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