Global attractors for 2D Navier–Stokes–Voight equations in an unbounded domain

Çelebi, A. Okay; Kalantarov, Varga K.; Polat, M.

doi:10.1080/00036810902766682

Cited by 29 publications

(7 citation statements)

References 14 publications

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“…[30,33,41], etc. The 2D Navier-Stokes-Voight equation in an unbounded strip-like domain is considered in [3] and the authors established that the semigroup generated by this equation has a global attractor in weighted Sobolev spaces. The authors in [26] considered the problem of existence of a finite dimensional global attractor and established the estimates for the number of determining modes on the global attractor of Kelvin-Voight fluids of order L ≥ 1.…”

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confidence: 99%

“…We also describe some properties of the memory kernel appearing in (1) in the same section. In section 3, we first consider the transformed system (autonomous) using the transformation given in (3), so that solution operator defines a one parameter family of semigroups. We establish the existence of an absorbing ball in V for the semigroup S(t), t ≥ 0 defined for the Kelvin-Voigt fluid flow equations with "fading memory" (section 3.1).…”

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Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory"

Mohan¹

2022

EECT

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<p style='text-indent:20px;'>The three-dimensional viscoelastic fluid flow equations, arising from the motion of Kelvin-Voigt (Kelvin-Voight) fluids in bounded domains is considered in this work. We investigate the long-term dynamics of such viscoelastic fluid flow equations with "fading memory" (non-autonomous). We first establish the existence of an absorbing ball in appropriate spaces for the semigroup defined for the Kelvin-Voigt fluid flow equations of order one with "fading memory" (transformed autonomous coupled system). Then, we prove that the semigroup is asymptotically compact, and hence we establish the existence of a global attractor for the semigroup. We provide estimates for the number of determining modes for both asymptotic as well as for trajectories on the global attractor. Once the differentiability of the semigroup with respect to initial data is established, we show that the global attractor has finite Hausdorff as well as fractal dimensions. We also show the existence of an exponential attractor for the semigroup associated with the transformed (equivalent) autonomous Kelvin-Voigt fluid flow equations with "fading memory". Finally, we show that the semigroup has Ladyzhenskaya's squeezing property and hence is quasi-stable, which also implies the existence of global as well as exponential attractor having finite fractal dimension.</p>

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Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory"

Mohan¹

2022

EECT

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“…[1][2][3][4][5][6][7] When the damping au + |u| 2 u is absent in (1.1), we call it the Navier-Stokes-Voigt equations, and some interesting conclusions on the existence of solutions and attractors can be found in previous studies. [8][9][10][11][12][13] In 2013, the existence of pullback- attractors to the system (1.1) is derived in Anh and Trang 14 in some unbounded domains, but there is no result on the large-time behavior of solutions to the system with delay.…”

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confidence: 99%

The pullback‐ attractors for the 3D Kelvin‐Voigt‐Brinkman‐Forchheimer system with delay

Qin

2018

Math Methods in App Sciences

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Under some assumptions on the external force f(t, u(t − (t))), the 3D Kelvin-Voigt-Brinkman-Forchheimer system with continuous delay is considered. After the existence of pullback- absorbing sets for the continuous solution process, the asymptotical compactness obtained by the decomposition method leads to the existence of pullback- attractors. KEYWORDScontinuous delay, Kelvin-Voigt-Brinkman-Forchheimer system, pullback- attractors 6122

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“…Kalantarov, Kalantarov, Levant and Titi, and Kalantarov and Titi proved the existence of global attractors in 3D. Çelebi, Kalantarov, and Polat studied the global attractors in 2D. Gao and Sun and Sun and Gao obtained the random attractors in 3D.…”

Section: Introductionmentioning

confidence: 99%

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

Cao

Qin

2017

Math Methods in App Sciences

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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.

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Global attractors for 2D Navier–Stokes–Voight equations in an unbounded domain

Abstract: We consider the 2D Navier-Stokes-Voight equation in an unbounded strip-like domain. It is shown that the semigroup generated by this equation has a global attractor in weighted Sobolev spaces.

Cited by 29 publications

References 14 publications

Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory"

Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory"

The pullback‐ attractors for the 3D Kelvin‐Voigt‐Brinkman‐Forchheimer system with delay

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

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