Lojasiewicz inequalities for maps of the 2-sphere

We prove a Lojasiewicz-Simon inequality $$\mid E(u)-4\pi n\mid \le C\mathrm{\parallel }\mathcal{T}(u){\mathrm{\parallel }}^{\alpha }$$ for maps $u\in W^{2,2}(S^2,S^2)\mathrm{.}$ 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 $t\to \mathrm{\infty }$ 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.