On the existence of canonical multi-phase Brakke flows

TL;DR

This paper proves the global existence of multi-phase Brakke flows from arbitrary initial data, showing that these flows satisfy volume change identities and addressing non-uniqueness issues under certain conditions.

Contribution

It establishes the existence of multi-phase Brakke flows that are also BV solutions, providing new insights into their volume evolution and uniqueness properties.

Findings

Proves global-in-time existence of multi-phase Brakke flows.

Shows boundaries evolve according to generalized mean curvature.

Addresses non-uniqueness of Brakke flows under certain conditions.

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.