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

TL;DR

This paper investigates the long-term behavior of finite energy solutions to Landau-Lifshitz flows from  into Ke4hler manifolds, proving convergence to constant maps under certain energy conditions and establishing a bubbling theorem for general cases.

Contribution

It provides new results on the asymptotic convergence of Landau-Lifshitz flows and introduces a bubbling theorem analogous to Struwe's results for heat flows.

Findings

Solutions with energy below a critical threshold converge to constant maps.

For the sphere target, solutions with energy below 4\u03c0 converge to a constant map.

A bubbling theorem is established for general Ke4hler targets and finite energy data.

Abstract

In this paper, we study the asymptotic behaviors of finite energy solutions to the Landau-Lifshitz flows from $\Bbb R^2$ 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 $t\to \infty$ 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.