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The uniform attractor for the 2D non-autonomous Navier-Stokes flow in some unbounded domain. (English) Zbl 1057.35031
The authors consider the Navier-Stokes equation on a region Ω 2 , where Ω only has to satisfy the PoincarĂ© inequality. The weak solutions define a semiprocess depending on an external force f. The authors suppose f, where is bounded (as a subset of L ( + ,V ' ) and V ' is the dual of the space of divergence free functions in H 0 1 (Ω)) and positive invariant (i.e. sf(s+h) for all h0, f). Under these assumptions they prove the existence of a compact uniform (with respect to ) attractor. In the case of f being a k-dimensional quasi-periodic external force, they also give an upper bound on the Hausdorff dimension of the attractor.
35Q30Stokes and Navier-Stokes equations
37L30Attractors and their dimensions, Lyapunov exponents
76D05Navier-Stokes equations (fluid dynamics)