Trajectory and Global Attractors of Three-Dimensional Navier–Stokes Systems

We construct the trajectory attractor $\mathfrak A$ of a three-dimensional Navier–Stokes system with exciting force $g(x)\in H$. The set $\mathfrak A$ consists of a class of solutions to this system which are bounded in $H$, defined on the positive semi-infinite interval $\mathbb R_+$ of the time axis, and can be extended to the entire time axis $\mathbb R$ so that they still remain bounded-in-$H$ solutions of the Navier–Stokes system. In this case any family of bounded-in-$L_\infty (\mathbb R_+;H)$ solutions of this system comes arbitrary close to the trajectory attractor $\mathfrak A$. We prove that the solutions $\{u(x,t), t\ge 0\}$ are continuous in $t$ if they are treated in the space of functions ranging in $H^{-\delta }$, $0<\delta \le 1$. The restriction of the trajectory attractor $\mathfrak A$ to $t=0$, $\mathfrak A|_{t=0}=:\mathscr A$, is called the global attractor of the Navier–Stokes system. We prove that the global attractor $\mathscr A$ thus defined possesses properties typical of well-known global attractors of evolution equations. We also prove that as $m\to \infty $ the trajectory attractors $\mathfrak A_m$ and the global attractors $\mathscr A_m$ of the $m$-order Galerkin approximations of the Navier–Stokes system converge to the trajectory and global attractors $\mathfrak A$ and $\mathscr A$, respectively. Similar problems are studied for the cases of an exciting force of the form $g=g(x,t)$ depending on time $t$ and of an external force $g$ rapidly oscillating with respect to the spatial variables or with respect to time $t$.