Existence of curvature flow with forcing in a critical Sobolev space

TL;DR

This paper proves the existence of a curvature flow with an external forcing term in a critical Sobolev space, allowing the flow to evolve through singularities starting from a rectifiable initial set.

Contribution

It establishes the existence of a non-trivial curvature flow with forcing in a critical Sobolev space, extending the theory to include singularities and external forces.

Findings

Flow exists starting from initial rectifiable set

Flow satisfies Brakke's motion law

Flow can pass through singularities

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 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.