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

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 8, page 467-487
- ISSN: 1292-8119

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topChepyzhov, V. V., and Vishik, M. I.. "Non-autonomous 2D Navier–Stokes system with a simple global attractor and some averaging problems." ESAIM: Control, Optimisation and Calculus of Variations 8 (2010): 467-487. <http://eudml.org/doc/90657>.

@article{Chepyzhov2010,

abstract = {
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.
},

author = {Chepyzhov, V. V., Vishik, M. I.},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Non-autonomous Navier–Stokes system;
global attractor; time averaging.; time averaging; Navier-Stokes system with rapidly oscillating external force; averaged NS system; small Grashof number; averaged global attractor},

language = {eng},

month = {3},

pages = {467-487},

publisher = {EDP Sciences},

title = {Non-autonomous 2D Navier–Stokes system with a simple global attractor and some averaging problems},

url = {http://eudml.org/doc/90657},

volume = {8},

year = {2010},

}

TY - JOUR

AU - Chepyzhov, V. V.

AU - Vishik, M. I.

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

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 8

SP - 467

EP - 487

AB -
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.

LA - eng

KW - Non-autonomous Navier–Stokes system;
global attractor; time averaging.; time averaging; Navier-Stokes system with rapidly oscillating external force; averaged NS system; small Grashof number; averaged global attractor

UR - http://eudml.org/doc/90657

ER -

## References

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- V.V. Chepyzhov and A.A. Ilyin, On the fractal dimension of invariant sets; applications to Navier-Stokes equations (to appear).
- M.I. Vishik and V.V. Chepyzhov, Averaging of trajectory attractors of evolution equations with rapidly oscillating terms. Mat. Sbornik192 (2001) 16-53. English transl.: Sbornik: Mathematics192 (2001).
- V.V. Chepyzhov and M.I. Vishik, Trajectory attractors for 2D Navier-Stokes systems and some generalizations. Topol. Meth. Nonl. Anal., J.Juliusz Schauder Center8 (1996) 217-243.
- J.W.S. Kassels, An introduction to Diophantine approximations. Cambridge University Press (1957).
- B. Fiedler and M.I. Vishik, Quantitative homogenization of global attractors for reaction-diffusion systems with rapidly oscillating terms. Preprint (2000).

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