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

Abstract

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 $\{ u(x,t),t \geqslant 0\} \in \mathfrak{A}$ are continuous in t if they are treated in the space of functions ranging in $H^{ - \delta } ,0 < \delta \leqslant 1$. The restriction of the trajectory attractor $\mathfrak{A}$ to $t = 0$, $\mathfrak{A}{\text{|}}_{t = 0} = :\mathcal{A}$, is called the global attractor of the Navier--Stokes system. We prove that the global attractor $\mathcal{A}$ 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 $\mathcal{A}_m$ of the $m$-order Galerkin approximations of the Navier--Stokes system converge to the trajectory and global attractors $\mathfrak{A}$ and $\mathcal{A}$, 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 $t$.