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

✍ Scribed by Josef Málek; Jindřich Nečas

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 827 KB
• Volume: 127
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1996.0080

✦ Synopsis

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 either p (2+n)Â2 and u 0 # H or p 1+2nÂ(n+2) and u 0 # W 1, 2 (C ) n & H. In the first case, the existence of a global attractor in H is 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 short $-trajectories with initial value in the attractor, is estimated in the L 2 (0, $; H ) topology. Having uniqueness only for more regular data in the second case, many trajectories can start from the initial value u 0 # H. This does not allow one to define a semigroup on the space H. Therefore, the set of short trajectories X s $ , closed in L 2 (0, $; 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.