A stationary minimal surface with a mildly flat singular point admits a non-trivial Brakke flow, so Brakke flow starting from it is non-unique.
A general regularity theory for weak mean curvature flow
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Non-uniqueness of Brakke flows starting from minimal surfaces with singularities
A stationary minimal surface with a mildly flat singular point admits a non-trivial Brakke flow, so Brakke flow starting from it is non-unique.