Weak solutions of the curve shortening flow (Q1374878)

The author formulates a parametric notion of weak solution for the curve shortening flow in arbitrary codimensions, and he proves the existence of such a solution which is global in time for an arbitrary smooth closed initial curve in \(\mathbb{R}^n\). The idea is to replace the problem by a simpler one which preserves the geometry of the evolving curves, and then to show that the modified problem has a weak solution in the author's sense by proving suitable a priori estimates for the solutions of a family of regularized problems and extracting a convergent subsequence. Alternative notions of generalized solution for mean curvature flow of arbitrary dimension and codimension have been developed by \textit{K. A. Brakke} [The motion of a surface by its mean curvature, Princeton University Press (1978; Zbl 0386.53047)] and by \textit{L. Ambrosio} and \textit{H. M. Soner} [J. Differ. Geom. 43, 693-737 (1996; Zbl 0868.35046)].