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Regularity criteria for the 3D MHD equations via partial derivatives

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  • In this paper, we establish two regularity criteria for the 3D MHD equations in terms of partial derivatives of the velocity field or the pressure. It is proved that if $\partial_3 u \in L^\beta(0,T; L^\alpha(\mathbb{R}^3)),~\mbox{with}~ \frac{2}{\beta}+\frac{3}{\alpha}\leq\frac{3(\alpha+2)}{4\alpha},~\alpha>2$, or $\nabla_h P \in L^\beta(0,T; L^{\alpha}(\mathbb{R}^3)),~\mbox{with}~\frac{2}{\beta}+\frac{3}{\alpha}< 3,~\alpha>\frac{9}{7},~\beta\geq 1$, then the weak solution $(u,b)$ is regular on $[0, T]$.
    Mathematics Subject Classification: Primary: 35Q35, 35B65; Secondary: 76D05.

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