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

35Q30Stokes and Navier-Stokes equations
37L30Attractors and their dimensions, Lyapunov exponents
76D05Navier-Stokes equations (fluid dynamics)
Full Text: DOI
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