Asymptotic behaviors of Landau-Lifshitz flows from ${\mathbb{R}}^2$ to K\"ahler manifolds

In this paper, we study the asymptotic behaviors of finite energy solutions to the Landau-Lifshitz flows from ${\mathbb{R}}^2$ into K"ahler manifolds. First, we prove that the solution with initial data below the critical energy converges to a constant map in the energy space as $t\to \mathrm{\infty }$ 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. Second, for general compact K"ahler manifolds and initial data of an arbitrary finite energy, we obtain a bubbling theorem analogous to the Struwe’s results on the heat flows.