Existence of curvature flow with forcing in a critical Sobolev space

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.