Existence of curvature flow with forcing in a critical Sobolev space

Abstract

Suppose that a closed 1-rectifiable set ${\Gamma }_0\subset {\mathbb{R}}^2$ of finite 1-dimensional Hausdorff measure and a vector field $u$ in a dimensionally critical Sobolev space are given. It is proved that, starting from ${\Gamma }_0$, there exists a non-trivial flow of curves with the normal velocity given by the sum of the curvature and the given vector field $u$. The motion law is satisfied in the sense of Brakke and the flow exists through singularities.

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