Existence and regularity of mean curvature flow with transport term in higher dimensions

Given an initial $C^1$ hypersurface and a time-dependent vector field in a Sobolev space, we prove a time-global existence of a family of hypersurfaces which start from the given hypersurface and which move by the velocity equal to the mean curvature plus the given vector field. We show that the hypersurfaces are $C^1$ for a short time and, even after some singularities occur, almost everywhere $C^1$ away from higher multiplicity region.Comment: 60 page