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.