Pullback attractors of 2D incompressible Navier‐Stokes‐Voight equations with delay

Abstract

In this paper, the 2D Navier-Stokes-Voight equations with 3 delays in R 2 is considered. By using the Faedo-Galerkin method, Lions-Aubin lemma, and Arzelà-Ascoli theorem, we establish the global well-posedness of solutions and the existence of pullback attractors in H 1 . KEYWORDScontinuous delay, distributed delay, Navier-Stokes-Voight equation, pullback attractors INTRODUCTIONIn this paper, the existence of pullback attractors for 2D Navier-Stokes-Voight (NSV) equations with delays will be discussed. This model has a close connection to the incompressible Navier-Stokes equations. First of all, we recall some results of the Navier-Stokes equations with delay.Krasovskii 1 first noticed the system with delay in 1963. He constructed the Navier-Stokes equations with delay and got the well-posedness of this model. Taniguchi 2 considered the nonautonomous Navier-Stokes equations with continuous delay and obtained the existence of absorbing sets. Hale 3 established the existence and uniqueness of weak solutions to the Navier-Stokes equations with delay. Caraballo and Real 4-6 derived some results such as the well-posedness of solutions, the large time behavior of solutions, and the existence of pullback attractors for the Navier-Stokes equations with delay on an open and bounded domain with regular boundary. Marín-Rubio and Real 7 established the existence of pullback attractors after expanding the domain to some unbounded domain in which the Poincaré inequality held. Garcín-Luengo, Marín-Rubio, and Planas 8 showed the existence of pullback attractors for 2D Navier-Stokes equations with double-time delay in the convective term and the external force. For more results about the fluid flow with delay, we can refer to the literature. 7,[9][10][11][12][13][14][15][16] The study of global well-posedness of the 3D Navier-Stokes equations is a challenging problem. Therefore, people hope to study the Navier-Stokes equations by considering some properties of the NSV equations. Thus, the study of the NSV equations is very important.The NSV equation models the dynamics of a Kelvin-Voight viscoelastic incompressible fluid and was introduced by Oskolkov in previous study 17 as a model of motion of linear, viscoelastic fluids. Kalantarov, 18 Kalantarov, Levant and Titi,19 and Kalantarov and Titi 20 proved the existence of global attractors in 3D. Çelebi, Kalantarov, and Polat 21 studied the global attractors in 2D.

Solvability of the Boussinesq Approximation for Water Polymer Solutions

“…Of course, in this passage-to-limit procedure, we used all the convergence results (22)- (26). It follows from ( [16] § 2) that the mapping R α defined by…”

“…Oskolkov [8][9][10][11][12][13][14], there is an ever growing list of contributions. An interested reader can see the papers [15][16][17][18][19][20][21][22][23][24][25][26][27][28]; this list is by no means exhaustive, but gives a number of current mathematical results obtained for these types of viscoelastic fluids.…”

Double stabilities of pullback random attractors for stochastic delayed p‐Laplacian equations

“…2,5,6,[22][23][24][25][26] It is worth pointing out that there is no paper studying the stochastic delayed p-Laplacian equation, although other delayed equations had been well investigated. 16,17,[27][28][29][30][31][32] For Equation (3) even for the simpler cases such as deterministic equation (b j = 0) or bounded domain, it has not been solved for the existence (let alone backward compactness) of a pullback attractor in X = C([− , 0], X), where X = L 2 (R n ).…”

Three‐dimensional Navier‐Stokes‐Voight equation with a memory and the Brinkman‐Forchheimer damping term

“…Their result improved the regularity of global attractor of the system under study. Cao and Qin 5 and Li and Qin 16 established the existence of pullback attractors for 2D and 3D Navier‐Stokes‐Voight equations with delays …”

“…In this article, we consider the 3D Navier-Stokes-Voight equation with a memory and a nonlinear damping, which is a generalized version proposed by Plinio et al 1 The domain Ω ⊂ R 3 is bounded with the smooth boundary Ω, > 0, and the damping coefficient , k > 0, p ∈ [2,5) are the given constants,…”