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

Abstract

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 ∞ ( R + 3 × ( 0 , ∞ ) ) under the critical assumption that u ∈ 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 ∞ 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 Hölder inequality, we find an integral estimate for L ∞ -norm of u.