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<p>This paper is devoted to investigating the Liouville-type theorems for the 3D stationary fractional MHD system in anisotropic Lebesgue spaces. We proved that the solution is trivial under certain anisotropic integrability conditions on the components of the velocity and magnetic fields. A Liouville theorem involving only the velocity field was also obtained. Our results are improvements of several previous results, such as the result of Luo and Yin (2017, Arch. Rational. Mech. Anal., <b>224</b>, 209–231, 2017), the result of Phan (Dyn. Partial Differ. Equ., <b>17</b>, 229–243, 2020) and the results of Chae (Appl. Math. Lett., <b>142</b>, 108655, 2023) on stationary Navier–Stokes system, the result of Wang and Xiao (J. Math. Fluid Mech, <b>20</b>, 485–498, 2018), the result of Yang (J. Math. Fluid Mech., <b>24</b>:81, 2022) and the result of Chamorro and Poggi (Publ. Mat. <b>69</b>, 27–43, 2025) on stationary fractional Navier–Stokes, the result of Fan and Wang (Dyn. Partial Differ. Equ., <b>18</b>, 327–340, 2021) and the result of Chae and Weng (Discret. Contin. Dyn. Syst., <b>36</b>, 5267–5285, 2016) on stationary MHD system.</p>

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On Liouville-type theorems for the 3D stationary fractional MHD system in anisotropic Lebesgue spaces

  • 期刊论文

  • 发表日期：

    2025年8月20日

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  • Yong Zeng

摘要

This paper is devoted to investigating the Liouville-type theorems for the 3D stationary fractional MHD system in anisotropic Lebesgue spaces. We proved that the solution is trivial under certain anisotropic integrability conditions on the components of the velocity and magnetic fields. A Liouville theorem involving only the velocity field was also obtained. Our results are improvements of several previous results, such as the result of Luo and Yin (2017, Arch. Rational. Mech. Anal., 224, 209–231, 2017), the result of Phan (Dyn. Partial Differ. Equ., 17, 229–243, 2020) and the results of Chae (Appl. Math. Lett., 142, 108655, 2023) on stationary Navier–Stokes system, the result of Wang and Xiao (J. Math. Fluid Mech, 20, 485–498, 2018), the result of Yang (J. Math. Fluid Mech., 24:81, 2022) and the result of Chamorro and Poggi (Publ. Mat. 69, 27–43, 2025) on stationary fractional Navier–Stokes, the result of Fan and Wang (Dyn. Partial Differ. Equ., 18, 327–340, 2021) and the result of Chae and Weng (Discret. Contin. Dyn. Syst., 36, 5267–5285, 2016) on stationary MHD system.