On the Regularity of Navier-Stokes Equations in Critical Space

This paper focuses on the regularity of the Navier-Stokes equations in critical space. Let $u(x,t)$ and $p(x,t)$ denote suitable weak solution of the Navier-Stokes equations in $Q_T=\mathbb{R}^3\times(-T, 0)$. We prove that if $u(x,t)$ is in the scaling invariant spaces $L_t^{\infty}L_{x_3}^{p_1}L_{x_h}^{p_2}(Q_T)$ , where $\frac{1}{p_1}+\frac{2}{p_2}=1$ , $p_1\geq 2$ and $x_h = (x_1, x_2)$ , then $u$ is a smooth solution in $Q_T$ and doesn't blow up at $t = 0$. In particular, if $u(x,t) \in L_t^{\infty}L_{x_3}^{\infty}L_{x_h}^{2}(Q_T)$, then $u(x,t)$ is a smooth solution in $Q_T$ and regular up to $t = 0$.