Finite fractal dimension of pullback attractors for non-autonomous 2D Navier--Stokes equations in some unbounded domains (Q858649)

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 \textit{V. V. Chepyzhov} and \textit{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.