Journal of Dynamics and Differential Equations

, Volume 19, Issue 3, pp 655–684 | Cite as

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

  • V. V. ChepyzhovEmail author
  • M. I. Vishik

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).


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

AMS 2000 Mathematics Subject Classification

35B40 35B41 35B45 35Q35 


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© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Institute for Information Transmission ProblemsRussian Academy of SciencesMoscowRussia

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