Trajectory and global attractors of the three-dimensional Navier-Stokes system.

*(English. Russian original)* Zbl 1130.37404
Summary: We construct the trajectory attractor $\frak{A}$ of a three-dimensional Navier-Stokes system with exciting force $g(x) \in H$. The set $\frak{A}$ consists of a class of solutions to this system which are bounded in $H$, defined on the positive semi-infinite interval $\Bbb{R}_ + $ of the time axis, and can be extended to the entire time axis $\Bbb{R}$ so that they still remain bounded-in-$H$ solutions of the Navier-Stokes system. In this case any family of bounded-in-$L_\infty (\Bbb{R}_ + ;H)$ solutions of this system comes arbitrary close to the trajectory attractor $\frak{A}$. We prove that the solutions $\{u(x,t),t \geqslant 0\} \in \frak{A}$ 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 $\frak{A}$ to $t = 0$, $\frak{A}{\text{|}}_{t = 0} = :\cal{A}$, is called the global attractor of the Navier-Stokes system. We prove that the global attractor $\cal{A}$ thus defined possesses properties typical of well-known global attractors of evolution equations. We also prove that as $m\to\infty$ the trajectory attractors $\frak{A}_m $ and the global attractors $\cal{A}_m $ of the $m$-order Galerkin approximations of the Navier-Stokes system converge to the trajectory and global attractors $\frak{A}$ and $\cal{A}$, 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 $t$.

##### MSC:

37L30 | Attractors and their dimensions, Lyapunov exponents |

35B41 | Attractors (PDE) |

35Q35 | PDEs in connection with fluid mechanics |

76D05 | Navier-Stokes equations (fluid dynamics) |