  In this paper, we study the evolution of submannifold moving by mean
curvature minus a external force field. We prove that the flow has a long-time
smooth solution for all time under almost optimal conditions.
  Those conditions are that the second fundamental form on the initial
submanifolds is not too large, the external force field, with its any order
derivatives, is bounded, and the field is convex with its eigenvalues
satisfying a pinch inequality.
