In this paper we establish a new regularity criterion for the 3D incompressible MHD equations via ${\partial }_3{u}_3$ and the magnetic field $b$. By considering different weights in spatial variables, we show in anisotropic Lebesgue spaces if ${\partial }_3{u}_3$ and $b$ satisfy certain space–time integrable conditions, which are almost optimal from the scaling invariant point of view, then a weak solution $(u,b)$ is actually regular. This result gives new insights into the understanding of regularity theory of weak solutions.

中文翻译：

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基于各向异性Lebesgue空间中一个速度分量的3D MHD方程的正则性

在本文中，我们通过以下方法为3D不可压缩MHD方程建立了新的正则性准则 ${\partial }_3{ü}_3$ 和磁场 $b$。通过考虑空间变量中的不同权重，我们证明了在各向异性Lebesgue空间中，如果${\partial }_3{ü}_3$ 和 $b$ 满足某些时空可积分条件，从定标不变性的角度来看，这些条件几乎是最佳的，然后是一个弱解 $（ü，b）$实际上是正常的。该结果为了解弱解的正则性理论提供了新的见解。