Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces

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, then $ u $ is regular up to the boundary in $ (0,T)\times {\mathbb{R}}^d_+ $. Furthermore, with a stronger uniform local condition on the pressure $ p $, we prove $ u $ is unique and tends to zero as $ t\rightarrow \infty $ if $ T = \infty $. This generalizes a result by Escauriaza, Seregin, and Šverák [14] to higher dimensions and domains with boundary. 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 Hölder continuous in the closure of the set $ Q^+(1/4) $.