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On the existence of canonical multi-phase Brakke flows

Salvatore Stuvard, Yoshihiro Tonegawa

Abstract

This paper establishes the global-in-time existence of a multi-phase mean curvature flow, evolving from an arbitrary closed rectifiable initial datum, which is a Brakke flow and a BV solution at the same time. In particular, we prove the validity of an explicit identity concerning the change of volume of the evolving grains, showing that their boundaries move according to the generalized mean curvature vector of the Brakke flow. As a consequence of the results recently established by Fischer et al. in arXiv:2003.05478, under suitable assumptions on the initial datum, such additional property resolves the non-uniqueness issue of Brakke flows.

On the existence of canonical multi-phase Brakke flows

Abstract

This paper establishes the global-in-time existence of a multi-phase mean curvature flow, evolving from an arbitrary closed rectifiable initial datum, which is a Brakke flow and a BV solution at the same time. In particular, we prove the validity of an explicit identity concerning the change of volume of the evolving grains, showing that their boundaries move according to the generalized mean curvature vector of the Brakke flow. As a consequence of the results recently established by Fischer et al. in arXiv:2003.05478, under suitable assumptions on the initial datum, such additional property resolves the non-uniqueness issue of Brakke flows.

Paper Structure

This paper contains 25 sections, 28 theorems, 250 equations.

Table of Contents

  1. Introduction
  2. Notation and main results
  3. Basic notation
  4. Three weak notions of MCF
  5. Main results
  6. Outline of the proofs and plan of the paper
  7. Further notation
  8. Existence of multi-phase Brakke flow
  9. Volume-controlled Lipschitz deformations
  10. The constructive scheme
  11. BV flow: proof of Theorem \ref{['main3']}(6)
  12. Characterization as $L^2$ flow
  13. Existence of measure-theoretic velocities
  14. Boundaries move by their mean curvature
  15. Ilmanen's density lower bound
  16. ...and 10 more sections

Key Result

Theorem A

Let $E_{0,1},\ldots, E_{0,N}\subset\mathbb R^{n+1}$ be mutually disjoint non-empty open sets with $N\geq 2$ such that $\Gamma_0:=\mathbb R^{n+1}\setminus\cup_{i=1}^N E_{0,i}$ is countably $n$-rectifiable. Assume that the $n$-dimensional Hausdorff measure of $\Gamma_0$ is finite or grows at most expo and satisfying the following properties. Writing $\Gamma(t):=\mathbb R^{n+1} \setminus \cup_{i=1}^N

Theorems & Definitions (52)

  • Theorem A
  • Definition \oldthetheorem: Brakke flow
  • Definition \oldthetheorem
  • Definition \oldthetheorem: $L^2$ flow
  • Definition \oldthetheorem: ${\rm BV}$ flow
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • ...and 42 more