The author considers the Navier-Stokes equations
with smooth and bounded, and a constant exterior force. One imposes on (1) a standard Hilbert space setting:
is the -closure of , while is the closure of with respect to the norm
A weak solution of (1) is an element in (for all ) subject to three classical conditions, one requiring that satisfies a standard variational form of (1). The set 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 on given by
admits a global attractor . 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 on a metric space . Since this notion is not directly applicable to the space , the author defines a notion of -samples as follows. With the restriction of to one sets:
and with one associates as follows:
One then sets:
The main results then are:
(I) is a generalized semiflow on ,
(II) with the attractor for and , is a global attractor for .
The author extends the above theory to a familiy of objects called “germs”, which are a kind of infinitesimal -samples .