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Some Liouville-Type Results for the 3D Incompressible MHD Equations

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Abstract

In this paper, we investigate some Liouville-type properties for the stationary incompressible MHD equations in three dimensions. Liouville-type properties are concerned with the uniqueness of trivial solutions to the stationary MHD equations. We show that the velocity field u and the magnetic field b are vanishing provided that \( u \in {BMO}^{-1}({\mathbb {R}}^3) \) and \( b\in L^{p,\infty }({\mathbb {R}}^3)\) with \(2 < p \le 6\). Moreover, we also prove that if

then u and b are vanishing for the stationary 3D MHD equations with partial viscosity and partial diffusivity.

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The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

The authors thank the reviewers for their helpful comments on the initial manuscript. The second author was partially supported by Natural Science Foundation of Jiangsu Province (BK20201478) and Qing Lan Project of Jiangsu Universities.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huizhou University, Huizhou, 516007, Guangdong, China

    Peng Wang

  2. School of Mathematics and Statistics, Huaiyin Normal University, Huaian, 223300, Jiangsu, China

    Zhengguang Guo

  3. Department of Mathematics, Wenzhou University, Wenzhou, 325035, Zhejiang, China

    Zhengguang Guo

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  1. Peng Wang
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Correspondence to Zhengguang Guo.

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Wang, P., Guo, Z. Some Liouville-Type Results for the 3D Incompressible MHD Equations. J Geom Anal 34, 39 (2024). https://doi.org/10.1007/s12220-023-01480-x

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