Chepyzhov, V. V.; Vishik, M. I. Non-autonomous 2D Navier-Stokes system with a simple global attractor and some averaging problems. (English) Zbl 1068.35089 ESAIM, Control Optim. Calc. Var. 8, 467-487 (2002). Summary: 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 (NS) 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 \(g(x,t,t/\varepsilon )\), which has the average as \(\varepsilon \rightarrow 0+\). 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 \(\varepsilon \). Cited in 13 Documents MSC: 35Q30 Navier-Stokes equations 37L30 Infinite-dimensional dissipative dynamical systems–attractors and their dimensions, Lyapunov exponents 34C29 Averaging method for ordinary differential equations 35B40 Asymptotic behavior of solutions to PDEs 35B41 Attractors 37N10 Dynamical systems in fluid mechanics, oceanography and meteorology 76D05 Navier-Stokes equations for incompressible viscous fluids Keywords:time averaging; Navier-Stokes system with rapidly oscillating external force; averaged NS system; small Grashof number; averaged global attractor PDF BibTeX XML Cite \textit{V. V. Chepyzhov} and \textit{M. I. Vishik}, ESAIM, Control Optim. Calc. Var. 8, 467--487 (2002; Zbl 1068.35089) Full Text: DOI Numdam EuDML References: [1] A. Haraux , Systèmes dynamiques dissipatifs et applications . Masson, Paris, Milan, Barcelona, Rome ( 1991 ). MR 1084372 | Zbl 0726.58001 · Zbl 0726.58001 [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 . MR 1273705 | Zbl 0838.58021 · Zbl 0838.58021 [3] V.V. Chepyzhov and M.I. Vishik , Attractors for equations of mathematical physics . AMS, Providence, Rhode Island ( 2002 ). MR 1868930 | Zbl 0986.35001 · Zbl 0986.35001 [4] J.-L. Lions , Quelques méthodes de résolution des problèmes aux limites non linéaires . Dunod Gauthier-Villars, Paris ( 1969 ). MR 259693 | Zbl 0189.40603 · Zbl 0189.40603 [5] O.A. Ladyzhenskaya , The mathematical theory of viscous incompressible flow . Moscow, Nauka ( 1970 ). English transl.: Gordon and Breach, New York ( 1969 ). MR 254401 | Zbl 0184.52603 · Zbl 0184.52603 [6] R. Temam , Infinite-dimensional dynamical systems in mechanics and physics . New York, Springer-Verlag, Appl. Math. Ser. 68 ( 1988 ), 2nd Ed. 1997. MR 1441312 | Zbl 0662.35001 · Zbl 0662.35001 [7] A.V. Babin and M.I. Vishik , Attractors of evolution equations . Nauka, Moscow ( 1989 ). English transl.: North Holland ( 1992 ). MR 1007829 | Zbl 0778.58002 · Zbl 0778.58002 [8] V.V. Chepyzhov and A.A. Ilyin , On the fractal dimension of invariant sets; applications to Navier-Stokes equations (to appear). Zbl 1049.37047 · Zbl 1049.37047 · doi:10.3934/dcds.2004.10.117 [9] M.I. Vishik and V.V. Chepyzhov , Averaging of trajectory attractors of evolution equations with rapidly oscillating terms . Mat. Sbornik 192 ( 2001 ) 16 - 53 . English transl.: Sbornik: Mathematics 192 ( 2001 ). MR 1830471 · Zbl 1011.35104 [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 Center 8 ( 1996 ) 217 - 243 . Zbl 0894.35011 · Zbl 0894.35011 [11] J.W.S. Kassels , An introduction to Diophantine approximations . Cambridge University Press ( 1957 ). Zbl 0077.04801 · Zbl 0077.04801 [12] B. Fiedler and M.I. Vishik , Quantitative homogenization of global attractors for reaction-diffusion systems with rapidly oscillating terms . Preprint ( 2000 ). MR 1992283 | Zbl 1135.35315 · Zbl 1135.35315 · iospress.metapress.com This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.