Hausdorff dimension of attractors for the two-dimensional Navier-Stokes equations with boundary conditions
The author considers attracting sets in a viscous incompressible fluid in a region \(\Omega\) of \({\mathbb{R}}^ 2\) where the time evolution of the fluid is described by the Navier-Stokes equations with boundary conditions. Two kinds of boundary conditions are investigated: a given velocity and an imposed force on the boundary of \(\Omega\). Using earlier results by \textit{D. Ruelle} [ibid. 87, 287-302 (1982; Zbl 0546.76083); 93, 285-300 (1984; Zbl 0565.76031)] and \textit{E. Lieb} [ibid. 92, 473-480 (1984; Zbl 0598.76054)] the author derives explicit bounds (depending only on the data) for the entropy (Kolmogorov-Sinai invariant) and the Hausdorff dimension of the attractors under consideration.
scientific article; zbMATH DE number 3984173