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

Abstract

We assume that $\Omega$ is either a smooth bounded domain in ${{\mathbb {R}}}^3$ or $\Omega ={{\mathbb {R}}}^3$, and $\Omega '$ is a sub-domain of $\Omega$. We prove that if $0\le T_1<T_2\le T\le \infty$, $({\mathbf {u}},{\mathbf {b}},p)$ is a suitable weak solution of the initial–boundary value problem for the MHD equations in $\Omega \times (0,T)$ and either ${\mathscr {F}}_{\gamma }(p_-)\in L^{\infty }(T_1,T_2;\, L^{3/2}(\Omega '))$ or ${\mathscr {F}}_{\gamma } ({\mathcal {B}}_+)\in L^{\infty }(T_1,T_2;\, L^{3/2}(\Omega '))$ for some $\gamma >0$, where ${\mathscr {F}}_{\gamma }(s)=s\, [\ln {}(1+ s)]^{1+\gamma }$, ${\mathcal {B}}= p+\frac{1}{2}|{\mathbf {u}}|^2+\frac{1}{2}|{\mathbf {b}}|^2$ 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 $\Omega '\times (T_1,T_2)$. If ${\mathbf {b}}\equiv {\mathbf {0}}$ then our result generalizes some previous known results from the theory of the Navier–Stokes equations.