Summary: In this paper, we establish new regularity criteria for the three-dimensional (3D) viscous incompressible magnetohydrodynamic (MHD) equations. It is proved that if the solution of the MHD equations satisfies \(u_3\in L^p(0, T; L^q(\mathbb{R}^3))\), \(j_3\in L^r(0, T; L^s(\mathbb{R}^3))\), \(\frac{2}{p} + \frac{3}{q} = \frac{13}{24}\), \(\frac{72}{13} \leq q \leq \infty\); \(\frac{2}{r} + \frac{3}{s} = 2\), \(\frac{3}{2} < s \leq \infty\) or \(u_3\in L^p(0, T; L^q(\mathbb{R}^3))\), \(w_3\in L^r(0, T; L^s(\mathbb{R}^3))\), \(\frac{2}{p} + \frac{3}{q} = \frac{13}{24}\), \(\frac{72}{13} \leq q \leq \infty\); \(\frac{2}{r} + \frac{3}{s} = 2\), \(\frac{3}{2}< s \leq \infty\), then the regularity of the solution on \((0, T)\), where \(u_3\), \(j_3\) and \(\omega_3\) are the third component of velocity \(\boldsymbol{u}\), current density \(\nabla\times\boldsymbol{b}\) and vorticity \(\nabla\times\boldsymbol{u}\), respectively. These results give new improvements of regularity theory of weak solutions.
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