Boundary regularity of weak solutions of the Navier-Stokes equations

Abstract

We prove that a solution to Navier-Stokes equations is in L-2(0, infinity: H-2(Ohm)) under the critical assumption that u is an element of L-r,L-r', 3/r + 2/r' less than or equal to 1 with r greater than or equal to 3. A boundary L-infinity estimate For the solution is derived if the pressure on the boundary is bounded. Here our estimate is local. Indeed, employing Moser type iteration and the reverse Holder inequality, we find an integral estimate for L-infinity-norm of u. Moreover the solution is C-1,C-alpha continuous up to boundary if the tangential derivatives of the pressure on the boundary are bounded. Then, from the bootstrap argument a local higher regularity theorem follows, that is, the velocity is as regular as the boundary data of the pressure. (C) 1998 Academic Press.