Inertial manifolds for the hyperviscous Navier-Stokes equations

We prove the existence of inertial manifolds for the incompressible hyperviscous Navier-Stokes equations on the two or three-dimensional torus:{ut+ν(−Δ)βu+(u⋅∇)u+∇p=f,(t,x)∈R+×Td,divu=0, where d=2 or 3 and β≥3/2. Since the spectral gap condition is not necessarily satisfied for the aforementioned problem in three dimensions, we employ the spatial averaging method introduced by Mallet-Paret and Sell in [26].