Local regularity and compactness for the p-harmonic map heat flows

Abstract

Abstract We study a geometric analysis and local regularity for the evolution of p {{p}} -harmonic maps, called p {{p}} -harmonic map heat flows. Our main result is to establish a criterion for a uniform local regularity estimate for regular p {{p}} -harmonic map heat flows, devising some new monotonicity-type formulas of a local scaled energy. The regularity criterion obtained is almost optimal, comparing with that of the corresponding stationary case. As application we show a compactness of regular p {{p}} -harmonic map heat flows with energy bound.