Mean Curvature Flow With External Force Field

In this thesis, we mainly investigate mean curvature flow with external force fieldswhich comes from the study of the vortex of a Ginzburg-Landau system. The Ginzburg-Landau system is a simple model which simulates inhomogeneous type II supercon-ducting materials as well as three-dimensional superconducting thin films having vari-able thickness. The vortex of this problem is moved by a mean curvature flow withexternal force fields.Firstly, we study the Cauchy problem of the Ginzburg-Landau system, and provethat the vortex (the set of zero points) is described by a mean curvature flow withexternal force field in the time interval at which the mean curvature flow is smooth.Secondly, we investigate the long-time existence of the smooth solution of themean curvature flow with external force field. For compact hypersurfaces and highcodimension compact submanifolds initial data respectively, we prove that the flow hasa smooth solution for all time under almost optimal conditions. Those conditions arethat the second fundamental form on the initial submanifolds is not too large, the exter-nal force field, with its any order derivatives, is bounded, and the field is convex with itseigenvalues satisfying a pinch inequality. In case of compact initial hypersurfaces, wegive several examples to show that the assumptions of small initial curvature and theconvexity of the external force field are necessary, that is, if the conditions of long-timeexistence are not satisfied, the flow will blow up in finite time. Especially, for a specialforce field which is of much interest in physics, we obtain the global smooth solutionto the flow. Besides, when initial hypersurface is a noncompact entire graph, we alsoobtain the long-time existence of smooth solution for some special external force fields.