*(English)*Zbl 1083.35094

The authors prove he existence of attractors for the 2d-Navier-Stokes equations with an exterior force $f\left(t\right)$ by using the theory of processes by Babin, Vishik and others. A process is based on a Banach space $X$, an index set ${\Sigma}$ (itself a topological vector space). A process is a collection ${U}_{\sigma}(t,\tau )$ of nonlinar operators, acting on $X$, labeled by $\sigma \in {\Sigma}$ such that

${\Sigma}$ is called the symbol space, $\sigma $ a symbol. Concepts such as uniform attractor, uniform absorbing set etc. are now introduced. E.g., ${B}_{0}\subset X$ is uniformly absorbing if given $C\in \mathbb{R}$ and a bounded set $B\subset X$ there is ${T}_{0}={T}_{0}(\tau ,B)\ge \tau $ such that

The authors now prove a number of preparatory lemmas concerning properties of processes. These results are then applied to the 2d-Navier-Stokes equation on a smooth bounded domain ${\Omega}$. To this end this equation is put into standard abstract form

based on the Hilbert spaces

The exterior force $f\left(t\right)=\sigma \left(t\right)$, $t\in \mathbb{R}$ is taken as symbol of the system (3), resp. of the induced process; one assumes

After recalling global existence and uniqueness of solutions of (3) the authors proceed to prove the existence of a uniform attractor ${A}_{0}$ of (3) and investigate its properties. The relevant Theorem 3.3 states among others (expressed somewhat losely) that if $f\left(t\right)$, $t\in \mathbb{R}$ has an additional property called “normal” then the ${A}_{0}$ associated with $f\left(t\right)$, $t\in \mathbb{R}$ coincides with the uniform attractor ${A}_{b}$ associated with $f(t+b)$, $t\in \mathbb{R}$ , for any $b\in \mathbb{R}$. Further results of this type are obtained (Theorems 4.1, 4.2).

##### MSC:

35Q30 | Stokes and Navier-Stokes equations |

35B40 | Asymptotic behavior of solutions of PDE |

35B41 | Attractors (PDE) |

37L30 | Attractors and their dimensions, Lyapunov exponents |

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