Abstract
This work establishes a new regularity criterion for the 3D incompressible MHD equations in term of one directional derivative of the pressure (i.e., \(\partial _{3}P\)) on framework of the anisotropic Lebesgue spaces. More precisely, it is proved that for \(T>0\), if \(\partial _{3}P\in L^{\beta }(0,T; L^{\alpha }(\mathbb {R}^{2}_{x_{1}x_{2}};L^{\gamma }(\mathbb {R}_{x_{3}})))\) with \(\frac{2}{\beta }+\frac{1}{\gamma }+\frac{2}{\alpha }=k\in [2,3)\) and \(\frac{3}{k}\le \gamma \le \alpha \le \frac{1}{k-2},\) then the corresponding solution (u, b) to the 3D MHD equations is regular on [0, T].
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Acknowledgements
The author would like to thank to his supervisor Prof. Song Jiang for many helpful suggestions. He also would like to acknowledge his sincere thanks to the editor and the referees for a careful reading of the work and many valuable comments and suggestions.
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This work is partially supported by the National Natural Science Foundation of China (11401202).
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Liu, Q. On Regularity for the 3D MHD Equations via One Directional Derivative of the Pressure. Bull Braz Math Soc, New Series 51, 157–167 (2020). https://doi.org/10.1007/s00574-019-00148-x
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DOI: https://doi.org/10.1007/s00574-019-00148-x