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 \(t\).