Attractors for autonomous and nonautonomous 3D Navier-Stokes-Voight equations

Pull-back attractors for three-dimensional Navier—Stokes—Voigt equations in some unbounded domains

“…In § § 4 and 5, following the general lines of the approach in [4,14] for 2D Navier-Stokes equations, we prove the existence and fractal dimension estimates of a minimal unique pull-back attractor for the process associated with the problem. In the last section, we give relationships between the pull-back attractor obtained in § 4, the uniform attractor obtained recently in [18] and the global attractor obtained when the external force f is independent of the time variable t.…”

“…Recently, in [11,12], Kalantarov and Titi improved this result, and proved the determining modes property and the Gevrey regularity of the global attractor. For an external force that is a translation bounded time-dependent function, the existence of a uniform attractor for (1.1) was proved very recently in [18]. However, to the best of our knowledge, all existing results for 3D Navier-Stokes-Voigt equations are devoted to the problem in bounded domains; only the work in [5] studied the problem in an unbounded two-dimensional domain.…”

“…More difficulty arises due to the lack of compactness of the Sobolev embeddings, since the considered domain is unbounded. This leads to the fact that the decomposition method of the solutions used for Navier-Stokes-Voigt equations in bounded domains [11,18] does not work here. To overcome these difficulties, we exploit the energy equations method introduced by Ball [2] to prove the pull-back asymptotic compactness of the process, and, as a consequence, the existence of a pull-back attractor.…”

Pullback attractors for three-dimensional non-autonomous Navier–Stokes–Voigt equations

“…We may also cite in this non-autonomous framework the paper [29], where the existence of uniform attractor for a Navier-Stokes-Voigt model is studied. However, there appears the same treatment with the fractional powers of the Stokes operator, and they require more regularity in the non-autonomous case that we need here.…”

Remarks on uniform attractors for the 3D non-autonomous Navier-Stokes-Voight equations

“…In this regard, Kalantarov and Titi [9] investigated the Navier-Stokes-Voight equations as an inviscid regularization of the 3D incompressible Navier-Stokes equations, and further obtained the existence of global attractors for Navier-Stokes-Voight equations. Recently, Qin, Yang and Liu [10] showed the existence of uniform attractors by uniform condition-(C) and weak continuous method to obtain uniformly asymptotical compactness in H 1 and H 2 , Yue and Zhong [11] investigated the attractors for autonomous and nonautonomous 3D Navier-Stokes-Voight equations in different methods. More details about the infinite-dimensional dynamics systems, we can refer to [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27].…”