  We prove a Lojasiewicz-Simon inequality $$ \left| E(u) - 4\pi n \right| \leq
C \| \mathcal{T}(u) \|^\alpha $$ for maps $u \in W^{2,2}\left( S^2, S^2
\right).$ The inequality holds with $\alpha = 1$ in general and with $\alpha >
1$ unless $u$ is nearly constant on an open set. We obtain polynomial
convergence of weak solutions of harmonic map flow $u(t) : S^2 \to S^2$ as $t
\to \infty$ on compact domains away from the singular set, assuming that the
body map is nonconstant. The proof uses Topping's repulsion estimates together
with polynomial lower bounds on the energy density coming from a bubble-tree
induction argument.

Differential Geometry;Analysis of PDEs