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arXiv · 2411.18284

Existence of curvature flow with forcing in a critical Sobolev space

Yuning Liu·Yoshihiro Tonegawa
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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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Yuning Liu, Yoshihiro Tonegawa. 2024-11-27. Existence of curvature flow with forcing in a critical Sobolev space. https://arxiv.org/abs/2411.18284

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