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

Abstract

We prove that if $0<T_0<T\leq\infty$, $(\mathbf{u},\mathbf{b},p)$ 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 $\mathbb{R}^3\times(0,T_0]$.