Global existence and partial regularity for the p-harmonic flow

We show a global existence for the Cauchy problem with large initial data for the p-harmonic flow between two smooth, compact Riemannian manifolds. We devise new monotonicity type formulas of a local scaled energy and establish a partial regularity for the solution. The partial regularity obtained is almost optimal, comparing with that of the corresponding stationary case. The p-harmonic flow obtained also converges to a p-harmonic map along a certain time sequence tending to infinity.