Inertial manifolds for the incompressible Navier-Stokes equations

TL;DR

This paper proves the existence of finite-dimensional inertial manifolds for the Navier-Stokes equations in two and three dimensions, using spatial averaging methods, advancing understanding of long-term dynamics in fluid flows.

Contribution

It constructs inertial manifolds for 2D Navier-Stokes and extends the spatial averaging method to hyperviscous 3D cases, including a specific hyperviscous index.

Findings

Existence of N-dimensional inertial manifold in 2D case.

Extension of spatial averaging method to abstract hyperviscous Navier-Stokes.

Verification of inertial manifold existence for hyperviscous 3D Navier-Stokes with index 5/4.

Abstract

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 $\mathbb{T}^{3}$.