The authors consider a viscous incompressible Newtonian fluid in a bounded open domain with smooth boundary , described by the classical Navier-Stokes equations
where is the velocity field, the pressure field, the kinematic viscosity, and the body force. Let , and be the space of generalized weak solutions. Let be the space of all that have polynomial growth at and satisfy the property
where . Let
be defined for all constants and all . Let be a function, defined on the backward orbit of (the set with values in , so that the subexponential growth condition is fulfilled,
For and a function satisfying the assumption (1) let us define the set
and let for
Let be the set of all measurable multifunctions contained in a ball , so that (1) is fulfilled with -probability 1. There exists an and a -invariant set of full -measure consisting of functions of polynomial growth so that
The main theorem of this paper says that for sufficiently large we have the following:
(i) The attractor defined by (2) for (for we set defines a -measurable multifunction.
(ii) is the unique attractor in .
(iii) For any , ,