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Remarks on Liouville-Type Theorems for the Steady MHD and Hall-MHD Equations

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Abstract

In this note, we investigate Liouville-type theorems for the steady three-dimensional MHD and Hall-MHD equations and show that the velocity field u and the magnetic field B are vanishing provided that \(B\in L^{6,\infty }(\mathbb {R}^3)\) and \(u\in BMO^{-1}(\mathbb {R}^3)\), which state that the velocity field plays an important role. Moreover, the similar result holds in the case of partial viscosity or diffusivity for the three-dimensional MHD equations.

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Acknowledgements

W. Wang was supported by NSFC under Grant 12071054 and 11671067.

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Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

    Xiaomeng Chen, Shuai Li & Wendong Wang

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  1. Xiaomeng Chen
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  2. Shuai Li
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  3. Wendong Wang
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Correspondence to Wendong Wang.

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Communicated by David Nicholls.

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Chen, X., Li, S. & Wang, W. Remarks on Liouville-Type Theorems for the Steady MHD and Hall-MHD Equations. J Nonlinear Sci 32, 12 (2022). https://doi.org/10.1007/s00332-021-09768-4

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