*(English)*Zbl 1111.35042

The authors consider the inhomogeneous Navier-Stokes equations

on a bounded Lipschitz domain ${\Omega}$ in ${\mathbb{R}}^{2}$. One assumes

where $E=\text{dom}\left({A}^{\frac{1}{4}}\right)$, with $A=-{P}_{{\Delta}}$ the Stokes operator associated with (1). The aim is to prove the existence of a global attractor for (1). To do so, the authors need several preparatory steps. First, using a suitable background flow $\psi $, eq. (1) is transformed into a new one, based on Dirichlet boundary conditions, i.e.:

where $F$ is an additional force term induced by the background flow $\psi $. In order to prove the existence of an attractor for (3), the authors have to rely on work of *V. V. Chepyzhov* and *M. I. Vishik* [Am. Math. Soc. Colloq. Publ. 49, 363 p. (2002; Zbl 0986.35001)]; they introduce a number of notions and discuss their properties. Thus one has the notion of indexed process $\{{U}_{\sigma}(t,\tau )\mid t\ge \tau ,\phantom{\rule{4pt}{0ex}}\tau \in \mathbb{R},\sigma \in {\Sigma}\}$ where ${\Sigma}$ is the index space (a metric space), $\sigma $ the symbol of the process and $\left\{{U}_{\sigma}(t,\tau )\right\}$ a family of mappings on a Banach space $E$ such that

In terms of this notion, the relevant topological concepts such as absorbing set, $\omega $-limit set, uniform attractor etc. are introduced, and some of their properties summarized. Criteria (Thms. 4.1, 4.2) for the existence of a uniform attractor are given. Finally, the index space is made precise: it is based on the translates $\left({T}_{h}f\right)\left(s\right)=f(h+s)$ induced by the exterior force $f$ in (1) resp. (3). In the main section 6 the existence of a uniform attractor in the sense of Chepyzhov and Vishik (loc. cit.) is proved. First, it is noted that existence of global solutions of (3) is guaranteed by a Galerkin method; for details the reader is referred to *R. M. Brown, P. A. Perry*, and *Zh. Shen* [Indiana Univ. Math. J. 49, 81–112 (2000; Zbl 0969.35105)] where a proof in a comparable situation is given. Then one proceeds to the proof of the main Theorem 6.1 which asserts the existence of a uniform attractor for (3). The proof involves lengthy estimates, based in part on the paper of Brown, Perry, Shen (loc. cit.). Theorem 6.2 finally asserts that if $f(x,s)$ is translation compact in $D\left({A}^{-\frac{1}{4}}\right)$ then the attractor in question is compact.

##### MSC:

35Q30 | Stokes and Navier-Stokes equations |

37L30 | Attractors and their dimensions, Lyapunov exponents |

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