  We study regularity criteria for the $d$-dimensional incompressible
Navier-Stokes equations. We prove if $u\in L_{\infty}^tL_d^x((0,T)\times
\mathbb{R}^d_+)$ is a Leray-Hopf weak solution vanishing on the boundary and
the pressure $p$ satisfies a local condition $\|p\|_{L_{2-1/d}(Q(z_0,1)\cap
(0,T)\times \mathbb{R}^d_+)}\leq K$ for some constant $K>0$ uniformly in $z_0$,
then $u$ is regular up to the boundary in $(0,T)\times \mathbb{R}^d_+$.
Furthermore, when $T=\infty$, $u$ tends to zero as $t\rightarrow \infty$. We
also study the local problem in half unit cylinder $Q^+$ and prove that if
$u\in L^t_{\infty}L^x_d(Q^+)$ and $ p\in L_{2-1/d}(Q^+)$, then $u$ is
H\"{o}lder continuous in the closure of the set $Q^+(1/4)$. This generalizes a
result by Escauriaza, Seregin, and \v{S}ver\'{a}k to higher dimensions and
domains with boundary.
