Inertial manifolds for the incompressible Navier-Stokes equations

In this article, we devote to the existence of an $N$-dimensional inertial manifold for the incompressible Navier-Stokes equations in $\mathbb{T}^{d}$ ($d=2,3$). Our results can be summarized as two aspects: Firstly, we construct an $N$-dimensional inertial manifold for the Navier-Stokes equations in $\mathbb{T}^{2}$; Secondly, we extend slightly the spatial averaging method to the abstract case: $\partial_{t}u+A^{1+\alpha}u+A^{\alpha}F(u)=f$ (here $0<\alpha<1$, $A>0$ is a self-adjoint operator with compact inverse and $F$ is Lipschitz from a Hilbert space $\mathbb{H}$ to $\mathbb{H}$), and then verify the existence of an $N$-dimensional inertial manifold for the hyperviscous Navier-Stokes equation with the hyperviscous index $5/4$ in $\mathbb{T}^{3}$.