Suppose we are given a closed set $M$ in the $n$-dimensional Euclidean space, which is of locally finite $n-1$-dimensional Hausdorff measure and countably $n-1$-rectifiable. Assume furthermore that $M$ is realized as a finite union of boundaries of mutually disjoint open sets. Typical $M$ would be, on a plane, arbitrary network of curves with finite number of junctions, and on 3-D, bubble clusters with complicated singularities. Taking such general $M$ as the initial surface, we prove that there exists a non-trivial Brakke's mean curvature flow which exists for all time. This may be seen as a proper weak solution describing motion of grain boundaries driven by excess surface energy. I will explain the results and outline of proof. This is a joint work with Lami Kim.

In this talk I'll present a length estimate for planar curve shortening flow and apply it to establish results about the evolution of nonsmooth objects. The estimate itself depends on a crude geometric quantity called the r-multiplicity, which is a type of coarse intersection profile. We'll also discuss extending these results to curve shortening flow in an arbitrary surface, the key step in which is an extension of a certain gradient estimate.