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.

我们假设\(\Omega \)是\({{\mathbb {R}}}^3\)或\(\Omega ={{\mathbb {R}}}^3\) 中的光滑有界域和\（\欧米茄“\）是一个子域\（\欧米茄\） 。我们证明如果\(0\le T_1<T_2\le T\le \infty \) , \(({\mathbf {u}},{\mathbf {b}},p)\)是一个合适的弱解在\(\Omega \times (0,T)\)和\({\mathscr {F}}_{\gamma }(p_-)\in L^{ \infty }(T_1,T_2;\, L^{3/2}(\Omega '))\)或\({\mathscr {F}}_{\gamma } ({\mathcal {B}}_+ )\in L^{\infty }(T_1,T_2;\, L^{3/2}(\Omega '))\)一些\(\gamma >0\)，其中\({\mathscr {F}}_{\gamma }(s)=s\, [\ln {}(1+ s)]^{1+\gamma }\ ) , \({\mathcal {B}}= p+\frac{1}{2}|{\mathbf {u}}|^2+\frac{1}{2}|{\mathbf {b}}| ^2\)和下标“-”和“ \(+\) ”分别表示负和非负部分，那么解\(({\mathbf {u}},{\mathbf {b}}, p)\)在\(\Omega '\times (T_1,T_2)\) 中没有奇异点。如果\({\mathbf {b}}\equiv {\mathbf {0}}\)那么我们的结果概括了一些先前已知的 Navier-Stokes 方程理论结果。