Existence and regularity theorems of one-dimensional Brakke flows

Abstract

Given a closed countably 1-rectifiable set in ${\mathbb{R}}^2$ 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 $W^{2,2}$ whose endpoints meet at junctions with angles of either 0, 60 or 120 degrees.