High-order Galerkin convergence and boundary characteristics of the 3-D Navier-Stokes equations on intervals of regularity

We consider the 3-D Navier-Stokes equations (NSE) on a bounded domain Ω⊂R3 with zero boundary data. Let P denote the Leray projection and let A=−PΔ denote the Stokes operator. For a natural number θ≥2 assume that the forcing data satisfies f∈C([0,T],D(A(θ−1)/2)), and let [0,T] be an interval over which the H1-norms of Galerkin solutions um are uniformly bounded. Then on any subinterval [τ,T] we show that supt∈[τ,T]‖Aθ/2(um(t)−u(t))‖2→0 as m→∞ where u is the unique regular strong solution of the NSE on [0,T]. From this convergence result we show that if f∈C([0,T];D(A(θ+1)/2)) then Aθ/2u(t)=0 on the boundary Γ of Ω for any t>0. When θ=2 applications of this boundary result include boundary data specification for the NSE pressure within the framework developed in [19,20], and corroboration for the choice of boundary values in [22] for the NS-α equation.