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 \(t\rightarrow \infty \) 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. 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.
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Acknowledgements
The authors thank Professor Youde Wang and Hao Yin for helpful discussions and encouragements. And we thank the referee for helpful comments and improvements of our paper. L. Zhao has been partially supported by the NSFC Grant of China (Nos. 10901148 and 11371337) and the Fundamental Research Funds for the Central Universities (WK3470000005). L. Zhao is also supported by Youth Innovation Promotion Association CAS.
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Li, Z., Zhao, L. Asymptotic behaviors of Landau–Lifshitz flows from \(\mathbb {R}^2\) to Kähler manifolds. Calc. Var. 56, 96 (2017). https://doi.org/10.1007/s00526-017-1182-0
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DOI: https://doi.org/10.1007/s00526-017-1182-0