Pressure representation and boundary regularity of the Navier-Stokes equations with slip boundary condition

We first represent the pressure in terms of the velocity in R-+(3). Using this representation we prove that a solution to the Navier-Stokes equations is in L-infinity(R-+(3) x (0, infinity)) under the critical assumption that u is an element of L-loc(r,r') ,3/r + 2/r'<= 1 with r >= 3, while for r = 3 the smallness is required. In [H.J. Choe, Boundary regularity of weak solutions of the Navier-Stokes equations, J. Differential Equations 149 (2) (1998) 211-247], a boundary L-infinity estimate for the solution is derived if the pressure on the boundary is bounded. In our work, we remove the boundedness assumption of the pressure. 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. (c) 2008 Elsevier Inc. All rights reserved.