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We propose a new weak solution concept for (two-phase) mean curvature flowwhich enjoys both (unconditional) existence and (weak-strong) uniquenessproperties. These solutions are evolving varifolds, just as in Brakke'sformulation, but are coupled to the phase volumes by a simple transportequation. 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 tothe Allen-Cahn equation is a varifold solution in our sense. Second, we provethat 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 theclass of our new varifold solutions. This is in sharp contrast to the case ofBrakke flows, which a priori may disappear at any given time and are thereforefatally non-unique. Finally, we propose an extension of the solution concept tothe multi-phase case which is at least guaranteed to satisfy a weak-stronguniqueness principle.

A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness