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Global well-posedness of the Landau–Lifshitz–Gilbert equation for initial data in Morrey spaces

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Abstract

We establish the global well-posedness of the Landau–Lifshitz–Gilbert equation in \(\mathbb R^n\) for any initial data \(\mathbf{m}_0\in H^1_*(\mathbb R^n,\mathbb S^2)\) whose gradient belongs to the scaling critical Morrey space \(M^{2,2}(\mathbb R^n)\) with small norm \(\displaystyle \Vert \nabla \mathbf{m}_0\Vert _{M^{2,2}(\mathbb R^n)}\). The method is based on priori estimates of a dissipative Schrödinger equation of Ginzburg-Landau types obtained from the Landau–Lifshitz–Gilbert equation by the moving frame technique.

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Notes

  1. \(\mathbf{m}\) satisfies (1.1) in the sense of distributions, if, for any \(\Phi \in C_0^\infty (P_{r_0}(z_0),\mathbb S^2)\), the following holds:

    $$\begin{aligned} \int _{P_{r_0}(z_0)} \big [\langle \partial _t \mathbf{m}, \Phi \rangle +\lambda \langle \nabla \mathbf{m}, \nabla \Phi \rangle -\langle \mathbf{m}\times \nabla \mathbf{m},\nabla \Phi \rangle -\lambda \langle |\nabla \mathbf{m}|^2\mathbf{m},\Phi \rangle \big ]\,dyds=0. \end{aligned}$$

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Acknowledgments

J. Lin has been partially supported by NSF of China (Grant 11001085) and the Ph.D. Programs Foundation of Ministry of Education of China (Grant 20100172120026). B. Lai has been partially supported by NSF of China (Grants 11201119 and 11126155). C. Wang has been partially supported by NSF grants 1001115 and 1265574, NSF of China grant 11128102, and a Simons Fellowship in Mathematics. The work was completed while both J. Lin and B. Lai visited Department of Mathematics, University of Kentucky. They would like to thank the Department for its hospitality and excellent research environment. Finally, the third author wishes to thank Professor Y. Giga for both his interest and useful suggestion on several important references on the study of Morrey spaces for evolution equations. The authors wish to thank annoymous referees for their constructive suggestions that improve the presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematics, South China University of Technology, Guangzhou, 510640, Guangdong, China

    Junyu Lin

  2. School of Mathematics and Information Science, Henan University, Kaifeng, 475004, Henan, China

    Baishun Lai

  3. Department of Mathematics, University of Kentucky, Lexington, KY, 40506, USA

    Changyou Wang

  4. Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN, 47907, USA

    Changyou Wang

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  1. Junyu Lin
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Correspondence to Changyou Wang.

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Communicated by Y. Giga.

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Lin, J., Lai, B. & Wang, C. Global well-posedness of the Landau–Lifshitz–Gilbert equation for initial data in Morrey spaces. Calc. Var. 54, 665–692 (2015). https://doi.org/10.1007/s00526-014-0801-2

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