We show that a suitable weak solution to the incompressible Navier–Stokes equations on ${\mathbb{R}}^3\times (-1,1)$ is regular on ${\mathbb{R}}^3\times (-1,0]$ if ${\partial }_{3}u$ belongs to $M^{2p∕(2p-3),\alpha }((-1,0);L^p\left({\mathbb{R}}^3\right))$ for any $\alpha >1$ and $p\in (3∕2,\infty )$, which is a logarithmic-type variation of a Morrey space in time. For each $\alpha >1$ this space is, up to a logarithm, critical with respect to the scaling of the equations, and contains all spaces $L^q((-1,0);L^p\left({\mathbb{R}}^3\right))$ that are subcritical, that is for which $2∕q+3∕p<2$.

中文翻译：

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整个指数范围内 3D 不可压缩 Navier-Stokes 方程的各向异性规则条件

我们证明了不可压缩 Navier-Stokes 方程的一个合适的弱解 ${\mathbb{电阻}}^3\times (-1,1)$ 经常在 ${\mathbb{电阻}}^3\times (-1,0]$ 如果 ${\partial }_3你$ 属于 ${米}^{2磷∕(2磷-3),\alpha }((-1,0);{升}^{磷}\left({\mathbb{电阻}}^3\right))$ 对于任何 $\alpha >1$ 和 $磷\in (3∕2,\infty )$，这是莫雷空间在时间上的对数型变体。对于每个$\alpha >1$ 这个空间对于方程的缩放至关重要，最多为对数，并且包含所有空间 ${升}^q((-1,0);{升}^{磷}\left({\mathbb{电阻}}^3\right))$ 是次临界的，也就是说 $2∕q+3∕磷<2$.