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Finite fractal dimension of pullback attractors for non-autonomous 2D Navier–Stokes equations in some unbounded domains. (English) Zbl 1113.37055
The authors analyze the long-term behavior of solutions of the non-autonomous two-dimensional Navier-Stokes equations on unbounded domains for which the Poincaré inequality holds true. They establish sufficient conditions which imply that the pullback attractor (a generalization of the concept of a global attractor in autonomous dynamical systems to the non-autonomous case; see V. V. Chepyzhov and M. I. Vishik [Attractors for equations of mathematical physics. Colloquium Publications. American Mathematical Society. 49. Providence, RI: American Mathematical Society (AMS) (2002; Zbl 0986.35001)]) of the associated evolution process has finite fractal dimension. The key assumption in this study is the boundedness of the non-autonomous terms in the past. This assumption makes the concept of pullback attractors applicable and allows to deduce “uniform pullback asymptotic compactness” (a weak form of compactness) for the evolution process. The compactness property replaces the lack of compact Sobolev embeddings due to unbounded domains.

37L30 Infinite-dimensional dissipative dynamical systems–attractors and their dimensions, Lyapunov exponents
35B41 Attractors
35Q30 Navier-Stokes equations
Full Text: DOI
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