A Finite-Dimensional Attractor for Three-Dimensional Flow of Incompressible Fluids

The asymptotic behavior of the flow for a system of the Navier–Stokes type is investigated. In the considered model, the viscous part of the stress tensor is generally a nonlinear function of the symmetric part of the velocity gradient. Provided that the function describing this dependence satisfies the polynomial (p−1) growth condition, a unique weak solution exists if eitherp (2+n)/2 andu0 Horp 1+2n/(n+2) andu0 W1, 2(C)n∩H. In the first case, the existence of a global attractor inHis proved. In order to indicate the finite-dimensional behavior of the flow at infinity, the fractal dimension of the new invariant set, composed from all shortδ-trajectories with initial value in the attractor, is estimated in theL2(0, δ; H) topology. Having uniqueness only for more regular data in the second case, many trajectories can start from the initial valueu0 H. This does not allow one to define a semigroup on the spaceH. Therefore, the set of short trajectoriesXsδ, closed inL2(0, δ; H), is introduced along with a semigroup working on this set. The existence of a global attractor with a finite fractal dimension is then demonstrated.