Heat flow of harmonic maps whose gradients belong to LxnLt∞

For any compact n-dimensional Riemannian manifold (M, g) without boundary, a compact Riemannian manifold N subset of R-k without boundary, and 0 < T <= + infinity, we prove that for n >= 4, if u : M x (0, T] -> N is a weak solution to the heat flow of harmonicmaps such that del u is an element of (LxLt infinity)-L-n (M x (0, T]), then u is an element of C-infinity (M x (0, T], N). As a consequence, we show that for n >= 3, if 0 < T < + infinity is the maximal time interval for the unique smooth solution u is an element of C-infinity (M x [0, T), N) of (1.1), then parallel to del u(t)parallel to(n)(L)((M)) blows up as t up arrow T.