Attractors for Navier-Stokes equations in domains with finite measure. (English) Zbl 0859.35090

The Navier-Stokes system \[ \partial_tu+\sum^2_{i=1} u^i\partial_iu=\nu\Delta u-\nabla p+f,\;\text{div }u=0,\quad u|_{\partial\Omega}=0 \tag{1} \] is considered in a domain \(\Omega\subset\mathbb{R}^2\). The system (1) may be unbounded, but has finite measure \(|\Omega|<\infty\). The following lower bounds of the spectrum of the Stokes problem are proved \[ \sum^m_{k=1}\lambda_k\geq {\pi m^2\over |\Omega|},\quad \lambda_1\geq{2\pi\over |\Omega|}, \] where \(|\Omega|\) is the area of \(\Omega\). The author proves the existence of a global attractor, and derives the explicit estimates of its Hausdorff and fractal dimension by using these bounds.


35Q30 Navier-Stokes equations
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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