A Regularity Criterion in Terms of Pressure for the 3D Viscous MHD Equations

Abstract

In this note, we are concerned with the regularity of solutions of the MHD equation in terms of the pressure. More precisely, it is proved that if the pressure satisfies the critical growth condition $$\begin{aligned} \pi (x,t)\in L^{\frac{2}{2+r}}\left( 0,T,\overset{.}{B}_{\infty ,\infty }^{r}(\mathbb {R}^{3})\right) \end{aligned}$$ for \(-1\le r\le 1\), then the solution remains smooth on (0, T]. The finding is mainly based on the innovative function decomposition methods together with Besov space techniques. Here \(\overset{.}{B}_{\infty ,\infty }^{r}\) denotes the homogeneous Besov space.