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

Abstract

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.

Finite-dimensional global and exponential attractors for the reaction–diffusion problem with an obstacle potential

“…Indeed, by our construction, M ⊂ S(1)Φ and, therefore, due to estimate (2.23), M is globally bounded in C 2−ν . Using now the following interpolation inequality 19) we see that the dimension of M is finite not only in L 2 , but also in L ∞ and that the attraction property holds in L ∞ as well. Thus, the desired exponential attractor M in the phase space Φ is constructed and Theorem 3.4 is proved.…”

“…We will use the so-called method of "l-trajectories (introduced by Málek and Nečas in [19], see also [20] and [29]) in order to construct the proper spaces H and H 1 and to verify the assumptions of Lemma 3.5.…”

“…This result is obtained using some modification of the so-called method of l-trajectories which was originally introduced by Málek and Nečas in [19] and is widely used nowadays in the attractors theory, see [20,23] and references therein.…”

Vorticity dynamics and turbulence models for Large-Eddy Simulations

“…Following Malek & Necas [36], we employed in [25] the method of short trajectories in order to prove that the attractor (36) has a finite fractal dimension.…”

Trajectory attractor for a non-autonomous magnetohydrodynamic equation of non-Newtonian fluids

“…One can also cite the paper [14] which extended the notion of pullback attractor to nonautonomous and stochastic multivalued dynamical systems. The third method is the theory of trajectory attractor which was introduced in [17] , [37], [49]. Due to its simplicity this method has become very popular and used in many works but we only mention [17], [18], [19], [31], and [57] for few relevant results.…”