Weak solutions of the curve shortening flow.

*(English)*Zbl 0990.35076The 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 K. A. Brakke [The motion of a surface by its mean curvature, Princeton University Press (1978; Zbl 0386.53047)] and by L. Ambrosio and H. M. Soner [J. Differ. Geom. 43, 693-737 (1996; Zbl 0868.35046)].

Alternative notions of generalized solution for mean curvature flow of arbitrary dimension and codimension have been developed by K. A. Brakke [The motion of a surface by its mean curvature, Princeton University Press (1978; Zbl 0386.53047)] and by L. Ambrosio and H. M. Soner [J. Differ. Geom. 43, 693-737 (1996; Zbl 0868.35046)].

Reviewer: J.Urbas (Bonn)

##### MSC:

35K65 | Degenerate parabolic equations |

35D05 | Existence of generalized solutions of PDE (MSC2000) |

58E10 | Variational problems in applications to the theory of geodesics (problems in one independent variable) |

49Q99 | Manifolds and measure-geometric topics |