Remarks on the Liouville type theorems for the 3D stationary MHD equations

Peng Wang; Shidi Zhou

doi:10.3934/cpaa.2023018

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2023, Volume 22, Issue 4: 996-1008. Doi: 10.3934/cpaa.2023018

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Limits of Spaces and Structures

Remarks on the Liouville type theorems for the 3D stationary MHD equations

Peng Wang^1, , and
Shidi Zhou^2,

1.
School of Mathematics and Statistics, Huizhou University, China
2.
School of Science, Jiangsu University of Science and Technology, China

^*Corresponding author: Shidi Zhou
^*Corresponding author: Shidi Zhou

Received: July 2022

Revised: December 2022

Early access: January 2023

Published: April 2023

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Abstract

In this paper we study the stationary incompressible MHD equations on $ {\mathbb{R}}^3 $. We show that the velocity field $ u $ and the magnetic field $ b $, which vanish at infinity, must be identically zero, provided that $ \nabla u $ and $ \nabla b $ belong to $ L^q ({{\Bbb R}}^3) $ for some $ q \in (\frac{3}{2}, 2] $, and $ u $ belongs to $ \operatorname{BMO}^{-1}({{\Bbb R}}^3) $. Moreover, motivated by the recent work [19], we also obtain the Liouville type theorem by assuming that $ (u, b) $ is axisymmetric and $ u $ satisfies some decay property.

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Mathematics Subject Classification: Primary: 35Q30, 35B53; Secondary: 76D05.

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References

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