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

*(English)* Zbl 1130.37404
Summary: We construct the trajectory attractor $\U0001d504$ of a three-dimensional Navier-Stokes system with exciting force $g\left(x\right)\in H$. The set $\U0001d504$ 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 $\U0001d504$. We prove that the solutions $\left\{u\right(x,t),t\u2a7e0\}\in \U0001d504$ are continuous in $t$ if they are treated in the space of functions ranging in ${H}^{-\delta}$, $0<\delta \u2a7d1$. The restriction of the trajectory attractor $\U0001d504$ to $t=0$, $\U0001d504{\text{|}}_{t=0}=:\mathcal{A}$, 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 ${\U0001d504}_{m}$ 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 $\U0001d504$ 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 $t$.

##### MSC:

37L30 | Attractors and their dimensions, Lyapunov exponents |

35B41 | Attractors (PDE) |

35Q35 | PDEs in connection with fluid mechanics |

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