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

Abstract

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

This version improves the results in original Theorem 1.2 by including the positive curvature case

References in corpus (1)

• Equivariant Schrödinger Maps in two spatial dimensions

Cited by in corpus (2)

• Convergence to harmonic maps for the Landau-Lifshitz flows on two dimensional hyperbolic spaces
• Blowup dynamics for smooth equivariant solutions to energy critical Landau-Lifschitz flow