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Existence and Stability of Cylindrical Symmetric Static Solutions to the Landau–Lifshitz Equation for Multidirectional Ferromagnets

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Abstract

We investigate the existence and stability of cylindrical symmetric static solutions to the Landau–Lifshitz equation for multidirectional ferromagnets in the three-dimensional (3D) cylindrical surfaces \({\varvec{H_c}} = (x,y,\sqrt{{c^2} - {x^2} - {y^2}} )\), supplemented with Dirichlet boundary condition. After introducing the cylindrical coordinates transformation, the Landau–Lifshitz equation in Cartesian coordinates, existing a multi-direct effective field, is transformed into that in cylindrical symmetric coordinates, and then the existence of static solutions of the cylindrical symmetric model is proved. The theory of elliptic PDE of second order and the Schauder fixed-point theorem are fundamental ingredients in the existence analysis. Here, the two most notable things are to cope with the multi-direct effective field and the increased terms derived from the transformation. What’s more, we observe that the linearized operator mentioned in this paper can’t be extended to the self-adjoint operator so that the spectral problem on the linearized operator is so complicated, and thus we need to provide more careful estimates to establish the Lyapunov stability of cylindrical symmetric solutions of the time-dependent Landau–Lifshitz equation in the 3D cylindrical surfaces \({\varvec{H_c}}\).

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Acknowledgements

The authors would like to express their appreciations to editors and reviewers for their insightful and valuable comments that helped to revise manuscript. This project is supported by National Natural Science Foundation of China (No. 12001415, No. 11961080).

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  1. School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, China

    Jie Zhang

  2. Department of Mathematics, Yunnan Minzu University, Chenggong Yuehua, Kunming, 650504, China

    Ganshan Yang

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  1. Jie Zhang
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  2. Ganshan Yang
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Zhang, J., Yang, G. Existence and Stability of Cylindrical Symmetric Static Solutions to the Landau–Lifshitz Equation for Multidirectional Ferromagnets. Appl Math Optim 87, 14 (2023). https://doi.org/10.1007/s00245-022-09919-3

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