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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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