  We prove existence, uniqueness and asymptotics of global smooth solutions for
the Landau-Lifshitz-Gilbert equation in dimension $n \ge 3$, valid under a
smallness condition of initial gradients in the $L^n$ norm. The argument is
based on the method of moving frames that produces a covariant complex
Ginzburg-Landau equation, and a priori estimates that we obtain by the method
of weighted-in-time norms as introduced by Fujita and Kato.
