On the existence of mean curvature flow with transport term

We prove the global-in-time existence of weak solution for a hypersurface evolution problem where the velocity is the sum of the mean curvature and arbitrarily given non-smooth vector field in a suitable Sobolev space. The approximate solution is obtained by the Allen-Cahn equation with transport term. By establishing the density ratio upper bound on the phase boundary measure it is shown that the limiting surface moves with the desired velocity in the sense of Brakke.