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The global attractor for the 2D Navier-Stokes flow on some unbounded domains. (English) Zbl 0901.35070
The author studies the Navier-Stokes equations in a domain \(\Omega \subset \mathbb{R}^2\), i.e. \[ u_t- \nu\Delta u+(u \cdot \nabla) u+\nabla p=f, \quad \text{div} u =0\quad\text{in } \Omega \times (0,\infty), \]
\[ u=0\quad\text{on }\partial \Omega \times (0,\infty), \quad u(0)= u_0\quad\text{in } \Omega. \] Here, \(\Omega\) is an arbitrary open set in \(\mathbb{R}^2\) having the property that the Poincaré inequality holds on it and \(f\) is a time independent external force. Assuming that \(f\in V'\) (the dual of \(V= \overline {\{v\in C^\infty_0 (\Omega) \mid \text{div} v= 0\}}^{\| \nabla \cdot \|_{L^2}})\) the author proves the existence of the global attractor improving on earlier work by Abergel and Babin. Furthermore he gives an estimate for its dimension in terms of the usual physical parameters. The proof uses a weak continuity property of the underlying semigroup \(S(t)\) which is employed to show that \(S(t)\) is asymptotically compact. The existence of the global attractor then follows from an abstract result.

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