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Regular solutions for multiplicative stochastic Landau-Lifshitz-Gilbert equation and blow-up phenomena

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Abstract

The two-dimensional Landau-Lifshitz-Gilbert equation of motion for a classical magnetic moment perturbed by a multiplicative noise is considered. This equation is highly nonlinear in nature and, for this reason, many mathematical results in stochastic partial differential equations (SPDEs) cannot be applied. The aim of this work is to introduce the difference method to handle SPDEs and prove the existence of regular martingale solutions in dimension two. Some blow-up phenomena are presented, which are drastically different from the deterministic case. Finally, to yield correct thermal-equilibrium properties, Stratonovitch integral is used instead of Itô integral.

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Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing University, Chongqing, 400044, China

    XueKe Pu

  2. Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing, 100088, China

    BoLing Guo

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  1. XueKe Pu
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  2. BoLing Guo
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Correspondence to XueKe Pu.

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Pu, X., Guo, B. Regular solutions for multiplicative stochastic Landau-Lifshitz-Gilbert equation and blow-up phenomena. Sci. China Math. 53, 3115–3130 (2010). https://doi.org/10.1007/s11425-010-4099-z

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  • DOI: https://doi.org/10.1007/s11425-010-4099-z

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