Existence of curvature flow with forcing in a critical Sobolev space

Suppose that a closed $1$-rectifiable set ${\mathrm{\Gamma }}_0\subset {\mathbb{R}}^2$ 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 ${\mathrm{\Gamma }}_0$, 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.