The interior regularity of pressure associated with a weak solution to the Navier-Stokes equations with the Navier-type boundary conditions

We prove that if u is a weak solution to the Navier-Stokes system with the Navier-type boundary conditions in Omega x (0, T), satisfying the strong energy inequality in Omega x (0, T) and Serrin's integrability conditions in Omega' x (t(1), t(2)) (where Omega' is a sub-domain of Omega and 0 <= t(1) < t(2) <= T) then p and partial derivative(t)u have spatial derivatives of all orders essentially bounded in Omega '' x (t(1) + epsilon, t(2) - epsilon) for any bounded sub-domain Omega '' subset of (Omega '') over bar subset of Omega' and epsilon > 0 so small that t(1) + epsilon < t(2) - epsilon. (See Theorem 1.) We show an application of Theorem 1 to the procedure of localization. (C) 2018 Elsevier Inc. All rights reserved.