  We introduce a rectified $n$-harmonic map flow from an n-dimensional closed
Riemannian manifold to another closed Riemannian manifold. We prove existence
of a global solution, which is regular except for a finite number of points, of
the rectified n-harmonic map flow and establish an energy identity for the flow
at each singular time. Finally, we present two applications of the rectified
n-harmonic map flow to minimizing the n-energy functional and the Dirichlet
energy functional in a homotopy class.
