Boundary Regularity of Weak Solutions of the Navier–Stokes Equations

We prove that a solution to Navier–Stokes equations is inL2(0, ∞: H2(Ω)) under the critical assumption thatu Lr, r′, 3/r+2/r′ 1 withr 3. A boundaryL∞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 Hölder inequality, we find an integral estimate forL∞-norm ofu. Moreover the solution isC1, αcontinuous 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.