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 $L_t^{\mathrm{\infty }}{L}_{x_3}^{p_1}{L}_{x_h}^{p_2}(Q_T)$ , where $\frac{1}{p_1}+\frac{2}{p_2}=1$ , $p_1\ge 2$ 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 $u(x,t)\in L_t^{\mathrm{\infty }}{L}_{x_3}^{\mathrm{\infty }}{L}_{x_h}^2(Q_T)$, then $u(x,t)$ is a smooth solution in $Q_T$ and regular up to $t=0$.