A Note on Regularity Criteria in Terms of Pressure for the 3D Viscous MHD Equations

This note is devoted to the study of the smoothness of weak solutions to the Cauchy problem for three-dimensional magneto-hydrodynamic system in terms of the pressure. It is proved that if the pressure $\pi$ belongs to $L^2(0,T,\dot B_{\infty,\infty}^{-1}(\mathbb R^3))$ or the gradient field of pressure $\nabla\pi$ belongs to $L^{2/3}(0,T,\mathrm{BMO}(\mathbb R^3))$, then the corresponding weak solution $(u,b)$ remains smooth on $[0,T]$.