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What is A Finite-Dimensional Attractor for Three-Dimensional Flow of Incompressible Fluids about?

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)2 and u 0 # H or p 1+2n(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.

Author

Josef Málek; Jindřich Nečas

Publisher

Elsevier Science; Elsevier ; Elsevier Inc.; Elsevier BV (ISSN 0022-0396)

Published

1996

Language

EN