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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 $$\Omega\subset\mathbb{R}^2$$, where $$\Omega$$ only has to satisfy the Poincaré inequality. The weak solutions define a semiprocess depending on an external force $$f$$. The authors suppose $$f\in \mathcal{F}$$, where $$\mathcal{F}$$ is bounded (as a subset of $$L^\infty(\mathbb{R}^+,V')$$ and $$V'$$ is the dual of the space of divergence free functions in $$H_0^1(\Omega)$$) and positive invariant (i.e. $$s\mapsto f(s+h)\in \mathcal{F}$$ for all $$h\geq 0$$, $$f\in \mathcal{F}$$). Under these assumptions they prove the existence of a compact uniform (with respect to $$\mathcal{F}$$) 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.

##### MSC:
 35Q30 Navier-Stokes equations 37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems 76D05 Navier-Stokes equations for incompressible viscous fluids
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