*(English)*Zbl 1075.35037

The author considers the Navier-Stokes equations

with ${\Omega}$ smooth and bounded, and $f$ a constant exterior force. One imposes on (1) a standard Hilbert space setting:

$H$ is the ${L}^{2}$-closure of ${V}_{0}=\{u/u\in {C}_{0}{\left({\Omega}\right)}^{3},\text{div}\left(u\right)=0\}$, while $V$ is the closure of ${V}_{0}$ with respect to the norm

A weak solution of (1) is an element $u$ in ${L}^{\infty}(0,\infty ;H)\cap {L}^{2}(0,T;V)$ (for all $T$) subject to three classical conditions, one requiring that $u$ satisfies a standard variational form of (1). The set $W$ of weak solutions is considered as a metric space under the norm

It was proved by *G. R. Sell* [J. Dyn. Differ. Eq. 8, No. 1, 1–33 (1996; Zbl 0855.35100)] that the semiflow ${S}_{t}$ on $W$ given by

admits a global attractor $A$. The author now takes a generalization of this situation given by *J. M. Ball* [J. Nonlinear Sci. 7, No. 5, 475–502 (1997; Zbl 0903.58020)] as starting point who introduced the notion of generalized semiflow $G$ on a metric space $X$. Since this notion is not directly applicable to the space $W$, the author defines a notion of $\epsilon $-samples as follows. With $u|\epsilon $ the restriction of $u\in {L}^{\infty}(0,\infty ;H)$ to $[0,\epsilon ]$ one sets:

and with $u\in W$ one associates ${\phi}_{u}^{\epsilon}:[0,\infty ]\to W$ as follows:

One then sets:

The main results then are:

(I) ${G}_{\epsilon}$ is a generalized semiflow on ${W}_{\epsilon}$,

(II) with $A$ the attractor for ${S}_{t}$ and ${A}_{\epsilon}=\{u\mid \epsilon \mid u\in A\}$, ${A}_{\epsilon}$ is a global attractor for ${G}_{\epsilon}$.

The author extends the above theory to a familiy of objects called “germs”, which are a kind of infinitesimal $\epsilon $-samples $(\epsilon =0)$.