# Long-time behavior of the three dimensional globally modified Navier-Stokes equations

## TL;DR

This paper studies the long-term dynamics of solutions to the three-dimensional globally modified Navier-Stokes equations, proving the existence of a finite-dimensional global attractor and demonstrating the solutions' eventual regularity.

## Contribution

It establishes the existence and regularity of a global attractor for the 3D globally modified Navier-Stokes equations and constructs an exponential attractor with finite fractal dimension.

## Key findings

- Existence of a global attractor in H
- Solutions become more regular over time
- Finite fractal dimension of the attractor

## Abstract

This paper is concerned with the long-time behavior of solutions for the three dimensional globally modified Navier-Stokes equations in a three-dimensional bounded domain. We prove the existence of a global attractor $\mathcal{A}_0$ in $H$ and investigate the regularity of the global attractors by proving that $\mathcal{A}_0=\mathcal{A}$ established in \cite{ct}, which implies the asymptotic smoothing effect of solutions for the three dimensional globally modified Navier-Stokes equations in the sense that the solutions will eventually become more regular than the initial data. Furthermore, we construct an exponential attractor in $H$ by verifying the smooth property of the difference of two solutions, which entails the fractal dimension of the global attractor is finite.