In this paper, we study the regularity criterion of weak solutions to the three-dimensional (3D) MHD equations. It is proved that the solution (u, b) becomes regular provided that one velocity and one current density component of the solution satisfy

$$\begin{aligned} u_{3}\in L^{\frac{30\alpha }{7\alpha -45}}\left( 0,T;L^{\alpha ,\infty }\left( {\mathbb {R}}^{3}\right) \right) \quad \text { with }\frac{45}{7} \le \alpha \le \infty , \end{aligned}$$(0.1)

and

$$\begin{aligned} j_{3}\in L^{\frac{2\beta }{2\beta -3}}\left( 0,T;L^{\beta ,\infty }\left( {\mathbb {R}}^{3}\right) \right) \quad \text {with }\frac{3}{2}\le \beta \le \infty , \end{aligned}$$(0.2)

which generalize some known results.

中文翻译：

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Lorentz空间中涉及一个速度和一个电流密度分量的3D MHD方程的正则性准则

在本文中，我们研究了三维（3D）MHD方程的弱解的正则性准则。证明只要溶液的一个速度和一个电流密度分量满足，溶液（u，  b）就成为正则。

$$ \ begin {aligned} u_ {3} \ in L ^ {\ frac {30 \ alpha} {7 \ alpha -45}} \ left（0，T; L ^ {\ alpha，\ infty} \ left（ {\ mathbb {R}} ^ {3} \ right）\ right）\ quad \ text {与} \ frac {45} {7} \ le \ alpha \ le \ infty，\ end {aligned} $$（0.1 ）

和

$$ \ begin {aligned} j_ {3} \ in L ^ {\ frac {2 \ beta} {2 \ beta -3}} \ left（0，T; L ^ {\ beta，\ infty} \ left（ {\ mathbb {R}} ^ {3} \ right）\ right）\ quad \ text {with} \ frac {3} {2} \ le \ beta \ le \ infty，\ end {aligned} $$（0.2 ）

归纳出一些已知结果。