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

Asymptotic behaviors of Landau–Lifshitz flows from $$\mathbb {R}^2$$ R 2 to Kähler... In this paper, we study the asymptotic behaviors of finite energy solutions to the Landau–Lifshitz flows from $$\mathbb {R}^2$$ 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 $$t\rightarrow \infty$$ t → ∞ 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 $$4\pi$$ 4 π 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 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Calculus of Variations and Partial Differential Equations Springer Journals

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

, Volume 56 (4) – Jun 10, 2017
35 pages

/lp/springer_journal/asymptotic-behaviors-of-landau-lifshitz-flows-from-mathbb-r-2-r-2-to-k-2tmKejli3t
Publisher
Springer Journals
Subject
Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
ISSN
0944-2669
eISSN
1432-0835
D.O.I.
10.1007/s00526-017-1182-0
Publisher site
See Article on Publisher Site

### Abstract

In this paper, we study the asymptotic behaviors of finite energy solutions to the Landau–Lifshitz flows from $$\mathbb {R}^2$$ 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 $$t\rightarrow \infty$$ t → ∞ 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 $$4\pi$$ 4 π 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 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.

### Journal

Calculus of Variations and Partial Differential EquationsSpringer Journals

Published: Jun 10, 2017

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