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Liouville-Type Theorems for the 3D Stationary MHD Equations

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In this paper, we consider the Liouville-type theorems for the 3D stationary incompressible MHD equations. Using the Caccioppoli type estimate, we proved the smooth solutions (ub) are identically equal to zero when \((u,b)\in L^{p}({\mathbb {R}}^{3}),\ p\in (\frac{3}{2},3).\) In addition, under an additional assumption in the setting of the Sobolev space of negative order \(\dot{H}^{-1}({\mathbb {R}}^{3}),\) we can extend the index \(p\in (3,+\infty ).\) In fact, our results combine with the result of Yuan and Xiao (J Math Anal Appl 491(2):124343, 2020) that \(p\in [2,\frac{9}{2}],\) which implies a very intriguing and novel result for the 3D stationary MHD equations with \( p\in (\frac{3}{2},+\infty ).\)

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This work was supported by Anhui Education Bureau under Grant No. KJ2019A0556 and Scientific Research Project for the Excellent Youth Scholars of Higher Education of Anhui Province under Grant No. 2023AH030073.

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Zhang Provides the main idea and write part of the content; Zu wrote part of the manuscript and correct some errors in article. All authors reviewed the manuscript.

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Correspondence to Hui Zhang.

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Zhang, H., Zu, Q. Liouville-Type Theorems for the 3D Stationary MHD Equations. Mediterr. J. Math. 21, 132 (2024).

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