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.

中文翻译：

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一维 Brakke 流的存在性和规律性定理

给定 $\mathbb R^2$ 中的闭可数 $1$-rectifiable 集合，具有局部有限 $1$-维 Hausdorff 测度，我们证明存在从给定集合开始的 Brakke 流，其具有以下正则性。几乎一直以来，局部流动都由有限数量的 $W^{2,2}$ 类嵌入曲线组成，其端点在交点处相交，角度为 0、60 或 120 度。