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Stability and optimal decay for the 3D magnetohydrodynamic equations with only horizontal dissipation

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Abstract

This paper develops an effective approach to establishing the optimal decay estimates on solutions of the 3D anisotropic magnetohydrodynamic (MHD) equations with only horizontal dissipation. As our first step, we prove the global existence and stability of solutions to the aforementioned MHD system emanating from any initial data with small \(H^1\)-norm. Due to the lack of dissipation in the vertical direction, the large-time behavior does not follow from the classical approaches. The analysis of the nonlinear terms are much more difficult than in the case of full dissipation. In particular, we need to represent the MHD equations in an integral form, exploit cancellations and other properties such as the incompressibility in order to control terms involving vertical derivatives.

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Acknowledgements

Shang was partially supported by Natural Science Foundation of Hebei Province under grant No. A2023501008 and National Natural Science Foundation of China under grant No. 12371232. Wu was partially supported by NSF grant DMS 2104682 and the AT &T Foundation at Oklahoma State University. Zhang was partially supported by the National Natural Science Foundation of China [grant number 12326416] and the Innovation Capacity Enhancement Program-Science and Technology Platform Project, Hebei Province (22567623H).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao, Hebei, 066004, People’s Republic of China

    Haifeng Shang

  2. Department of Mathematics, University of Notre Dame, South Bend, IN, 46556, USA

    Jiahong Wu

  3. Hebei Key Laboratory of Machine Learning and Computational Intelligence, School of Mathematics and Information Science, Hebei University, Baoding, 071002, People’s Republic of China

    Qian Zhang

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Shang, H., Wu, J. & Zhang, Q. Stability and optimal decay for the 3D magnetohydrodynamic equations with only horizontal dissipation. J. Evol. Equ. 24, 12 (2024). https://doi.org/10.1007/s00028-023-00940-9

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