Journal of Dynamics and Differential Equations

, Volume 19, Issue 3, pp 655–684

# Non-autonomous 2D Navier–Stokes System with Singularly Oscillating External Force and its Global Attractor

• V. V. Chepyzhov
• M. I. Vishik
Article

We study the global attractor $$\mathcal{A}^{\varepsilon}$$ of the non-autonomous 2D Navier–Stokes (N.–S.) system with singularly oscillating external force of the form $$g_{0}(x,t)+\frac{1} {\varepsilon^{\rho }} g_{1} (\frac{x} {\varepsilon}, t), x \in \Omega \Subset \mathbb{R}^{2}, t \in \mathbb{R}, 0\, \leqslant\, \rho\, \leqslant\, 1$$. If the functions g 0(x, t) and g 1 (z, t) are translation bounded in the corresponding spaces, then it is known that the global attractor $$\mathcal{A}^{\varepsilon}$$ is bounded in the space H, however, its norm $$\Vert \mathcal{A}^{\varepsilon }\Vert _{H}$$ may be unbounded as $$\varepsilon \rightarrow 0+$$ since the magnitude of the external force is growing. Assuming that the function g 1 (z, t) has a divergence representation of the form $$g_{1} ( z,t) =\partial _{z_{1}}G_{1}(z,t)+\partial_{z_{2}}G_{2}(z,t),z=(z_{1},z_{2})\in \mathbb{R}^{2},$$ where the functions $$G_{j}(z,t)\in L_{2}^{b}(\mathbb{R};Z)$$ (see Section 3), we prove that the global attractors $$\mathcal{A}^{\varepsilon}$$ of the N.–S. equations are uniformly bounded with respect to $$\varepsilon :\;\Vert \mathcal{A}^{\varepsilon }\Vert _{H}\,\leqslant\,C$$ for all $$0 < \varepsilon\,\leqslant\, 1$$. We also consider the “limiting” 2D N.–S. system with external force g 0(x, t). We have found an estimate for the deviation of a solution $$u^{\varepsilon }(x,t)$$ of the original N.–S. system from a solution u 0(x, t) of the “limiting” N.–S. system with the same initial data. If the function g 1 (z, t) admits the divergence representation, the functions g 0(x, t) and g 1 (z, t) are translation compact in the corresponding spaces, and $$0\, \leqslant\, \rho < 1$$ , then we prove that the global attractors $$\mathcal{A}^{\varepsilon }$$ converges to the global attractor $$\mathcal{A}^{0}$$ of the “limiting” system as $$\varepsilon \rightarrow 0+$$ in the norm of H. In the last section, we present an estimate for the Hausdorff deviation of $$\mathcal{A}^{\varepsilon}$$ from $$\mathcal{A}^{0}$$ of the form: $$\mathrm{dist}_{H}(\mathcal{A}^{\varepsilon }, \mathcal{A}^{0}) \leqslant C(\rho )\varepsilon^{1-\rho}$$ in the case, when the global attractor $$\mathcal{A}^{0}$$ is exponential (the Grashof number of the “limiting” 2D N.–S. system is small).

## Keywords

Non-autonomous 2D Navier–Stokes system global attractor singularly oscillating terms homogenization translation compact functions

## AMS 2000 Mathematics Subject Classification

35B40 35B41 35B45 35Q35

## References

1. 1.
Amerio L., Prouse G. (1971) Abstract Almost Periodic Functions And Functional Equations. Van Nostrand, New-YorkGoogle Scholar
2. 2.
Babin A.V., Vishik M.I. (1992) Attractors of Evolution Equations, Nauka, Moscow, 1989. English transl.: North Holland, AmsterdamGoogle Scholar
3. 3.
Chepyzhov V.V., Goritsky Yu.A., Vishik M.I. (2005) Integral manifolds and attractors with exponential rate for nonautonomous hyperbolic equations with dissipation. Russ. J. Math. Phys. 12(1): 17–39
4. 4.
Chepyzhov V.V., Ilyin A.A. (2004) On the fractal dimension of invariant sets; applications to Navier Stokes equations. Discr. and Cont. Dyn. Syst. 10(1 and 2): 117–135
5. 5.
Chepyzhov, V. V., and Vishik, M. I. (2002). Attractors for Equations of Mathematical Physics, AMS Coll. Publ., Vol 49, Providence, RI.Google Scholar
6. 6.
Chepyzhov V.V., Vishik M.I. (2002) Non-autonomous 2D Navier–Stokes system with a simple global attractor and some averaging problems. El. J. ESAIM: COCV 8: 467–487
7. 7.
Chepyzhov, V. V., and Vishik, M. I. (2006). Global attractors for non-autonomous Ginzburg–Landau equation with singularly oscillating terms. Rend. Acad. Naz. Sci. XL, Mem. Mat. Appl. 123 Vol.XXIX, fasc. 1, 123–148.Google Scholar
8. 8.
Chepyzhov V.V., Vishik M.I., Wendland W.L. (2005) On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging. Discr. Cont. Dyn. Syst. 12(1): 27–38
9. 9.
Constantin, P., and Foias, C. (1989). Navier–Stokes Equations, The University of Chicago Press, Chicago and London, 1989.Google Scholar
10. 10.
Efendiev M., Zelik S. (2002) Attractors of the reaction–diffusion systems with rapidly oscillating coefficients and their homogenization. Ann. I.H.Poincaré 19(6): 961–989
11. 11.
Hale J.K., Verduyn Lunel S.M. (1990) Averaging in infinite dimensions. J. Int. Eq. Appl. 2(4): 463–494
12. 12.
Ilyin A.A. (1996) Averaging principle for dissipative dynamical systems with rapidly oscillating right–hand sides. Sbornik: Math. 187(5): 635–677
13. 13.
Ladyzhenskaya, O. A. (1969). The Mathematical Theory of Viscous Incompressible Flow, Moscow, Nauka, 1970. English transl.: Gordon and Breach, New York.Google Scholar
14. 14.
Ladyzhenskaya O.A. (1991) Attractors for Semigroups and Evolution Equations, Lezioni Lincei. Cambridge University Press, Cambridge New-YorkGoogle Scholar
15. 15.
Levitan B., Zhikov V. (1982) Almost Periodic Functions and Differential Equations, Moscow State University, Moscow, 1978; English transl. Cambridge Univ. Press, CambridgeGoogle Scholar
16. 16.
Lions J.-L. (1969) Quelques Méthodes de Résolution des Problèmes Aux Limites Non Linéaires. Dunod Gauthier-Villars, Paris
17. 17.
Sell G., You Y. (2002) Dynamics of Evolutionary Equations. Springer-Verlag, New York
18. 18.
Temam, R. (1984). Navier–Stokes Equations, Theory and Numerical Analysis, 3rd. rev. ed., North-Holland, Amsterdam.Google Scholar
19. 19.
Temam, R. (1988). Infinite-dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematics Series, 68, New York, Springer-Verlag, 2nd ed. 1997.Google Scholar
20. 20.
Vishik M.I., Chepyzhov V.V. (2001) Averaging of trajectory attractors of evolution equations with rapidly oscillating terms. Sbornik: Math. 192(1): 11–47
21. 21.
Vishik M.I., Chepyzhov V.V. (2003) Approximation of trajectories lying on a global attractor of a hyperbolic equation with exterior force rapidly oscillating in time. Sbornik: Math. 194(9): 1273–1300
22. 22.
Vishik M.I., Fiedler B. (2002) Quantitative homogenization of global attractors for hyperbolic wave equations with rapidly oscillating terms. Russ. Math. Surv. 57(4): 709–728
23. 23.
Weinstein M. (1983) Nonlinear Schrödinger equations and sharp interpolation estimates. Commun. Math. Phys. 87: 567–576