain the Brakke limit. This strengthens the link between phase-field models and geometric flows and provides a framework for BV-flow-type uniqueness up to topological changes in this setting.

Abstract

We show that a mean curvature flow obtained as the limit of the Allen--Cahn equation is not only a Brakke flow but also a generalized BV flow proposed by Stuvard and Tonegawa.
Paper Structure (10 sections, 13 theorems, 52 equations)

This paper contains 10 sections, 13 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Main Results
  3. Basic Notation
  4. Weak Notions of Mean Curvature Flow
  5. Assumptions and main result
  6. Proof of Theorem \ref{['mainresult1']}
  7. Absolute continuity of phase boundary measure
  8. Basic Properties of $L^2$ Flow and Set of Finite Perimeter
  9. Boundaries move by mean curvature
  10. Measure-Function Pairs

Key Result

Theorem 1

Let $\{ \mu_t \}_{ t \geq 0 }$ be the Brakke flow and $\varphi ( x , t ) = \chi_{ E_t } ( x )$ be the phase function obtained as a limit of (AC). Then, for all test function $\phi \in C^1_c ( \mathbb{R}^n )$ and $0 \leq t_1 < t_2 < \infty$, we have where $h ( \cdot , t )$ is the generalized mean curvature vector of $\mu_t$, $| \nabla \varphi ( \cdot , t ) |$ is the perimeter measure of the phase

Theorems & Definitions (23)

  • Theorem
  • Definition 2.1
  • Definition 2.2: $L^2$ flow
  • Definition 2.3: Generalized BV flow
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • Theorem 2.7
  • Lemma 3.1
  • Lemma 3.2
  • ...and 13 more