Upper bounds for the attractor dimension of damped Navier-Stokes equations in $\mathbb R^2$

TL;DR

This paper establishes upper bounds on the fractal dimension of the global attractor for damped, driven 2D Navier-Stokes equations in the plane, using Sobolev space norms of the forcing term.

Contribution

It provides new upper bounds for the attractor dimension in 2D damped Navier-Stokes equations based on the forcing term's Sobolev space regularity.

Findings

Existence of a global attractor for the system.

Upper bounds depend on the Sobolev space norm of the forcing.

Results apply to a full scale of homogeneous Sobolev spaces from -1 to 1.

Abstract

We consider finite energy solutions for the damped and driven two-dimensional Navier--Stokes equations in the plane and show that the corresponding dynamical system possesses a global attractor. We obtain upper bounds for its fractal dimension when the forcing term belongs to the whole scale of homogeneous Sobolev spaces from -1 to 1