Given a closed countably 1-rectifiable set in ℝ2 with locally finite 1-dimensional Hausdorff measure, we prove that there exists a Brakke flow starting from the given set with the following regularity property. For almost all time, the flow locally consists of a finite number of embedded curves of class W2,2 whose endpoints meet at junctions with angles of either 0, 60 or 120 degrees.
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