  In this paper, we first investigate the integral curvature condition to
extend the mean curvature flow of submanifolds in a Riemannian manifold with
codimension $d\geq1$, which generalizes the extension theorem for the mean
curvature flow of hypersurfaces due to Le-\v{S}e\v{s}um \cite{LS} and the
authors \cite{XYZ1,XYZ2}. Using the extension theorem, we prove two convergence
theorems for the mean curvature flow of closed submanifolds in ${R}^{n+d}$
under suitable integral curvature conditions.
