New Regularity Criteria for Weak Solutions to the MHD Equations in Terms of an Associated Pressure

Abstract

We prove that if $0<T_0<T\le \infty$, $(\mathbf{u},\mathbf{b},p)$ is a suitable weak solution of the MHD equations in ${\mathbb{R}}^3\times (0,T)$ and either ${\mathcal{F}}_{\gamma }(p_{-})\in L^{\infty }(0,T_0;L^{3/2}({\mathbb{R}}^3))$ or ${\mathcal{F}}_{\gamma }((|\mathbf{u}{|}^2+|\mathbf{b}{|}^2+2p{)}_{+})\in L^{\infty }(0,T_0;L^{3/2}({\mathbb{R}}^3))$ for some $\gamma >0$, where ${\mathcal{F}}_{\gamma }(s)=s[\ln (1+s){]}^{1+\gamma }$ and the subscripts "$-$" and "$+$" denote the negative and the nonnegative part, respectively, then the solution $(\mathbf{u},\mathbf{b},p)$ has no singular points in ${\mathbb{R}}^3\times (0,T_0]$.