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

Abstract

We prove that if u is a weak solution to the Navier–Stokes system with the Navier-type boundary conditions in Ω×(0,T), satisfying the strong energy inequality in Ω×(0,T) and Serrin's integrability conditions in Ω′×(t1,t2) (where Ω′ is a sub-domain of Ω and 0≤t1<t2≤T) then p and ∂tu have spatial derivatives of all orders essentially bounded in Ω″×(t1+ϵ,t2−ϵ) for any bounded sub-domain Ω″⊂Ω″‾⊂Ω′ and ϵ>0 so small that t1+ϵ<t2−ϵ. (See Theorem 1.) We show an application of Theorem 1 to the procedure of localization.