*(English)*Zbl 1131.35056

The authors study the asymptotic behaviour of weak solutions of the Navier-Stokes equations

considered on a smooth bounded domain ${\Omega}\subseteq {\mathbb{R}}^{3}$, and subject to homogeneous Dirichlet boundary conditions; here $A=-{P}_{{\Delta}}$ is the Stokes operator, while $B(u,u)=P\left(u\nabla \right)u$ is the nonlinearity. Eq. (1) is studied in an ${L}^{2}$-setting based on the familiar spaces $H$, $V$ [*R. Temam*, Navier-Stokes equations. Theory and numerical analysis. Amsterdam etc.: North-Holland (1979; Zbl 0426.35003)]. Following these lines, a standard definition of weak solution is given. Starting point is an unproved assumption (H) imposed on the exterior force $f\left(t\right)$:

(H) Let $f\in {L}_{\text{for}}^{2}([\tau ,\infty ),H)$; there exists a function $F(\alpha ,\beta ,\gamma )$, continuous and nondecreasing in $\alpha $ and nonincreasing in $\beta $, such that for any ${u}_{\tau}\in V$ there exists a globally defined weak solution $u\left(\phantom{\rule{4pt}{0ex}}\right)$ such that $u\left(\tau \right)={u}_{\tau}$ and

It then follows that the weak solutions $u\left(\phantom{\rule{4pt}{0ex}}\right)$ provided by assumption (H) satisfy

where

and are unique in the class of solutions which satisfy (3). The authors then prove a series of theorems, all expressing properties of weak solutions provided by (H). Based on these preparations, the authors then proceed to study a class of attractors associated with these weak solutions. They set $\tau =0$ in (1)–(3), assume that $f\in H$ in (1) is autonomous and define a set valued mapping $G$ from ${\mathbb{R}}_{+}\times H$ into the set $P\left(H\right)$ of all nonempty subsets of $H$ as follows:

(4) $G(t,{u}_{0})$ is the set of all $u\left(t\right)$, where $u\left(\phantom{\rule{4pt}{0ex}}\right)$ is a globally defined weak solution subject to (3), and with $u\left(0\right)={u}_{0}\xb7$ It is proved that $G$ is a strict multivalued semiflow, i.e.

For such semiflows the notion of global attractor can be defined along established lines. The authors prove that the semiflow $G$ has a global attractor $\mathcal{A}$, which is minimal. The case of nonautonomous $f\left(t\right)$ is more involved and requires the introduction of multivalued processes in the sense of *A. V. Babin* and *M. I. Vishik* [Attractors of evolution equations. Moscow: Nauka (1989; Zbl 0804.58003)]. The authors prove the existence of a global attractor in this more general situation. The paper concludes with a theory of attractors for multivalued processes on topological spaces, extending results obtained by *V. S. Melnik* and *J. Valero* [Set-Valued Anal. 8, No. 4, 375–403 (2000; Zbl 1063.35040)] for metric spaces.

##### MSC:

35Q30 | Stokes and Navier-Stokes equations |

37L30 | Attractors and their dimensions, Lyapunov exponents |

35B41 | Attractors (PDE) |

35K55 | Nonlinear parabolic equations |

37B25 | Lyapunov functions and stability; attractors, repellers |

58C06 | Set-valued and function-space valued mappings on manifolds |

35B40 | Asymptotic behavior of solutions of PDE |