Dynamics of weak solutions for the three dimensional Navier-Stokes equations with nonlinear damping

Abstract

The main objective of this paper is to study the existence of a finite dimensional global attractor for the three dimensional Navier-Stokes equations with nonlinear damping for r > 4. Motivated by the idea of [1], even though we can obtain the existence of a global attractor for r ≥ 2 by the multi-valued semiflow, it is very difficult to provide any information about its fractal dimension. Therefore, we prove the existence of a global attractor in H and provide the upper bound of its fractal dimension by the methods of-trajectories in this paper.

Pullback exponential attractors for the three dimensional non-autonomous Navier-Stokes equations with nonlinear damping

“…The existence of an exponential attractor in V for problem (1) was proved by using the squeezing property in [35]. In [23], the authors have established the existence of a finite dimensional global attractor for problem (1) with r > 4 by the methods of -trajectories, which extends the results established in [33].…”

Asymptotic analysis of the 2D convective Brinkman-Forchheimer equations in unbounded domains: Global attractors and upper semicontinuity

“…The existence of a global attractor for the Brinkman-Forchheimer equations in bounded domains have been discussed in the works [45,46], etc. The existence of global attractors for the dynamical systems generated by weak as well as strong solutions of the 3D NSE with damping in bounded domains has been obtained in the works [27,35,39,40], etc. As mentioned in the paper [22], the major difficulty in working with bounded domains Ω ⊂ R 3 is that P H (|u| r−1 u) (here P H : L p (Ω) → H, p ∈ [2, ∞), is the Helmholtz-Hodge projection) need not be zero on the boundary, and P H and −∆ are not necessarily commuting (for a counter example, see Example 2.19, [38]).…”

Dynamics for the 3D incompressible Navier-Stokes equations with double time delays and damping

“…For g = 0, f = 0, the decay rate or the asymptotic stability also have been established in [17,45]. The existence of the attractors were built up in [25,36]. The authors in [25] showed for β > 3, the existence of a global attractor in H and provided the upper bound of its fractal dimension.…”

“…The existence of the attractors were built up in [25,36]. The authors in [25] showed for β > 3, the existence of a global attractor in H and provided the upper bound of its fractal dimension. Song and Hou obtained the existence and the uniqueness of global attractors in V and H 2 (Ω) in [36], they also in [37] proved the existence of (V, V ) uniform attractor and (V, H 2 (Ω)) uniform attractor.…”