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We apply our abstract gradient inequalities developed by the authors inarXiv:1510.03817 to prove Lojasiewicz--Simon gradient inequalities for theharmonic map energy function using Sobolev spaces which impose minimalregularity requirements on maps between closed, Riemannian manifolds. OurLojasiewicz--Simon gradient inequalities for the harmonic map energy functiongeneralize those of Kwon (2002), Liu and Yang (2010), Simon (1983, 1985), andTopping (1997).