On conditional regularity criteria for the 3D generalized MHD system via partial components

This paper focuses on the regularity criteria for the 3D generalized magnetohydrodynamic (MHD) equations, particularly those involving the partial components of the velocity $u_3$ and the magnetic field $(b_1,b_2,b_3)$. For any $l,i,j,k\in \{1,2,3\}$, we establish regularity criteria via ${\mathrm{\partial }}_l{u}_3$ and $({\mathrm{\partial }}_i{b}_1,{\mathrm{\partial }}_j{b}_2,{\mathrm{\partial }}_k{b}_3)$, $u_3$ and $({\mathrm{\partial }}_i{b}_1,{\mathrm{\partial }}_j{b}_2,{\mathrm{\partial }}_k{b}_3)$, or ${\mathrm{\partial }}_l{u}_3$ and $(b_1,b_2,b_3)$. Furthermore, a comprehensive regularity condition is obtained via $(\xi u_3,\eta b_1,\zeta b_2,\sigma b_3)$ and $(\xi {}^{\prime }{\mathrm{\partial }}_l{u}_3,\eta {}^{\prime }{\mathrm{\partial }}_i{b}_1,\zeta {}^{\prime }{\mathrm{\partial }}_j{b}_2,\sigma {}^{\prime }{\mathrm{\partial }}_k{b}_3)$ with $\xi +\xi {}^{\prime }=\eta +\eta {}^{\prime }=\zeta +\zeta {}^{\prime }=\sigma +\sigma {}^{\prime }=1$ and $\xi ,\eta ,\zeta ,\sigma ,\xi {}^{\prime },\eta {}^{\prime },\zeta {}^{\prime },\sigma {}^{\prime }\in \{0,1\}$.