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Blow-Up Criterion of the 3D Magnetic Bénard Equations via the Gradient of Pressure

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Abstract

In this paper, we address the problem raised by Q. Liu (Appl. Math. Lett. 104: 106255, 2020) to successfully establish a Serrin-type blow-up criterion for local smooth solutions to the 3D magnetic Bénard equations in terms of the gradient of pressure. More precisely, we prove that if the gradient of pressure \(\nabla P\) satisfies

$$ \nabla P\in L^{\frac{2}{3-r}}(0,T; L^{\frac{3}{r}}(\mathbb {R}^{3}))\quad \text { with }~0<r\le 1, $$

then the corresponding solution \((u,b,\theta )\) to the 3D magnetic Bénard equations can be extended beyond the time \(t= T\).

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Acknowledgements

This work does not have any conflicts of interest. The author would like to acknowledge his great gratitude to the referees for their valuable comments and suggestions. This paper is supported by the National Natural Science Foundation of China (nos. 12361034 and 11961030) and the Natural Science Foundation of Shaanxi Province (no. 2022JM-034).

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  1. School of Mathematics and Information Science, Baoji University of Arts and Sciences, Baoji, Shaanxi, 721013, China

    Jihong Zhao

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Zhao, J. Blow-Up Criterion of the 3D Magnetic Bénard Equations via the Gradient of Pressure. Vietnam J. Math. (2024). https://doi.org/10.1007/s10013-024-00691-1

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