Heat Flow of Harmonic Maps Whose Gradients Belong to $$L^{n}_{x}L^{\infty}_{t}$$

For any compact n-dimensional Riemannian manifold (M, g) without boundary, a compact Riemannian manifold $$N \subset {\mathbb{R}}^{k}$$ without boundary, and 0 < T ≦ +∞, we prove that for n ≧ 4, if u : M × (0, T] → N is a weak solution to the heat flow of harmonic maps such that $$\nabla u \in L^{n}_{x}L^{\infty}_{t}(M \times (0, T])$$ , then u ∈C∞(M × (0, T], N). As a consequence, we show that for n ≧3, if 0 < T < +∞ is the maximal time interval for the unique smooth solution u ∈C∞(M × [0, T), N) of (1.1), then $$||{\nabla}u(t)||_{L^{n}(M)}$$ blows up as t ↑ T.