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 is an element of (L infinity Ldx)-L-t((0,T) x R-+(d)) is a Leray-Hopf weak solution vanishing on the boundary, then u is regular up to the boundary in (0,T) x R-+(d). Furthermore, with a stronger uniform local condition on the pressure p, we prove u is unique and tends to zero as t -> infinity if T = infinity. This generalizes a result by Escauriaza, Seregin, and Sverak [14] to higher dimensions and domains with boundary. We also study the local problem in half unit cylinder Q(+) and prove that if u is an element of (L infinity Ldx)-L-t(Q(+))and p is an element of L2-1/d(Q(+)), then u is Holder continuous in the closure of the set Q(+) (1/4).