  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.
