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Weak solutions of the curve shortening flow. (English) Zbl 0990.35076
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 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)

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
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