Regularity criteria in weak $L^3$ for 3D incompressible Navier-Stokes equations

We study the regularity of a distributional solution $(u,p)$ of the 3D incompressible evolution Navier-Stokes equations. Let $B_r$ denote concentric balls in $\mathbb{R}^3$ with radius $r$. We will show that if $p\in L^{m} (0,1; L^1(B_2))$, $m>2$, and if $u$ is sufficiently small in $L^{\infty} (0,1; L^{3,\infty}(B_2))$, without any assumption on its gradient, then $u$ is bounded in $B_1\times (\frac{1}{10},1)$. It is an endpoint case of the usual Serrin-type regularity criteria, and extends the steady-state result of Kim-Kozono to the time dependent setting. In the appendix we also show some nonendpoint borderline regularity criteria.