On uniqueness of heat flow of harmonic maps


In this paper, we establish the uniqueness of heat flow of harmonic maps into (N, h) that have sufficiently small renormalized energies, provided that N is either a unit sphere Skβˆ’1S^{k-1} or a compact Riemannian homogeneous manifold without boundary. For such a class of solutions, we also establish the convexity property of the Dirichlet energy for tβ‰₯t0>0t\ge t_0>0 and the unique limit property at time infinity. As a corollary, the uniqueness is shown for heat flow of harmonic maps into any compact Riemannian manifold N without boundary whose gradients belong to LtqLxlL^q_t L^l_x for q>2q>2 and l>nl>n satisfying the Serrin's condition.Comment: 24 pages. Two errors of proof of lemma 2.3 have been fixe

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