This thesis studies problems derived from nonlinear partial differential equations of parabolic type. Part I. A mass reducing flow for integral currents. A mass reducing flow of integral current is constructed. The current flow has the property that it is Holder continuous under the flat norm and reduces the mass of the initial current while keeping the boundary fixed. Part II. Estimate of singular set of the evolution problems for harmonic maps. Let u: M × R\sb+ → N be a weak solution to the evolution problem for harmonic maps. We prove that the singular set of u has at most finite m − 2 dimensional Hausdorff measure on each time slice M × {t}.