Asymptotic behaviors of Landau-Lifshitz flows from R2 to Kahler manifolds

In this paper, we study the asymptotic behaviors of finite energy solutions to the Landau-Lifshitz flows from R-2 into Kahler manifolds. First, we prove that the solution with initial data below the critical energy converges to a constant map in the energy space as t -> infinity for the compact Riemannian surface targets. In particular, when the target is a two dimensional sphere, we prove that the solution to the Landau-Lifshitz-Gilbert equation with initial data having an energy below 4 pi converges to some constant map in the energy space. The proof bases on the method of induction on energy and geometric renormalizations. Second, for general compact Kahler manifolds and initial data of an arbitrary finite energy, we obtain a bubbling theorem analogous to the Struwe's results on the heat flows.