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

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.