Pullback attractor of 2D non-autonomous g-Navier-Stokes equations on some bounded domains

Abstract

The existence of the pullback attractor for the 2D non-autonomous g-NavierStokes equations on some bounded domains is investigated under the general assumptions of pullback asymptotic compactness. A new method to prove the existence of the pullback attractor for the 2D g-Navier-Stokes equations is given.

The Long-Time Behavior of 2D Nonautonomous g -Navier-Stokes Equations with Weak Dampness and Time Delay

“…The understanding of the behavior with dynamical systems was one of the most important problems of modern mathematical physics (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]). In the last decades, g-Navier-Stokes equations have received increasing attention due to their importance in the fluid motion.…”

“…The Hausdorff and fractal dimension of the global attractor about the 2D g-Navier-Stokes equation for the periodic and Dirichlet boundary conditions and the global attractor of the 2D g-Navier-Stokes equation on some unbounded domains were researched in [5]. In [6][7][8][9][10], the finite dimensional global attractor and the pullback attractor for g-Navier-Stokes equation were studied. Moreover, Anh et al studied long-time behavior for 2D nonautonomous g-Navier-Stokes equations and the stability of solutions to stochastic 2D g-Navier-Stokes equation with finite delays in [11,12]; Quyee researched the stationary solutions to 2D g-Navier-Stokes equation and pullback attractor for 2D g-Navier-Stokes equation with infinite delays in [13].…”

Large-domain stability of random attractors for stochastic g-Navier–Stokes equations with additive noise

“…studied in both autonomous and non-autonomous cases without the stochastic situations, see [10,11,18]. As described in [18], the 2D g-NS equations arise in a natural way when we study the standard 3D Navier-Stokes problem in a 3D thin domain O g = O × (0, εg), (O ⊂ R 2 ) which was introduced by [9,17], and we do not claim that the g-NS equations form a model of any fluid flow.…”

Regular measurable backward compact random attractor for <inline-formula><tex-math id="M1">\begin{document}$ g $\end{document}</tex-math></inline-formula>-Navier-Stokes equation

“…(1.1) reduces to the standard 2D Navier-Stokes equation. In the last decade, the long-time behavior of solutions available in terms of the existence of attractors for deterministic 2D g-NS equation has been studied in, e.g., [1,10,11,22]. As described in [21,13], the 2D g-NS equation arises in a natural way when we study the standard 3D Navier-Stokes problem in a 3D thin domain O g = O × (0, g), (O ⊂ R 2 ) which introduced by [9,19,20], and good properties of the 2D g-NS equation can lead to an initial study of 3D Navier-Stokes equations in the thin domain O g .…”

“…Proof. Given τ ∈ R, ω ∈ Ω, we define 11) where ρ 1 (τ, ω), ρ 3 (τ, ω) is given by Lemma 4.1 and Lemma 4.2, respectively. Therefore, by Lemma 5.1, φ has a D-pullback backward uniform absorbing set in H g (O).…”