## Non-autonomous 2D Navier–Stokes system with a simple global attractor and some averaging problems

Institute for Information Transmission Problems, Russian Academy of Sciences, B. Karetniy 19, Moscow 101447, GSP-4, Russia; vishik@iitp.ru.

Received:
3
March
2002

We study the global attractor of the non-autonomous 2D
Navier–Stokes system with time-dependent external force
*g(x,t)*. We assume that *g(x,t)* is a translation
compact function and the corresponding Grashof number is small.
Then the global attractor has a simple structure: it is the
closure of all the values of the unique bounded complete
trajectory of the Navier–Stokes system. In particular, if
*g(x,t)* is a quasiperiodic function with respect to *t*,
then the attractor is a continuous image of a torus. Moreover
the global attractor attracts all the solutions of the NS system
with exponential rate, that is, the attractor is exponential.
We also consider the 2D Navier–Stokes system with rapidly oscillating
external force *g(x,t,t/ε)*, which has the
average as ε → 0+. We assume that the
function *g(x,t,z)* has a bounded primitive with respect to *z*
and the averaged NS system has a small Grashof number that
provides a simple structure of the averaged global attractor.
Then we prove that the distance from the global attractor of the
original NS system to the attractor of the averaged
NS system is less than a small power of *ε*.

Mathematics Subject Classification: 35B40 / 35Q30 / 34C29

Key words: Non-autonomous Navier–Stokes system / global attractor / time averaging.

*© EDP Sciences, SMAI, 2002*