  We propose a new weak solution concept for (two-phase) mean curvature flow
which enjoys both (unconditional) existence and (weak-strong) uniqueness
properties. These solutions are evolving varifolds, just as in Brakke's
formulation, but are coupled to the phase volumes by a simple transport
equation. First, we show that, in the exact same setup as in Ilmanen's proof
[J. Differential Geom. 38, 417-461, (1993)], any limit point of solutions to
the Allen-Cahn equation is a varifold solution in our sense. Second, we prove
that any calibrated flow in the sense of Fischer et al. [arXiv:2003.05478] -
and hence any classical solution to mean curvature flow - is unique in the
class of our new varifold solutions. This is in sharp contrast to the case of
Brakke flows, which a priori may disappear at any given time and are therefore
fatally non-unique. Finally, we propose an extension of the solution concept to
the multi-phase case which is at least guaranteed to satisfy a weak-strong
uniqueness principle.
