  In this paper, we prove that the solution of the Landau-Lifshitz flow
$u(t,x)$ from $\mathbb{H}^2$ to $\mathbb{H}^2$ converges to some harmonic map
as $t\to\infty$. The essential observation is that although there exist
infinite numbers of harmonic maps from $\Bbb H^2$ to $\Bbb H^2$, the heat flow
initiated from $u(t,x)$ for any given $t>0$ converges to the same harmonic map
as the heat flow initiated from $u(0,x)$. This observation enables us to
construct a variant of Tao's caloric gauge to reduce the convergence to
harmonic maps for the Landau-Lifshitz flow to the decay of the corresponding
heat tension field. The advantage of the strategy used in this paper is that we
can see the limit harmonic map directly by evolving $u(0,x)$ along a heat flow
without evolving the Landau-Lifshitz flow to the infinite time.
