Pullback attractors in V for non-autonomous 2D-Navier–Stokes equations and their tempered behaviour

García-Luengo, Julia; Marín-Rubio, Pedro; Real, José

doi:10.1016/j.jde.2012.01.010

Cited by 84 publications

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“…Now, we establish several estimates in finite intervals of time when the initial time is sufficiently shifted in a pullback sense (cf. [8,9] for similar results in a context without delays).…”

Section: Remark 42 Observe That For Anysupporting

confidence: 53%

“…In this section we recall briefly some results from [9] about the existence of minimal pullback attractors (see also [2,3,18]). In particular, we consider the process U being closed (see Definition 3.1 below).…”

Section: Abstract Results On Minimal Pullback Attractorsmentioning

confidence: 99%

“…Namely, if the map µ → σ µ is non-decreasing, we look for µ < µ < 2νλ 1 (this may involve a smallness condition on the delay); otherwise, we seek for 0 < µ < µ. Under any of these conditions, we would obtain new families of pullback attractors and new relations among them (see [9,Remark 4.15] …”

Section: Remark 42 Observe That For Anymentioning

confidence: 99%

“…We will prove that u n → u strongly in C([t − h, t]; V), (5.14) using an energy method for continuous functions analogous to that employed in the proof of Proposition 4.2, but starting with the energy equality (2.5) as in [9]. Indeed, if (5.14) is false, there exist ε > 0, a value t * ∈ [t − h, t], and subsequences (which we relabel the same) {u n } and {t n } ⊂ [t − h, t], with lim n→∞ t n = t * , such that u n (t n ) − u(t * ) ≥ ε ∀ n ≥ On the other hand, using the energy equality (2.5) for u and all u n , and reasoning as for the obtention of (5.2), we have that for all t − h − 1 ≤ s 1 ≤ s 2 ≤ t, u n (s 2 ) 2 + ν some 0 < µ < 2νλ 1 such that µ > 2e µh L g .…”

Section: Remark 42 Observe That For Anymentioning

confidence: 99%

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Pullback Attractors for 2D Navier-Stokes Equations with Delays and Their Regularity

García-Luengo

Marín-Rubio

Real

2013

Advanced Nonlinear Studies

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In this paper we obtain some results on the existence of solution, and of pullback attractors, for a 2D Navier-Stokes model with finite delay studied in [4] and [6]. Actually, we prove a result of existence and uniqueness of solution under less restrictive assumptions than in [4]. More precisely, we remove a condition on square integrable control of the memory terms, which allows us to consider a bigger class of delay terms (for instance, just under a measurability condition on the delay function leading the delayed time). After that, we deal with dynamical systems in suitable phase spaces within two metrics, the L 2 norm and the H 1 norm. Moreover, we prove that under these assumptions, pullback attractors not only of fixed bounded sets but also of a set of tempered universes do exist. Finally, from comparison results of attractors we establish relations among them, and under suitable additional assumptions we conclude that these families of attractors are in fact the same object.

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Section: Remark 42 Observe That For Anysupporting

confidence: 53%

Section: Abstract Results On Minimal Pullback Attractorsmentioning

confidence: 99%

Section: Remark 42 Observe That For Anymentioning

confidence: 99%

Section: Remark 42 Observe That For Anymentioning

confidence: 99%

See 2 more Smart Citations

Pullback Attractors for 2D Navier-Stokes Equations with Delays and Their Regularity

García-Luengo

Marín-Rubio

Real

2013

Advanced Nonlinear Studies

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“…Indeed, we will not study the pullback attractors of "fixed" bounded sets, but for a class (called universe) of families which can move in time and fulfil a tempered condition (cf. [11,9,33]). …”

Section: Introduction and Statement Of The Problemmentioning

confidence: 99%

Robustness of nonautonomous attractors for a family of nonlocal reaction–diffusion equations without uniqueness

2015

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In this paper, we consider a non-autonomous nonlocal reactiondiffusion equation with a small perturbation in the nonlocal diffusion term and the non-autonomous force. Under the assumptions imposed on the viscosity function, the uniqueness of weak solutions cannot be guaranteed. In this multi-valued framework, the existence of weak solutions and minimal pullback attractors in the L 2 -norm are analysed. In addition, some relationships between the attractors of the universe of fixed bounded sets and those associated to a universe given by a tempered condition are established. Finally, the upper semicontinuity property of pullback attractors w.r.t. the parameter is proved. Indeed, under suitable assumptions, we prove that the family of pullback attractors converges to the corresponding global compact attractor associated to the autonomous nonlocal limit problem when the parameter goes to zero.Keywords nonlocal diffusion · reaction-diffusion equations without uniqueness · pullback attractors · upper semicontinuity of attractors · multivalued dynamical systems.

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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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Pullback attractors in V for non-autonomous 2D-Navier–Stokes equations and their tempered behaviour

Cited by 84 publications

References 22 publications

Pullback Attractors for 2D Navier-Stokes Equations with Delays and Their Regularity

Pullback Attractors for 2D Navier-Stokes Equations with Delays and Their Regularity

Robustness of nonautonomous attractors for a family of nonlocal reaction–diffusion equations without uniqueness

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

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