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

V. V. Chepyzhov; M. I. Vishik

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

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

Abstract

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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.

How to cite

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Chepyzhov, 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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  1. A. Haraux, Systèmes dynamiques dissipatifs et applications. Masson, Paris, Milan, Barcelona, Rome (1991).  
  2. V.V. Chepyzhov and M.I. Vishik, Attractors of non-autonomous dynamical systems and their dimension. J. Math. Pures Appl.73 (1994) 279-333.  
  3. V.V. Chepyzhov and M.I. Vishik, Attractors for equations of mathematical physics. AMS, Providence, Rhode Island (2002).  
  4. J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod Gauthier-Villars, Paris (1969).  
  5. O.A. Ladyzhenskaya,The mathematical theory of viscous incompressible flow. Moscow, Nauka (1970). English transl.: Gordon and Breach, New York (1969).  
  6. R. Temam, Infinite-dimensional dynamical systems in mechanics and physics. New York, Springer-Verlag, Appl. Math. Ser.68 (1988), 2nd Ed. 1997.  
  7. A.V. Babin and M.I. Vishik,Attractors of evolution equations. Nauka, Moscow (1989). English transl.: North Holland (1992).  
  8. V.V. Chepyzhov and A.A. Ilyin, On the fractal dimension of invariant sets; applications to Navier-Stokes equations (to appear).  
  9. 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).  
  10. 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.  
  11. J.W.S. Kassels, An introduction to Diophantine approximations. Cambridge University Press (1957).  
  12. 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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