On endpoint regularity criterion of the 3D Navier–Stokes equations

Let $(u, \pi)$ with $u=(u_1,u_2,u_3)$ be a suitable weak solution of the three dimensional Navier-Stokes equations in $\mathbb{R}^3\times [0, T]$. Denote by $\dot{\mathcal{B}}^{-1}_{\infty,\infty}$ the closure of $C_0^\infty$ in $\dot{B}^{-1}_{\infty,\infty}$. We prove that if $u\in L^\infty(0, T; \dot{B}^{-1}_{\infty,\infty})$, $u(x, T)\in \dot{\mathcal{B}}^{-1}_{\infty,\infty})$, and $u_3\in L^\infty(0, T; L^{3, \infty})$ or $u_3\in L^\infty(0, T; \dot{B}^{-1+3/p}_{p, q})$ with $3