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Research Article Approximate Kelvin-Voigt Fluid Driven by an External Force Depending on Velocity with Distributed Delay Yantao Guo, 1,2 Shuilin Cheng, 3 and Yanbin Tang 1 1 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China 2 School of Mathematics and Statistics, Xuchang University, Xuchang, Henan 461000, China 3 School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan, Hubei 430073, China Correspondence should be addressed to Yanbin Tang; [email protected] Received 23 March 2015; Accepted 2 May 2015 Academic Editor: Luca Gori Copyright © 2015 Yantao Guo et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider the approximate 3D Kelvin-Voigt fluid driven by an external force depending on velocity with distributed delay. We investigate the long time behavior of solutions to Navier-Stokes-Voigt equation with a distributed delay external force depending on the velocity of fluid on a bounded domain. By a prior estimate and a contractive function, we give a sufficient condition for the existence of pullback attractor of NSV equation. 1. Introduction In this paper, we consider 3D Navier-Stokes-Voigt (NSV) equation with a distributed delay external force depending on the velocity of the fluid: − 2 Δ − ]Δ + ( ⋅ ∇) + ∇ =∫ 0 −ℎ (, ( + )) , div (, ) = 0, (, ) ∈ [, +∞) × Ω, (, ) = 0, (, ) ∈ [, +∞) × Ω, (, ) = ( − , ) , ∈ [ − ℎ, ] , ∈ Ω, (1) where = ( 1 , 2 , 3 ) is the velocity field of the fluid, is the pressure, ] > 0 is the kinematic viscosity, > 0 is the length scale parameter of the elasticity of the fluid, the external force and initial velocity field are defined in the interval of time [−ℎ, 0], where ℎ is a fixed positive number and Ω is a bounded smooth domain of 3 . e NSV equation was introduced by Oskolkov [1] to give an approximate description of the Kelvin-Voigt fluid and was proposed as a regularization of 3D Navier-Stokes equation for the purpose of direct numerical simulations in [2]. Since the term − 2 Δ changes the parabolic character of the equation, the NSV equation being well posed in 3D, many authors have studied the long time dynamics of this model. Kalantarov and Titi [3] investigated the existence of the global attractor, the estimation of the upper bounds for the number of determining modes, and the dimension of global attractor of the semigroup generated by the equations. By a useful decomposition method, Yue and Zhong [4] proved the asymptotic regularity of solution of NSV equation and obtained the existence of the uniform attractor; they also described the structure of the uniform attractor and its regularity. Garc´ ıa-Luengo et al. [5] investigated the existence and relationship between minimal pullback attractor for the universe of fixed bounded sets and universe given by a tempered condition. Partial differential equations with delays arise from var- ious fields, like physics, control theory, and so on (see, e.g., [6–10]); the unknown functions depend on not only present stage but also some past stage. e existence and stability of solution and global attractor for Navier-Stokes equation with discrete delay were established in [11–13]. e existence of pullback attractors in 1 0 and 1 0 ∩ 2 was proved for the processes associated with nonclassical diffusion equations with variable bounded delay in [14, 15]. Delay effect has been considered on an unbounded domain in [16]. e existence of Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2015, Article ID 721673, 9 pages http://dx.doi.org/10.1155/2015/721673
Transcript
Page 1: Research Article Approximate Kelvin-Voigt Fluid …downloads.hindawi.com/journals/ddns/2015/721673.pdfWe consider the approximate D Kelvin-Voigt uid driven by an external force depending

Research ArticleApproximate Kelvin-Voigt Fluid Driven by an External ForceDepending on Velocity with Distributed Delay

Yantao Guo12 Shuilin Cheng3 and Yanbin Tang1

1School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan Hubei 430074 China2School of Mathematics and Statistics Xuchang University Xuchang Henan 461000 China3School of Statistics and Mathematics Zhongnan University of Economics and Law Wuhan Hubei 430073 China

Correspondence should be addressed to Yanbin Tang tangybhustsinacom

Received 23 March 2015 Accepted 2 May 2015

Academic Editor Luca Gori

Copyright copy 2015 Yantao Guo et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We consider the approximate 3D Kelvin-Voigt fluid driven by an external force depending on velocity with distributed delay Weinvestigate the long time behavior of solutions to Navier-Stokes-Voigt equation with a distributed delay external force dependingon the velocity of fluid on a bounded domain By a prior estimate and a contractive function we give a sufficient condition for theexistence of pullback attractor of NSV equation

1 Introduction

In this paper we consider 3D Navier-Stokes-Voigt (NSV)equationwith a distributed delay external force depending onthe velocity of the fluid

119906119905minus120572

2Δ119906119905minus ]Δ119906+ (119906 sdot nabla) 119906 +nabla119901

= int

0

minusℎ

119866 (119904 119906 (119905 + 119904)) 119889119904

div119906 (119905 119909) = 0 (119905 119909) isin [120591 +infin) times Ω

119906 (119905 119909) = 0 (119905 119909) isin [120591 +infin) times 120597Ω

119906 (119905 119909) = 120601 (119905 minus 120591 119909) 119905 isin [120591 minus ℎ 120591] 119909 isin Ω

(1)

where 119906 = (1199061 1199062 1199063) is the velocity field of the fluid 119901 is thepressure ] gt 0 is the kinematic viscosity 120572 gt 0 is the lengthscale parameter of the elasticity of the fluid the external force119866 and initial velocity field 120601 are defined in the interval of time[minusℎ 0] where ℎ is a fixed positive number andΩ is a boundedsmooth domain of 1198773

The NSV equation was introduced by Oskolkov [1] togive an approximate description of the Kelvin-Voigt fluidand was proposed as a regularization of 3D Navier-Stokesequation for the purpose of direct numerical simulations in

[2] Since the term minus1205722Δ119906119905changes the parabolic character

of the equation the NSV equation being well posed in 3Dmany authors have studied the long time dynamics of thismodel Kalantarov and Titi [3] investigated the existence ofthe global attractor the estimation of the upper bounds forthe number of determining modes and the dimension ofglobal attractor of the semigroup generated by the equationsBy a useful decomposition method Yue and Zhong [4]proved the asymptotic regularity of solution of NSV equationand obtained the existence of the uniform attractor theyalso described the structure of the uniform attractor and itsregularity Garcıa-Luengo et al [5] investigated the existenceand relationship between minimal pullback attractor for theuniverse of fixed bounded sets and universe given by atempered condition

Partial differential equations with delays arise from var-ious fields like physics control theory and so on (see eg[6ndash10]) the unknown functions depend on not only presentstage but also some past stage The existence and stabilityof solution and global attractor for Navier-Stokes equationwith discrete delay were established in [11ndash13] The existenceof pullback attractors in 119862

11986710and 119862

11986710cap119867

2 was proved forthe processes associatedwith nonclassical diffusion equationswith variable bounded delay in [14 15] Delay effect has beenconsidered on an unbounded domain in [16]The existence of

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2015 Article ID 721673 9 pageshttpdxdoiorg1011552015721673

2 Discrete Dynamics in Nature and Society

pullback attractor for a Navier-Stokes equation with infinitediscrete delay effect was studied in [17]

The aim of this paper is to investigate the NSV equationwith a distributed delay instead of the discussions with finitedelays in the referencesOur purpose is twofoldWefirst showthe existence and uniqueness of solution to NSV equation(1) with a distributed delay then we prove the existenceof pullback attractor for the process generated by the NSVequation (1)

This paper is organized as follows In Section 2 we givesome preliminary results and prove existence of solution toNSV equationwith a distributed delay In Section 3 we derivethe existence of pullback attractor by prior estimates andcontractive functions

2 Existence of Solutions

In order to prove the existence of solutions to problem (1) wedefine the function spaces

V = 119906 isin (119862infin

0 (Ω))3 div119906= 0 (2)

119867 is the closure ofV in (1198712(Ω))3 with the inner product (sdot sdot)and associate norm | sdot |119881 is the closure ofV in (1198671

0 (Ω))3 with

scalar product ((sdot sdot)) and associate norm sdot where

((119906 V)) =3sum

119894119895=1int

Ω

120597119906119895

120597119909119894

120597V119895

120597119909119894

119889119909 forall119906 V isin (11986710 (Ω))

3 (3)

it follows that 119881 sub 119867 equiv 1198671015840sub 1198811015840 where the injections are

dense and compact We will use sdot lowastfor the norm in 1198811015840 and

⟨sdot sdot⟩ for the duality pairing between 119881 and 1198811015840Define the linear continuous operator 119860 119881 rarr 119881

1015840 as

⟨119860119906 V⟩ = ((119906 V)) forall119906 V isin 119881 (4)

We denote 119863(119860) = 119906 isin 119881119860119906 isin 119867 one has that 119863(119860) =(119867

2(Ω))

3cap 119881 and 119860119906 = minus119875Δ119906 for all 119906 isin 119863(119860) is the Stokes

operator where 119875 is the orthoprojector from (1198712(Ω))

3 onto119867 also denote 119862

119867= 119862

0([minusℎ 0]119867) and 119862

119881= 119862

0([minusℎ 0] 119881)

Define the trilinear form 119887 on 119881 times 119881 times 119881 by

119887 (119906 V 119908) =3sum

119894119895=1int

Ω

119906119894

120597V119895

120597119909119894

119908119895119889119909 forall119906 V 119908 isin 119881 (5)

and the operator 119861 119881 times 119881 rarr 1198811015840 as

⟨119861 (119906 V) 119908⟩ = 119887 (119906 V 119908) forall119906 V 119908 isin 119881 (6)

and denote 119861(119906) = 119861(119906 119906)The trilinear form 119887 satisfies that

119887 (119906 V 119908) = minus 119887 (119906 119908 V)

119887 (119906 V V) = 0

forall119906 V 119908 isin 119881

(7)

We also recall that there exists a constant 119862 depending onlyonΩ such that

|119887 (119906 V 119908)| le 119862 11990612

|119860119906|12

V |119908|

119906 isin 119863 (119860) V isin 119881 119908 isin 119867

|119887 (119906 V 119908)| le 119862 119906 V |119908|12 11990812

119906 isin 119881 V isin 119881 119908 isin 119881

|119887 (119906 V 119908)| le 119862 |119906|12

11990612

V 119908

119906 isin 119881 V isin 119881 119908 isin 119881

(8)

For the term containing the time delay 119866 119877 times 119867 rarr 119867

satisfies that

(1198671) 119866(sdot 119906) 119877 rarr 119867 is a measurable function

(1198672) 119866(119905 0) = 0 for all 119905 isin 119877

(1198673) there exists a positive constant 119871 such that forall119877 gt

0 if |119906| lt 119877 and |V| lt 119877 then

|119866 (119905 119906) minus119866 (119905 V)|2 le 119871 |119906 minus V|2 (9)

Remark 1 Hypotheses (1198672)-(1198673) imply that |119866(119905 119906)|2 le

119871|119906|2 so we have |119866(119905 119906)|2 isin 119871infin(120591 119879) for |119906| lt 119877

Problem (1) can be rewritten as

120597

120597119905

(119906 + 1205722119860119906)+ ]119860119906+119861 (119906 119906)

= int

0

minusℎ

119866 (119904 119906 (119905 + 119904)) 119889119904

119906 (119905 119909) = 120601 (119905 minus 120591 119909) 119905 isin [120591 minus ℎ 120591] 119909 isin Ω

(10)

then we get the existence of solution to problem (10)

Theorem 2 Let 120601 isin 119862119881 let 119866 119877times119867 rarr 119867 satisfy the hypo-

theses (1198671)ndash(1198673) and let 120591 isin R Then forall119879 gt 120591 there exists aunique weak solution to (10) such that

119906 (119905 119909) isin 119862 ([120591 minus ℎ 119879] 119881)

120597119906 (119905 119909)

120597119905

isin 1198712(120591 119879 119881)

(11)

Moreover if 120601 isin 119862119863(119860)

then problem (10) admits a strong sol-ution

Proof Consider the Galerkin approximations for problem(10)

119889

119889119905

(119906119898+120572

2119860119906119898) + ]119860119906119898 +119861 (119906119898 119906119898)

= int

0

minusℎ

119866 (119904 119906119898(119905 + 119904)) 119889119904

(12)

Discrete Dynamics in Nature and Society 3

where 119906119898 = sum119898

119895=1 119906119898

119895119890119895119860119906119898 = sum

119898

119895=1 120582119895119906119898

119895119890119895 and 119890

119895and120582

119895are

the corresponding orthonormal eigenfunctions and eigen-values of operator 119860 respectively then

10038171003817100381710038171199061198981003817100381710038171003817

2=

119898

sum

119895=1120582119895(119906119898

119895)

2

10038161003816100381610038161198601199061198981003816100381610038161003816

2=

119898

sum

119895=11205822119895(119906119898

119895)

2

10038161003816100381610038161199061198981003816100381610038161003816

2=

119898

sum

119895=1(119906119898

119895)

2

(13)

We now derive a prior estimate for the Galerkin approx-imate solution Multiplying (12) by 119906119898

119895 summing from 119895 = 1

to119898 and using the fact

(119861 (119906119898 119906119898) 119906119898) = 119887 (119906

119898 119906119898 119906119898) = 0 (14)

we obtain that for ae 119905 gt 120591119889

119889119905

(10038161003816100381610038161199061198981003816100381610038161003816

2+120572

2 10038171003817100381710038171199061198981003817100381710038171003817

2) + 2] 100381710038171003817

10038171199061198981003817100381710038171003817

2

= 2(int0

minusℎ

119866 (119904 119906119898(119905 + 119904)) 119889119904 119906

119898)

le

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906119898(119905 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

+10038161003816100381610038161199061198981003816100381610038161003816

2

(15)

Integrating (15) from 120591 to 119905 we obtain that

1003816100381610038161003816119906119898(119905)1003816100381610038161003816

2+120572

2 1003817100381710038171003817119906119898(119905)1003817100381710038171003817

2+ 2]int

119905

120591

1003817100381710038171003817119906119898(119904)1003817100381710038171003817

2119889119904

le1003816100381610038161003816119906119898(120591)

1003816100381610038161003816

2+120572

2 1003817100381710038171003817119906119898(120591)

1003817100381710038171003817

2

+int

119905

120591

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906119898(119903 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

119889119903

+int

119905

120591

1003816100381610038161003816119906119898(119903)1003816100381610038161003816

2119889119903

(16)

Remark 1 implies that

int

119905

120591

10038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906119898(119903 + 119904)) 119889119904

10038161003816100381610038161003816100381610038161003816

2

119889119903

le int

0

minusℎ

int

119905

120591

1003816100381610038161003816119866 (119904 119906

119898(119903 + 119904))

1003816100381610038161003816

2119889119903 119889119904

le int

0

minusℎ

int

119905

120591

1198711003816100381610038161003816119906119898(119903 + 119904)

1003816100381610038161003816

2119889119903 119889119904

le 119871int

0

minusℎ

int

119905+119904

120591+119904

1003816100381610038161003816119906119898(119903)1003816100381610038161003816

2119889119903 119889119904

le 119871int

0

minusℎ

(int

120591

120591minusℎ

1003816100381610038161003816119906119898(119903)1003816100381610038161003816

2119889119903 +int

119905

120591

1003816100381610038161003816119906119898(119903)1003816100381610038161003816

2119889119903) 119889119904

le 119871ℎ2 10038171003817100381710038171206011003817100381710038171003817

2119862119867

+119871ℎint

119905

120591

1003816100381610038161003816119906119898(119903)1003816100381610038161003816

2119889119903

(17)

Then forall119905 isin (120591 119879) and

1003816100381610038161003816119906119898(119905)1003816100381610038161003816

2+120572

2 1003817100381710038171003817119906119898(119905)1003817100381710038171003817

2+ 2]int

119905

120591

1003817100381710038171003817119906119898(119904)1003817100381710038171003817

2119889119904

le (119871ℎ + 2) int119905

120591

1003816100381610038161003816119906119898(119904)1003816100381610038161003816

2119889119904 + 119871ℎ

2 10038171003817100381710038171206011003817100381710038171003817

2119862119867

+10038161003816100381610038161206011003816100381610038161003816

2119862119867

+1205722 10038171003817100381710038171206011003817100381710038171003817

2119862119881

(18)

So we have

1003816100381610038161003816119906119898(119905)1003816100381610038161003816

2le 119862int

119905

120591

1003816100381610038161003816119906119898(119904)1003816100381610038161003816

2119889119904 +119862

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+10038161003816100381610038161206011003816100381610038161003816

2119862119867

+1205722 10038171003817100381710038171206011003817100381710038171003817

2119862119881

(19)

The Gronwall inequality implies that

1003816100381610038161003816119906119898(119905)1003816100381610038161003816

2le 119862 (20)

Putting (20) into the right-hand side of (18) we have

1205722 1003817100381710038171003817119906119898(119905)1003817100381710038171003817

2+ 2]int

119905

120591

1003817100381710038171003817119906119898(119904)1003817100381710038171003817

2119889119904 le 119862 forall119905 isin (120591 119879) (21)

This implies that

119906119898 is bounded in 119871

infin(120591 119879 119881) cap 119871

2(120591 119879 119881) (22)

Now multiplying (12) by 120597119905119906119898 and integrating overΩ we

have

10038161003816100381610038161205971199051199061198981003816100381610038161003816

2+120572

2 10038171003817100381710038171205971199051199061198981003817100381710038171003817

2+

]2119889

119889119905

10038171003817100381710038171199061198981003817100381710038171003817

2

le1003816100381610038161003816119887 (119906119898 119906119898 120597119905119906119898)1003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

(int

0

minusℎ

119866 (119904 119906119898(119905 + 119904)) 119889119904 120597

119905119906119898)

10038161003816100381610038161003816100381610038161003816

le 11988810038171003817100381710038171199061198981003817100381710038171003817

32 10038161003816100381610038161199061198981003816100381610038161003816

12 10038171003817100381710038171205971199051199061198981003817100381710038171003817

+

12

10038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906119898(119905 + 119904)) 119889119904

10038161003816100381610038161003816100381610038161003816

2

+

1210038161003816100381610038161205971199051199061198981003816100381610038161003816

2

(23)

since

10038171003817100381710038171199061198981003817100381710038171003817

32 10038161003816100381610038161199061198981003816100381610038161003816

12 10038171003817100381710038171205971199051199061198981003817100381710038171003817le 119888

10038171003817100381710038171199061198981003817100381710038171003817

3 10038161003816100381610038161199061198981003816100381610038161003816+

1205722

210038171003817100381710038171205971199051199061198981003817100381710038171003817

2 (24)

then

10038161003816100381610038161205971199051199061198981003816100381610038161003816

2+120572

2 10038171003817100381710038171205971199051199061198981003817100381710038171003817

2+ ]

119889

119889119905

10038171003817100381710038171199061198981003817100381710038171003817

2

le 11988810038171003817100381710038171199061198981003817100381710038171003817

3 10038161003816100381610038161199061198981003816100381610038161003816+

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906119898(119905 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

(25)

4 Discrete Dynamics in Nature and Society

integrating the above inequality from 120591 to 119905 by (17) (20) and(22) we have

int

119905

120591

(10038161003816100381610038161205971199051199061198981003816100381610038161003816

2+120572

2 10038171003817100381710038171205971199051199061198981003817100381710038171003817

2) 119889119904 + ] 100381710038171003817

1003817119906119898(119905)1003817100381710038171003817

2

le ] 1003817100381710038171003817119906119898(120591)

1003817100381710038171003817

2+ 119888int

119905

120591

10038171003817100381710038171199061198981003817100381710038171003817

3 10038161003816100381610038161199061198981003816100381610038161003816119889119904

+int

119905

120591

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906119898(119903 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

119889119903

le ] 1003817100381710038171003817120601 (120591)

1003817100381710038171003817

2+ 119888int

119905

120591

10038171003817100381710038171199061198981003817100381710038171003817

3 10038161003816100381610038161199061198981003816100381610038161003816119889119904 + 119888

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+ 119888int

119905

120591

1003816100381610038161003816119906119898(119903)1003816100381610038161003816

2119889119903

le ] 10038171003817100381710038171206011003817100381710038171003817

2119862119881+ 119888int

119905

120591

10038171003817100381710038171199061198981003817100381710038171003817

3 10038161003816100381610038161199061198981003816100381610038161003816119889119904 + 119888

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+ 119888int

119905

120591

1003816100381610038161003816119906119898(119903)1003816100381610038161003816

2119889119903

(26)

Since 119906119898 is bounded in 119871infin(120591 119879 119881) cap 1198712(120591 119879 119881) we obtainthat

120597119905119906119898 is bounded in 119871

2(120591 119879 119881) (27)

By the Faedo-Galerkin scheme for example see [14 18]according to the estimates (22) and (27) we can get existenceof the weak solution here we omit the details

We next consider the uniqueness of solution Let 119906 V betwo solutions to problem (10) corresponding the initial data120601 and 120595 respectively

Denote 119908 = 119906 minus V then we have

120597

120597119905

(119908+1205722119860119908)+ ]119860119908+119861 (119906 119906) minus 119861 (V V)

= int

0

minusℎ

119866 (119904 119906 (119905 + 119904)) 119889119904 minusint

0

minusℎ

119866 (119904 V (119905 + 119904)) 119889119904(28)

Multiplying (28) by 119908 and integrating overΩ we obtain

12119889

119889119905

(|119908|2+120572

2119908

2) + ] 1199082 + (119861 (119906 119906) 119908)

minus (119861 (V V) 119908)

= (int

0

minusℎ

119866 (119904 119906 (119905 + 119904)) 119889119904 minusint

0

minusℎ

119866 (119904 V (119905 + 119904)) 119889119904 119908)

(29)

Notice that

(119861 (119906 119906) 119908) minus (119861 (V V) 119908)

= 119887 (119906 119906 119906 minus V) minus 119887 (V 119906 119906 minus V) + 119887 (V 119906 119906 minus V)

minus 119887 (V V 119906 minus V)

= 119887 (119906 minus V 119906 119906 minus V) minus 119887 (V 119906 minus V 119906 minus V)

= 119887 (119908 119906 119908)

(30)

Substituting (30) into (29) and integrating from 120591 to 119905 weget

|119908 (119905)|2+120572

2119908 (119905)

2+ 2]int

119905

120591

119908 (119904)2119889119904 minus |119908 (120591)|

2

minus1205722119908 (120591)

2le int

119905

120591

|119887 (119908 119906 119908)| 119889119904 +int

119905

120591

|119908 (119904)|2119889119904

+int

119905

120591

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119903 + 119904)) 119889119904

minusint

0

minusℎ

119866 (119904 V (119903 + 119904)) 119889119904100381610038161003816100381610038161003816100381610038161003816

2

119889119903

(31)

(1198673) implies that

int

119905

120591

10038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119903 + 119904)) 119889119904 minusint

0

minusℎ

119866 (119904 V (119903 + 119904)) 11988911990410038161003816100381610038161003816100381610038161003816

2

119889119903

le int

0

minusℎ

int

119905

120591

|119866 (119904 119906 (119903 + 119904)) minus119866 (119904 V (119903 + 119904))|2 119889119903 119889119904

le int

0

minusℎ

int

119905

120591

|119866 (119904 119906 (119903 + 119904)) minus119866 (119904 V (119903 + 119904))|2 119889119903 119889119904

le 119871int

0

minusℎ

int

119905

120591

|119906 (119903 + 119904) minus V (119903 + 119904)|2 119889119903 119889119904

le 119871int

0

minusℎ

int

119905+119904

120591+119904

|119906 (119903) minus V (119903)|2 119889119903 119889119904

le 119871int

0

minusℎ

(int

120591

120591minusℎ

|119906 (119903) minus V (119903)|2 119889119903

+int

119905

120591

|119906 (119903) minus V (119903)|2 119889119903) 119889119904 le 119871ℎ2 1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119867

+119871ℎint

119905

120591

|119908 (119904)|2119889119904

(32)

As the property of operator 119887 and Poincare we have

int

119905

120591

|119887 (119908 119906 119908)| 119889119904 le 119862int

119905

120591

11990832

|119908|12

119906 119889119904

le 119862int

119905

120591

1199082119906 119889119904

(33)

Discrete Dynamics in Nature and Society 5

Substituting (32) and (33) into (31) we get

|119908 (119905)|2+120572

2119908 (119905)

2+ 2]int

119905

120591

119908 (119904)2119889119904

le int

119905

120591

1199082119906 119889119904 + 119871ℎ

2 1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119867

+ (119871ℎ + 1) int119905

120591

|119908 (119904)|2119889119904 + |119908 (120591)|

2

+1205722119908 (120591)

2

le int

119905

120591

1199082119906 119889119904 + 119871ℎ

2 1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119867

+ (119871ℎ + 1) int119905

120591

|119908 (119904)|2119889119904 +

1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119867

+1205722 1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119881

le 119862int

119905

120591

119908 (119904)2119889119904 +119862

1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119881

(34)

The last inequality comes from Poincare inequality andthe boundedness of 119906 Therefore the Gronwall inequalityimplies the uniqueness of the solution The proof is com-plete

3 Existence of Pullback Attractor

In this section we will prove the existence of pullbackattractor to problem (10) First we give existence of pullbackabsorbing set for the process 119880(119905 120591) generated by the globalsolution to problem (10)

Lemma 3 Assume (1198671)ndash(1198673) hold and 119871 le ]2120582211205764119890120576ℎ then

the process 119880(119905 120591) is pullback dissipative where 0 lt 120576 lt

min]1205722 ]12058214

Proof Multiplying (10) by119906 and integrating overΩ we obtain

120597

120597119905

(|119906|2+120572

2119906

2) + 2] 1199062

le 120578 |119906|2+

1120578

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119905 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

(35)

where 120578 is a constant determined laterBy Poincare inequality we have

120597

120597119905

(|119906|2+120572

2119906

2) + ] 1199062 + (]1205821 minus 120578) |119906|

2

le

1120578

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119905 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

(36)

Since

119890minus120576119905 120597

120597119905

[119890120576119905(|119906|

2+120572

2119906

2)]

= 120576 (|119906|2+120572

2119906

2) +

120597

120597119905

(|119906|2+120572

2119906

2)

(37)

then

119890minus120576119905 120597

120597119905

[119890120576119905(|119906|

2+120572

2119906

2)]

le (120576 + 120578 minus ]1205821) |119906|2+ (120576120572

2minus ]) 1199062

+

1120578

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119905 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

(38)

Integrating (38) from 120591 to 119905 we get

119890120576119905(|119906 (119905)|

2+120572

2119906 (119905)

2)

minus 119890120576120591(|119906 (120591)|

2+120572

2119906 (120591)

2)

le int

119905

120591

119890120576119904[(120576 + 120578 minus ]1205821) |119906|

2+ (120576120572

2minus ]) 1199062] 119889119904

+

1120578

int

119905

120591

119890120576119903

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119903 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

119889119903

(39)

Assumptions (1198671)ndash(1198673) imply that

int

119905

120591

119890120576119903

10038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119903 + 119904)) 119889119904

10038161003816100381610038161003816100381610038161003816

2

119889119903

le int

119905

120591

119890120576119903int

0

minusℎ

|119866 (119904 119906 (119903 + 119904))|2119889119904 119889119903

le 119871int

0

minusℎ

int

119905

120591

119890120576119903|119906 (119903 + 119904)|

2119889119903 119889119904

le 119871int

0

minusℎ

int

119905+119904

120591+119904

119890120576(119903minus119904)

|119906 (119903)|2119889119903 119889119904

le 119871int

0

minusℎ

int

119905

120591minusℎ

119890120576(119903minus119904)

|119906 (119903)|2119889119903 119889119904

le 119871int

0

minusℎ

119890minus120576119904

int

119905

120591minusℎ

119890120576119903|119906 (119903)|

2119889119903 119889119904

le 119871int

0

minusℎ

119890minus120576119904

(int

120591

120591minusℎ

119890120576119903|119906 (119903)|

2119889119903

+int

119905

120591

119890120576119903|119906 (119903)|

2119889119903) 119889119904 le

119871119890120576ℎ

120576

(

1120576

10038171003817100381710038171206011003817100381710038171003817

2119862119867

119890120576120591

+int

119905

120591

119890120576119903|119906 (119903)|

2119889119903) le

119871119890120576(120591+ℎ)

1205762

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+

119871119890120576ℎ

120576

sdot int

119905

120591

119890120576119903|119906 (119903)|

2119889119903

(40)

Substituting (40) into (39) we have

119890120576119905(|119906 (119905)|

2+120572

2119906 (119905)

2) minus 119890120576120591(|119906 (120591)|

2+120572

2119906 (120591)

2)

le int

119905

120591

119890120576119904[(120576 + 120578 minus ]1205821 +

119871119890120576ℎ

2120578120576) |119906|

2

+ (1205761205722minus ]) 1199062]119889119904 +

119871119890120576(120591+ℎ)

2120578120576210038171003817100381710038171206011003817100381710038171003817

2119862119867

(41)

6 Discrete Dynamics in Nature and Society

Let 120578 = (12)]1205821 choosing 0 lt 120576 lt min]1205722 ]12058214 119871 le

]2120582211205764119890120576ℎ implies that

max120576+ 120578minus ]1205821 +120578minus1119871119890120576ℎ

2120576 120576120572

2minus ] lt 0 (42)

then

119890120576119905(|119906 (119905)|

2+120572

2119906 (119905)

2)

le

119871119890120576(120591+ℎ)

1205762]1205821

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+ 119890120576120591(|119906 (120591)|

2+120572

2119906 (120591)

2)

(43)

which implies

|119906 (119905)|2+120572

2119906 (119905)

2

le

119871119890120576(120591+ℎminus119905)

1205762]1205821

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+ 119890120576(120591minus119905)

(1003816100381610038161003816120601 (120591)

1003816100381610038161003816

2+120572

2 1003817100381710038171003817120601 (120591)

1003817100381710038171003817

2)

(44)

Now if we take 119905 ge 120591 + ℎ then for 120579 isin [minusℎ 0] we have

|119906 (119905 + 120579)|2+120572

2119906 (119905 + 120579)

2

le

119871119890120576(120591+ℎminus119905minus120579)

1205762]1205821

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+ 119890120576(120591minus119905minus120579)

(10038171003817100381710038171206011003817100381710038171003817

2119862119867

+1205722 10038171003817100381710038171206011003817100381710038171003817

2119862119881)

(45)

We denote byR the set of all functions 119903 (minusinfin +infin) rarr

(0 +infin) such that

lim120591rarrminusinfin

1198901205721205911199032(120591) = 0 (46)

Then the closed ball in 119862119881defined by

119861 = 120593 isin11986211988110038171003817100381710038171205931003817100381710038171003817

2119862119881le 1 (47)

is pullback absorbing set for 119880(119905 120591) The proof is complete

We next prove the asymptotic compactness of solution toproblem (10) by contractive functions see [19 20]

Let 119883 be a Banach space and let 119861 be a bounded subsetof 119883 We call a function Φ(sdot sdot) which defined on 119883 times 119883 is acontractive function on 119861times119861 if for any sequence 119909

119899infin

119899=1sub 119861

there is a subsequence 119909119899119896infin

119896=1 sub 119909119899infin

119899=1 such that

lim119897rarrinfin

lim119896rarrinfin

Φ(119909119899119896 119909119899119897) = 0 (48)

Denote all such contractive functions on 119861 times 119861 by 119862(119861)

Theorem 4 (see [19]) Let 119878(119905)119905ge120591

be a semigroup on aBanach space (119883 sdot ) and have a bounded absorbing set 119861forall120598 gt 0 there exist 119879 = 119879(119861 120598) and Φ isin 119862(119861) such that

1003817100381710038171003817119878 (119879) 119909 minus 119878 (119879) 119910

1003817100381710038171003817le 120598 +Φ

119879(119909 119910) forall119909 119910 isin 119861 (49)

where Φ119879depends on 119879 Then 119878(119905)

119905ge120591is asymptotically

compact in119883

Lemma 5 Assume that (1198671)ndash(1198673) hold the process 119880(119905 120591)119905ge120591generated by the global solution to problem (10) is asymptoti-cally compact

Proof Let 119906119894(119905) be the solution to problem (10) with initialdata 120601119894 isin 119861 (119894 = 1 2) respectively Denote V(119905) = 119906

1(119905)minus119906

2(119905)

then V(119905) satisfies the equivalent abstract equation

120597

120597119905

(V+1205722119860V) + ]119860V+119861 (1199061 1199061) minus119861 (1199062 1199062)

= int

0

minusℎ

[119866 (119904 1199061(119905 + 119904)) minus119866 (s 1199062 (119905 + 119904))] 119889119904

(50)

with the initial condition V(119905) = 1206011(119905 minus 120591) minus 120601

2(119905 minus 120591) 119905 isin

[120591 minus ℎ 120591]Set an energy function

119864V (119905) =12int

Ω

|V|2 119889119909+1205722

2int

Ω

|nablaV|2 119889119909 (51)

Multiplying (50) by V and integrating over [119904 119879]timesΩwith 119879 gt

119905 + 120591 119904 ge 120591 we have

119864V (119879) minus119864V (119904) + ]int119879

119904

int

Ω

|nablaV|2 119889119909 119889119903

+int

119879

119904

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

= int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

(52)

then we have

]int119879

119904

int

Ω

|nablaV|2 119889119909 119889119903 le 119864V (119904) minusint119879

119904

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

+int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

(53)

Discrete Dynamics in Nature and Society 7

Using Poincare inequality and (51) and (53) we have

int

119879

120591

119864V (119904) 119889119904 =12int

119879

120591

int

Ω

|V|2 119889119909 119889119903 +1205722

2int

119879

120591

int

Ω

|nablaV|2 119889119909 119889119903 le 119862int

119879

120591

int

Ω

|nablaV|2 119889119909 119889119903

le 119862119864V (120591) minus119862int119879

120591

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

+119862int

119879

120591

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

(54)

Integrating (52) from 120591 to 119879 with respect to 119904 we obtain

119879119864V (119879) + ]int119879

120591

int

119879

119904

int

Ω

|nablaV|2 119889119909 119889119903 119889119904 = int

119879

120591

119864V (119904) 119889119904 minusint119879

120591

int

119879

119904

int

Ω

[119861 (1199061) minus 119861 (119906

2)] V (119903) 119889119909 119889119903 119889119904

+int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903 119889119904

(55)

Substituting (54) into (55) we get

119879119864V (119879) + ]int119879

120591

int

119879

119904

int

Ω

|nablaV|2 119889119909 119889119903 119889119904 le 119862119864V (120591) minus119862int119879

120591

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

+119862int

119879

120591

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

minusint

119879

120591

int

119879

119904

int

Ω

[119861 (1199061) minus 119861 (119906

2)] V (119903) 119889119909 119889119903 119889119904

+int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903 119889119904

(56)

Set

1198620 = 119862119864V (120591) =119862

2int

Ω

100381610038161003816100381610038161206011(0) minus 1206012 (0)1003816100381610038161003816

1003816

2119889119909+

1198621205722

2int

Ω

10038161003816100381610038161003816nabla120601

1(0) minus nabla1206012 (0)1003816100381610038161003816

1003816

2119889119909 (57)

Φ(1199061 119906

2) = minus119862int

119879

120591

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

+119862int

119879

120591

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

minusint

119879

120591

int

119879

119904

int

Ω

[119861 (1199061) minus 119861 (119906

2)] V (119903) 119889119909 119889119903 119889119904

+int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903 119889119904

(58)

8 Discrete Dynamics in Nature and Society

then by (56) we have

119864V (119879) le1198620119879

+

1119879

Φ(1199061 119906

2) (59)

One has forall120598 gt 0 we choose 119879 gt 0 large enough such that1198620119879 lt 120598 By Theorem 4 it suffices to prove that Φ(1199061 1199062) is

a contractive function since119861 is a bounded positive invariantset

If 119906119898 rarr 119906 (119898 rarr infin) we have the limits

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

(119861 (119906119899) minus 119861 (119906

119898)) (119906119899minus 119906119898) 119889119909 119889119903

= lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

[((119906119899minus 119906119898) sdot nabla) 119906

119899minus (119906119898sdot nabla) (119906

119899minus 119906119898)] (119906119899minus 119906119898) 119889119909 119889119903 = 0

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

(119861 (119906119899) minus 119861 (119906

119898)) (119906119899minus 119906119898) 119889119909 119889119903 119889119904

= lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

[((119906119899minus 119906119898) sdot nabla) 119906

119899minus (119906119898sdot nabla) (119906

119899minus 119906119898)] (119906119899minus 119906119898) 119889119909 119889119903 119889119904 = 0

(60)

By (1198671)ndash(1198673) we have

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

int

0

minusℎ

(119866 (120590 119906119899(119903 + 120590)) minus 119866 (120590 119906

119898(119903 + 120590))) V (119903) 119889120590 119889119909 119889119903

le lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

int

0

minusℎ

1198711003816100381610038161003816119906119899(119903 + 120590) minus 119906

119898(119903 + 120590)

1003816100381610038161003816

1003816100381610038161003816119906119899(119903) minus 119906

119898(119903)1003816100381610038161003816119889120590 119889119909 119889119903 = 0

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

(119866 (120590 119906119899(119903 + 120590)) minus 119866 (120590 119906

119898(119903 + 120590))) V (119903) 119889120590 119889119909 119889119903 119889119904

le lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

1198711003816100381610038161003816(119906119899minus 119906119898) (119903 + 120590)

1003816100381610038161003816

1003816100381610038161003816(119906119899minus 119906119898) (119903)

1003816100381610038161003816119889120590 119889119909 119889119903 119889119904 = 0

(61)

Combining (60)-(61) with (58) we have that Φ(1199061 1199062) is acontractive function

The proof is complete

Theorem 6 Assume that (1198671)ndash(1198673) hold and 119871 le ]2120582211205764119890120576ℎ then the process 119880(119905 120591) generated by the global solutionfor problem (10) has a pullback attractor

Proof Lemma 3 implies 119880(119905 120591) has a pullback absorbing set119861 and Lemma 5 implies 119880(119905 120591) is asymptotically compactwe obtain the conclusion immediately

According to Theorem 6 0 lt 120576 lt min]1205722 ]12058214 and119871 le ]2120582211205764119890

120576ℎ Under the assumptions (1198671)ndash(1198673) the NSVequation (1) with a distributed delay has a pullback attractor

Conflict of Interests

The authors declare that they have no competing interests

Authorsrsquo Contribution

Yantao Guo carried out the long time behavior of solutionsShuilin Cheng carried out the pullback attractor YanbinTang carried out the distributed delay All authors read andapproved the final paper

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China (Grant nos 11471129 and 11272277)

References

[1] A P Oskolkov ldquoThe uniqueness and solvability in the largeof boundary value problems for the equations of motion ofaqueous solutions of polymersrdquo Zapiski Nauchnykh SeminarovLeningrad Otdel Mathematics Institute Stekov (LOMI) vol 38pp 98ndash136 1973

Discrete Dynamics in Nature and Society 9

[2] Y Cao E M Lunasin and E S Titi ldquoGlobal well-posedness ofthe three-dimensional viscous and inviscid simplified Bardinaturbulence modelsrdquo Communications in Mathematical Sciencesvol 4 no 4 pp 823ndash848 2006

[3] V K Kalantarov and E S Titi ldquoGlobal attractors and determin-ing modes for the 3D Navier-Stokes-Voight equationsrdquo ChineseAnnals ofMathematics Series B vol 30 no 6 pp 697ndash714 2009

[4] G-C Yue and C-K Zhong ldquoAttractors for autonomous andnonautonomous 3D Navier-Stokes-Voight equationsrdquo Discreteand Continuous Dynamical Systems Series B vol 16 no 3 pp985ndash1002 2011

[5] J Garcıa-Luengo P Marın-Rubio and J Real ldquoPullback attrac-tors for three-dimensional non-autonomous Navier-Stokes-Voigt equationsrdquo Nonlinearity vol 25 no 4 pp 905ndash930 2012

[6] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977

[7] Y Tang and M Wang ldquoA remark on exponential stabilityof time-delayed Burgers equationrdquo Discrete and ContinuousDynamical Systems Series B vol 12 no 1 pp 219ndash225 2009

[8] Y Tang and L Zhou ldquoStability switch and Hopf bifurcationfor a diffusive prey-predator system with delayrdquo Journal ofMathematical Analysis and Applications vol 334 no 2 pp1290ndash1307 2007

[9] J H Wu Theory and Applications of Partial Differential Equa-tions Springer New York NY USA 1996

[10] L Zhou Y B Tang and S Hussein ldquoStability and Hopf bifur-cation for a delay competition diffusion systemrdquoChaos Solitonsamp Fractals vol 14 no 8 pp 1201ndash1225 2002

[11] T Caraballo and J Real ldquoNavier-Stokes equations with delaysrdquoThe Royal Society of London Proceedings Series A Mathemat-ical Physical and Engineering Sciences vol 457 no 2014 pp2441ndash2453 2001

[12] T Caraballo and J Real ldquoAsymptotic behaviour of two-dimen-sional Navier-Stokes equations with delaysrdquo The Royal Societyof London Proceedings Series A Mathematical Physical andEngineering Sciences vol 459 no 2040 pp 3181ndash3194 2003

[13] T Caraballo and J Real ldquoAttractors for 2D-Navier-Stokesmodels with delaysrdquo Journal of Differential Equations vol 205no 2 pp 271ndash297 2004

[14] Z Hu and Y Wang ldquoPullback attractors for a nonautonomousnonclassical diffusion equation with variable delayrdquo Journal ofMathematical Physics vol 53 no 7 Article ID 072702 17 pages2012

[15] H Y Li and Y M Qin ldquoPullback attractors for three-dimen-sional navier-stokes-voigt equations with delaysrdquo BoundaryValue Problems vol 2013 article 191 2013

[16] P Marın-Rubio and J Real ldquoAttractors for 2D-Navier-Stokesequations with delays on some unbounded domainsrdquoNonlinearAnalysis Theory Methods amp Applications vol 67 no 10 pp2784ndash2799 2007

[17] P Marın-Rubio J Real and J Valero ldquoPullback attractors for atwo-dimensional Navier-Stokesmodel in an infinite delay caserdquoNonlinear Analysis Theory Methods amp Applications vol 74 no5 pp 2012ndash2030 2011

[18] P Marın-Rubio and J Real ldquoPullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linearoperatorsrdquo Discrete and Continuous Dynamical Systems SeriesA vol 26 no 3 pp 989ndash1006 2010

[19] AKKhanmamedov ldquoGlobal attractors forwave equationswithnonlinear interior damping and critical exponentsrdquo Journal ofDifferential Equations vol 230 no 2 pp 702ndash719 2006

[20] L Yang and C-K Zhong ldquoGlobal attractor for plate equationwith nonlinear dampingrdquo Nonlinear Analysis Theory Methodsamp Applications vol 69 no 11 pp 3802ndash3810 2008

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 2: Research Article Approximate Kelvin-Voigt Fluid …downloads.hindawi.com/journals/ddns/2015/721673.pdfWe consider the approximate D Kelvin-Voigt uid driven by an external force depending

2 Discrete Dynamics in Nature and Society

pullback attractor for a Navier-Stokes equation with infinitediscrete delay effect was studied in [17]

The aim of this paper is to investigate the NSV equationwith a distributed delay instead of the discussions with finitedelays in the referencesOur purpose is twofoldWefirst showthe existence and uniqueness of solution to NSV equation(1) with a distributed delay then we prove the existenceof pullback attractor for the process generated by the NSVequation (1)

This paper is organized as follows In Section 2 we givesome preliminary results and prove existence of solution toNSV equationwith a distributed delay In Section 3 we derivethe existence of pullback attractor by prior estimates andcontractive functions

2 Existence of Solutions

In order to prove the existence of solutions to problem (1) wedefine the function spaces

V = 119906 isin (119862infin

0 (Ω))3 div119906= 0 (2)

119867 is the closure ofV in (1198712(Ω))3 with the inner product (sdot sdot)and associate norm | sdot |119881 is the closure ofV in (1198671

0 (Ω))3 with

scalar product ((sdot sdot)) and associate norm sdot where

((119906 V)) =3sum

119894119895=1int

Ω

120597119906119895

120597119909119894

120597V119895

120597119909119894

119889119909 forall119906 V isin (11986710 (Ω))

3 (3)

it follows that 119881 sub 119867 equiv 1198671015840sub 1198811015840 where the injections are

dense and compact We will use sdot lowastfor the norm in 1198811015840 and

⟨sdot sdot⟩ for the duality pairing between 119881 and 1198811015840Define the linear continuous operator 119860 119881 rarr 119881

1015840 as

⟨119860119906 V⟩ = ((119906 V)) forall119906 V isin 119881 (4)

We denote 119863(119860) = 119906 isin 119881119860119906 isin 119867 one has that 119863(119860) =(119867

2(Ω))

3cap 119881 and 119860119906 = minus119875Δ119906 for all 119906 isin 119863(119860) is the Stokes

operator where 119875 is the orthoprojector from (1198712(Ω))

3 onto119867 also denote 119862

119867= 119862

0([minusℎ 0]119867) and 119862

119881= 119862

0([minusℎ 0] 119881)

Define the trilinear form 119887 on 119881 times 119881 times 119881 by

119887 (119906 V 119908) =3sum

119894119895=1int

Ω

119906119894

120597V119895

120597119909119894

119908119895119889119909 forall119906 V 119908 isin 119881 (5)

and the operator 119861 119881 times 119881 rarr 1198811015840 as

⟨119861 (119906 V) 119908⟩ = 119887 (119906 V 119908) forall119906 V 119908 isin 119881 (6)

and denote 119861(119906) = 119861(119906 119906)The trilinear form 119887 satisfies that

119887 (119906 V 119908) = minus 119887 (119906 119908 V)

119887 (119906 V V) = 0

forall119906 V 119908 isin 119881

(7)

We also recall that there exists a constant 119862 depending onlyonΩ such that

|119887 (119906 V 119908)| le 119862 11990612

|119860119906|12

V |119908|

119906 isin 119863 (119860) V isin 119881 119908 isin 119867

|119887 (119906 V 119908)| le 119862 119906 V |119908|12 11990812

119906 isin 119881 V isin 119881 119908 isin 119881

|119887 (119906 V 119908)| le 119862 |119906|12

11990612

V 119908

119906 isin 119881 V isin 119881 119908 isin 119881

(8)

For the term containing the time delay 119866 119877 times 119867 rarr 119867

satisfies that

(1198671) 119866(sdot 119906) 119877 rarr 119867 is a measurable function

(1198672) 119866(119905 0) = 0 for all 119905 isin 119877

(1198673) there exists a positive constant 119871 such that forall119877 gt

0 if |119906| lt 119877 and |V| lt 119877 then

|119866 (119905 119906) minus119866 (119905 V)|2 le 119871 |119906 minus V|2 (9)

Remark 1 Hypotheses (1198672)-(1198673) imply that |119866(119905 119906)|2 le

119871|119906|2 so we have |119866(119905 119906)|2 isin 119871infin(120591 119879) for |119906| lt 119877

Problem (1) can be rewritten as

120597

120597119905

(119906 + 1205722119860119906)+ ]119860119906+119861 (119906 119906)

= int

0

minusℎ

119866 (119904 119906 (119905 + 119904)) 119889119904

119906 (119905 119909) = 120601 (119905 minus 120591 119909) 119905 isin [120591 minus ℎ 120591] 119909 isin Ω

(10)

then we get the existence of solution to problem (10)

Theorem 2 Let 120601 isin 119862119881 let 119866 119877times119867 rarr 119867 satisfy the hypo-

theses (1198671)ndash(1198673) and let 120591 isin R Then forall119879 gt 120591 there exists aunique weak solution to (10) such that

119906 (119905 119909) isin 119862 ([120591 minus ℎ 119879] 119881)

120597119906 (119905 119909)

120597119905

isin 1198712(120591 119879 119881)

(11)

Moreover if 120601 isin 119862119863(119860)

then problem (10) admits a strong sol-ution

Proof Consider the Galerkin approximations for problem(10)

119889

119889119905

(119906119898+120572

2119860119906119898) + ]119860119906119898 +119861 (119906119898 119906119898)

= int

0

minusℎ

119866 (119904 119906119898(119905 + 119904)) 119889119904

(12)

Discrete Dynamics in Nature and Society 3

where 119906119898 = sum119898

119895=1 119906119898

119895119890119895119860119906119898 = sum

119898

119895=1 120582119895119906119898

119895119890119895 and 119890

119895and120582

119895are

the corresponding orthonormal eigenfunctions and eigen-values of operator 119860 respectively then

10038171003817100381710038171199061198981003817100381710038171003817

2=

119898

sum

119895=1120582119895(119906119898

119895)

2

10038161003816100381610038161198601199061198981003816100381610038161003816

2=

119898

sum

119895=11205822119895(119906119898

119895)

2

10038161003816100381610038161199061198981003816100381610038161003816

2=

119898

sum

119895=1(119906119898

119895)

2

(13)

We now derive a prior estimate for the Galerkin approx-imate solution Multiplying (12) by 119906119898

119895 summing from 119895 = 1

to119898 and using the fact

(119861 (119906119898 119906119898) 119906119898) = 119887 (119906

119898 119906119898 119906119898) = 0 (14)

we obtain that for ae 119905 gt 120591119889

119889119905

(10038161003816100381610038161199061198981003816100381610038161003816

2+120572

2 10038171003817100381710038171199061198981003817100381710038171003817

2) + 2] 100381710038171003817

10038171199061198981003817100381710038171003817

2

= 2(int0

minusℎ

119866 (119904 119906119898(119905 + 119904)) 119889119904 119906

119898)

le

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906119898(119905 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

+10038161003816100381610038161199061198981003816100381610038161003816

2

(15)

Integrating (15) from 120591 to 119905 we obtain that

1003816100381610038161003816119906119898(119905)1003816100381610038161003816

2+120572

2 1003817100381710038171003817119906119898(119905)1003817100381710038171003817

2+ 2]int

119905

120591

1003817100381710038171003817119906119898(119904)1003817100381710038171003817

2119889119904

le1003816100381610038161003816119906119898(120591)

1003816100381610038161003816

2+120572

2 1003817100381710038171003817119906119898(120591)

1003817100381710038171003817

2

+int

119905

120591

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906119898(119903 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

119889119903

+int

119905

120591

1003816100381610038161003816119906119898(119903)1003816100381610038161003816

2119889119903

(16)

Remark 1 implies that

int

119905

120591

10038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906119898(119903 + 119904)) 119889119904

10038161003816100381610038161003816100381610038161003816

2

119889119903

le int

0

minusℎ

int

119905

120591

1003816100381610038161003816119866 (119904 119906

119898(119903 + 119904))

1003816100381610038161003816

2119889119903 119889119904

le int

0

minusℎ

int

119905

120591

1198711003816100381610038161003816119906119898(119903 + 119904)

1003816100381610038161003816

2119889119903 119889119904

le 119871int

0

minusℎ

int

119905+119904

120591+119904

1003816100381610038161003816119906119898(119903)1003816100381610038161003816

2119889119903 119889119904

le 119871int

0

minusℎ

(int

120591

120591minusℎ

1003816100381610038161003816119906119898(119903)1003816100381610038161003816

2119889119903 +int

119905

120591

1003816100381610038161003816119906119898(119903)1003816100381610038161003816

2119889119903) 119889119904

le 119871ℎ2 10038171003817100381710038171206011003817100381710038171003817

2119862119867

+119871ℎint

119905

120591

1003816100381610038161003816119906119898(119903)1003816100381610038161003816

2119889119903

(17)

Then forall119905 isin (120591 119879) and

1003816100381610038161003816119906119898(119905)1003816100381610038161003816

2+120572

2 1003817100381710038171003817119906119898(119905)1003817100381710038171003817

2+ 2]int

119905

120591

1003817100381710038171003817119906119898(119904)1003817100381710038171003817

2119889119904

le (119871ℎ + 2) int119905

120591

1003816100381610038161003816119906119898(119904)1003816100381610038161003816

2119889119904 + 119871ℎ

2 10038171003817100381710038171206011003817100381710038171003817

2119862119867

+10038161003816100381610038161206011003816100381610038161003816

2119862119867

+1205722 10038171003817100381710038171206011003817100381710038171003817

2119862119881

(18)

So we have

1003816100381610038161003816119906119898(119905)1003816100381610038161003816

2le 119862int

119905

120591

1003816100381610038161003816119906119898(119904)1003816100381610038161003816

2119889119904 +119862

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+10038161003816100381610038161206011003816100381610038161003816

2119862119867

+1205722 10038171003817100381710038171206011003817100381710038171003817

2119862119881

(19)

The Gronwall inequality implies that

1003816100381610038161003816119906119898(119905)1003816100381610038161003816

2le 119862 (20)

Putting (20) into the right-hand side of (18) we have

1205722 1003817100381710038171003817119906119898(119905)1003817100381710038171003817

2+ 2]int

119905

120591

1003817100381710038171003817119906119898(119904)1003817100381710038171003817

2119889119904 le 119862 forall119905 isin (120591 119879) (21)

This implies that

119906119898 is bounded in 119871

infin(120591 119879 119881) cap 119871

2(120591 119879 119881) (22)

Now multiplying (12) by 120597119905119906119898 and integrating overΩ we

have

10038161003816100381610038161205971199051199061198981003816100381610038161003816

2+120572

2 10038171003817100381710038171205971199051199061198981003817100381710038171003817

2+

]2119889

119889119905

10038171003817100381710038171199061198981003817100381710038171003817

2

le1003816100381610038161003816119887 (119906119898 119906119898 120597119905119906119898)1003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

(int

0

minusℎ

119866 (119904 119906119898(119905 + 119904)) 119889119904 120597

119905119906119898)

10038161003816100381610038161003816100381610038161003816

le 11988810038171003817100381710038171199061198981003817100381710038171003817

32 10038161003816100381610038161199061198981003816100381610038161003816

12 10038171003817100381710038171205971199051199061198981003817100381710038171003817

+

12

10038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906119898(119905 + 119904)) 119889119904

10038161003816100381610038161003816100381610038161003816

2

+

1210038161003816100381610038161205971199051199061198981003816100381610038161003816

2

(23)

since

10038171003817100381710038171199061198981003817100381710038171003817

32 10038161003816100381610038161199061198981003816100381610038161003816

12 10038171003817100381710038171205971199051199061198981003817100381710038171003817le 119888

10038171003817100381710038171199061198981003817100381710038171003817

3 10038161003816100381610038161199061198981003816100381610038161003816+

1205722

210038171003817100381710038171205971199051199061198981003817100381710038171003817

2 (24)

then

10038161003816100381610038161205971199051199061198981003816100381610038161003816

2+120572

2 10038171003817100381710038171205971199051199061198981003817100381710038171003817

2+ ]

119889

119889119905

10038171003817100381710038171199061198981003817100381710038171003817

2

le 11988810038171003817100381710038171199061198981003817100381710038171003817

3 10038161003816100381610038161199061198981003816100381610038161003816+

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906119898(119905 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

(25)

4 Discrete Dynamics in Nature and Society

integrating the above inequality from 120591 to 119905 by (17) (20) and(22) we have

int

119905

120591

(10038161003816100381610038161205971199051199061198981003816100381610038161003816

2+120572

2 10038171003817100381710038171205971199051199061198981003817100381710038171003817

2) 119889119904 + ] 100381710038171003817

1003817119906119898(119905)1003817100381710038171003817

2

le ] 1003817100381710038171003817119906119898(120591)

1003817100381710038171003817

2+ 119888int

119905

120591

10038171003817100381710038171199061198981003817100381710038171003817

3 10038161003816100381610038161199061198981003816100381610038161003816119889119904

+int

119905

120591

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906119898(119903 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

119889119903

le ] 1003817100381710038171003817120601 (120591)

1003817100381710038171003817

2+ 119888int

119905

120591

10038171003817100381710038171199061198981003817100381710038171003817

3 10038161003816100381610038161199061198981003816100381610038161003816119889119904 + 119888

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+ 119888int

119905

120591

1003816100381610038161003816119906119898(119903)1003816100381610038161003816

2119889119903

le ] 10038171003817100381710038171206011003817100381710038171003817

2119862119881+ 119888int

119905

120591

10038171003817100381710038171199061198981003817100381710038171003817

3 10038161003816100381610038161199061198981003816100381610038161003816119889119904 + 119888

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+ 119888int

119905

120591

1003816100381610038161003816119906119898(119903)1003816100381610038161003816

2119889119903

(26)

Since 119906119898 is bounded in 119871infin(120591 119879 119881) cap 1198712(120591 119879 119881) we obtainthat

120597119905119906119898 is bounded in 119871

2(120591 119879 119881) (27)

By the Faedo-Galerkin scheme for example see [14 18]according to the estimates (22) and (27) we can get existenceof the weak solution here we omit the details

We next consider the uniqueness of solution Let 119906 V betwo solutions to problem (10) corresponding the initial data120601 and 120595 respectively

Denote 119908 = 119906 minus V then we have

120597

120597119905

(119908+1205722119860119908)+ ]119860119908+119861 (119906 119906) minus 119861 (V V)

= int

0

minusℎ

119866 (119904 119906 (119905 + 119904)) 119889119904 minusint

0

minusℎ

119866 (119904 V (119905 + 119904)) 119889119904(28)

Multiplying (28) by 119908 and integrating overΩ we obtain

12119889

119889119905

(|119908|2+120572

2119908

2) + ] 1199082 + (119861 (119906 119906) 119908)

minus (119861 (V V) 119908)

= (int

0

minusℎ

119866 (119904 119906 (119905 + 119904)) 119889119904 minusint

0

minusℎ

119866 (119904 V (119905 + 119904)) 119889119904 119908)

(29)

Notice that

(119861 (119906 119906) 119908) minus (119861 (V V) 119908)

= 119887 (119906 119906 119906 minus V) minus 119887 (V 119906 119906 minus V) + 119887 (V 119906 119906 minus V)

minus 119887 (V V 119906 minus V)

= 119887 (119906 minus V 119906 119906 minus V) minus 119887 (V 119906 minus V 119906 minus V)

= 119887 (119908 119906 119908)

(30)

Substituting (30) into (29) and integrating from 120591 to 119905 weget

|119908 (119905)|2+120572

2119908 (119905)

2+ 2]int

119905

120591

119908 (119904)2119889119904 minus |119908 (120591)|

2

minus1205722119908 (120591)

2le int

119905

120591

|119887 (119908 119906 119908)| 119889119904 +int

119905

120591

|119908 (119904)|2119889119904

+int

119905

120591

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119903 + 119904)) 119889119904

minusint

0

minusℎ

119866 (119904 V (119903 + 119904)) 119889119904100381610038161003816100381610038161003816100381610038161003816

2

119889119903

(31)

(1198673) implies that

int

119905

120591

10038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119903 + 119904)) 119889119904 minusint

0

minusℎ

119866 (119904 V (119903 + 119904)) 11988911990410038161003816100381610038161003816100381610038161003816

2

119889119903

le int

0

minusℎ

int

119905

120591

|119866 (119904 119906 (119903 + 119904)) minus119866 (119904 V (119903 + 119904))|2 119889119903 119889119904

le int

0

minusℎ

int

119905

120591

|119866 (119904 119906 (119903 + 119904)) minus119866 (119904 V (119903 + 119904))|2 119889119903 119889119904

le 119871int

0

minusℎ

int

119905

120591

|119906 (119903 + 119904) minus V (119903 + 119904)|2 119889119903 119889119904

le 119871int

0

minusℎ

int

119905+119904

120591+119904

|119906 (119903) minus V (119903)|2 119889119903 119889119904

le 119871int

0

minusℎ

(int

120591

120591minusℎ

|119906 (119903) minus V (119903)|2 119889119903

+int

119905

120591

|119906 (119903) minus V (119903)|2 119889119903) 119889119904 le 119871ℎ2 1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119867

+119871ℎint

119905

120591

|119908 (119904)|2119889119904

(32)

As the property of operator 119887 and Poincare we have

int

119905

120591

|119887 (119908 119906 119908)| 119889119904 le 119862int

119905

120591

11990832

|119908|12

119906 119889119904

le 119862int

119905

120591

1199082119906 119889119904

(33)

Discrete Dynamics in Nature and Society 5

Substituting (32) and (33) into (31) we get

|119908 (119905)|2+120572

2119908 (119905)

2+ 2]int

119905

120591

119908 (119904)2119889119904

le int

119905

120591

1199082119906 119889119904 + 119871ℎ

2 1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119867

+ (119871ℎ + 1) int119905

120591

|119908 (119904)|2119889119904 + |119908 (120591)|

2

+1205722119908 (120591)

2

le int

119905

120591

1199082119906 119889119904 + 119871ℎ

2 1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119867

+ (119871ℎ + 1) int119905

120591

|119908 (119904)|2119889119904 +

1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119867

+1205722 1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119881

le 119862int

119905

120591

119908 (119904)2119889119904 +119862

1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119881

(34)

The last inequality comes from Poincare inequality andthe boundedness of 119906 Therefore the Gronwall inequalityimplies the uniqueness of the solution The proof is com-plete

3 Existence of Pullback Attractor

In this section we will prove the existence of pullbackattractor to problem (10) First we give existence of pullbackabsorbing set for the process 119880(119905 120591) generated by the globalsolution to problem (10)

Lemma 3 Assume (1198671)ndash(1198673) hold and 119871 le ]2120582211205764119890120576ℎ then

the process 119880(119905 120591) is pullback dissipative where 0 lt 120576 lt

min]1205722 ]12058214

Proof Multiplying (10) by119906 and integrating overΩ we obtain

120597

120597119905

(|119906|2+120572

2119906

2) + 2] 1199062

le 120578 |119906|2+

1120578

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119905 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

(35)

where 120578 is a constant determined laterBy Poincare inequality we have

120597

120597119905

(|119906|2+120572

2119906

2) + ] 1199062 + (]1205821 minus 120578) |119906|

2

le

1120578

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119905 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

(36)

Since

119890minus120576119905 120597

120597119905

[119890120576119905(|119906|

2+120572

2119906

2)]

= 120576 (|119906|2+120572

2119906

2) +

120597

120597119905

(|119906|2+120572

2119906

2)

(37)

then

119890minus120576119905 120597

120597119905

[119890120576119905(|119906|

2+120572

2119906

2)]

le (120576 + 120578 minus ]1205821) |119906|2+ (120576120572

2minus ]) 1199062

+

1120578

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119905 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

(38)

Integrating (38) from 120591 to 119905 we get

119890120576119905(|119906 (119905)|

2+120572

2119906 (119905)

2)

minus 119890120576120591(|119906 (120591)|

2+120572

2119906 (120591)

2)

le int

119905

120591

119890120576119904[(120576 + 120578 minus ]1205821) |119906|

2+ (120576120572

2minus ]) 1199062] 119889119904

+

1120578

int

119905

120591

119890120576119903

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119903 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

119889119903

(39)

Assumptions (1198671)ndash(1198673) imply that

int

119905

120591

119890120576119903

10038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119903 + 119904)) 119889119904

10038161003816100381610038161003816100381610038161003816

2

119889119903

le int

119905

120591

119890120576119903int

0

minusℎ

|119866 (119904 119906 (119903 + 119904))|2119889119904 119889119903

le 119871int

0

minusℎ

int

119905

120591

119890120576119903|119906 (119903 + 119904)|

2119889119903 119889119904

le 119871int

0

minusℎ

int

119905+119904

120591+119904

119890120576(119903minus119904)

|119906 (119903)|2119889119903 119889119904

le 119871int

0

minusℎ

int

119905

120591minusℎ

119890120576(119903minus119904)

|119906 (119903)|2119889119903 119889119904

le 119871int

0

minusℎ

119890minus120576119904

int

119905

120591minusℎ

119890120576119903|119906 (119903)|

2119889119903 119889119904

le 119871int

0

minusℎ

119890minus120576119904

(int

120591

120591minusℎ

119890120576119903|119906 (119903)|

2119889119903

+int

119905

120591

119890120576119903|119906 (119903)|

2119889119903) 119889119904 le

119871119890120576ℎ

120576

(

1120576

10038171003817100381710038171206011003817100381710038171003817

2119862119867

119890120576120591

+int

119905

120591

119890120576119903|119906 (119903)|

2119889119903) le

119871119890120576(120591+ℎ)

1205762

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+

119871119890120576ℎ

120576

sdot int

119905

120591

119890120576119903|119906 (119903)|

2119889119903

(40)

Substituting (40) into (39) we have

119890120576119905(|119906 (119905)|

2+120572

2119906 (119905)

2) minus 119890120576120591(|119906 (120591)|

2+120572

2119906 (120591)

2)

le int

119905

120591

119890120576119904[(120576 + 120578 minus ]1205821 +

119871119890120576ℎ

2120578120576) |119906|

2

+ (1205761205722minus ]) 1199062]119889119904 +

119871119890120576(120591+ℎ)

2120578120576210038171003817100381710038171206011003817100381710038171003817

2119862119867

(41)

6 Discrete Dynamics in Nature and Society

Let 120578 = (12)]1205821 choosing 0 lt 120576 lt min]1205722 ]12058214 119871 le

]2120582211205764119890120576ℎ implies that

max120576+ 120578minus ]1205821 +120578minus1119871119890120576ℎ

2120576 120576120572

2minus ] lt 0 (42)

then

119890120576119905(|119906 (119905)|

2+120572

2119906 (119905)

2)

le

119871119890120576(120591+ℎ)

1205762]1205821

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+ 119890120576120591(|119906 (120591)|

2+120572

2119906 (120591)

2)

(43)

which implies

|119906 (119905)|2+120572

2119906 (119905)

2

le

119871119890120576(120591+ℎminus119905)

1205762]1205821

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+ 119890120576(120591minus119905)

(1003816100381610038161003816120601 (120591)

1003816100381610038161003816

2+120572

2 1003817100381710038171003817120601 (120591)

1003817100381710038171003817

2)

(44)

Now if we take 119905 ge 120591 + ℎ then for 120579 isin [minusℎ 0] we have

|119906 (119905 + 120579)|2+120572

2119906 (119905 + 120579)

2

le

119871119890120576(120591+ℎminus119905minus120579)

1205762]1205821

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+ 119890120576(120591minus119905minus120579)

(10038171003817100381710038171206011003817100381710038171003817

2119862119867

+1205722 10038171003817100381710038171206011003817100381710038171003817

2119862119881)

(45)

We denote byR the set of all functions 119903 (minusinfin +infin) rarr

(0 +infin) such that

lim120591rarrminusinfin

1198901205721205911199032(120591) = 0 (46)

Then the closed ball in 119862119881defined by

119861 = 120593 isin11986211988110038171003817100381710038171205931003817100381710038171003817

2119862119881le 1 (47)

is pullback absorbing set for 119880(119905 120591) The proof is complete

We next prove the asymptotic compactness of solution toproblem (10) by contractive functions see [19 20]

Let 119883 be a Banach space and let 119861 be a bounded subsetof 119883 We call a function Φ(sdot sdot) which defined on 119883 times 119883 is acontractive function on 119861times119861 if for any sequence 119909

119899infin

119899=1sub 119861

there is a subsequence 119909119899119896infin

119896=1 sub 119909119899infin

119899=1 such that

lim119897rarrinfin

lim119896rarrinfin

Φ(119909119899119896 119909119899119897) = 0 (48)

Denote all such contractive functions on 119861 times 119861 by 119862(119861)

Theorem 4 (see [19]) Let 119878(119905)119905ge120591

be a semigroup on aBanach space (119883 sdot ) and have a bounded absorbing set 119861forall120598 gt 0 there exist 119879 = 119879(119861 120598) and Φ isin 119862(119861) such that

1003817100381710038171003817119878 (119879) 119909 minus 119878 (119879) 119910

1003817100381710038171003817le 120598 +Φ

119879(119909 119910) forall119909 119910 isin 119861 (49)

where Φ119879depends on 119879 Then 119878(119905)

119905ge120591is asymptotically

compact in119883

Lemma 5 Assume that (1198671)ndash(1198673) hold the process 119880(119905 120591)119905ge120591generated by the global solution to problem (10) is asymptoti-cally compact

Proof Let 119906119894(119905) be the solution to problem (10) with initialdata 120601119894 isin 119861 (119894 = 1 2) respectively Denote V(119905) = 119906

1(119905)minus119906

2(119905)

then V(119905) satisfies the equivalent abstract equation

120597

120597119905

(V+1205722119860V) + ]119860V+119861 (1199061 1199061) minus119861 (1199062 1199062)

= int

0

minusℎ

[119866 (119904 1199061(119905 + 119904)) minus119866 (s 1199062 (119905 + 119904))] 119889119904

(50)

with the initial condition V(119905) = 1206011(119905 minus 120591) minus 120601

2(119905 minus 120591) 119905 isin

[120591 minus ℎ 120591]Set an energy function

119864V (119905) =12int

Ω

|V|2 119889119909+1205722

2int

Ω

|nablaV|2 119889119909 (51)

Multiplying (50) by V and integrating over [119904 119879]timesΩwith 119879 gt

119905 + 120591 119904 ge 120591 we have

119864V (119879) minus119864V (119904) + ]int119879

119904

int

Ω

|nablaV|2 119889119909 119889119903

+int

119879

119904

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

= int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

(52)

then we have

]int119879

119904

int

Ω

|nablaV|2 119889119909 119889119903 le 119864V (119904) minusint119879

119904

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

+int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

(53)

Discrete Dynamics in Nature and Society 7

Using Poincare inequality and (51) and (53) we have

int

119879

120591

119864V (119904) 119889119904 =12int

119879

120591

int

Ω

|V|2 119889119909 119889119903 +1205722

2int

119879

120591

int

Ω

|nablaV|2 119889119909 119889119903 le 119862int

119879

120591

int

Ω

|nablaV|2 119889119909 119889119903

le 119862119864V (120591) minus119862int119879

120591

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

+119862int

119879

120591

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

(54)

Integrating (52) from 120591 to 119879 with respect to 119904 we obtain

119879119864V (119879) + ]int119879

120591

int

119879

119904

int

Ω

|nablaV|2 119889119909 119889119903 119889119904 = int

119879

120591

119864V (119904) 119889119904 minusint119879

120591

int

119879

119904

int

Ω

[119861 (1199061) minus 119861 (119906

2)] V (119903) 119889119909 119889119903 119889119904

+int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903 119889119904

(55)

Substituting (54) into (55) we get

119879119864V (119879) + ]int119879

120591

int

119879

119904

int

Ω

|nablaV|2 119889119909 119889119903 119889119904 le 119862119864V (120591) minus119862int119879

120591

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

+119862int

119879

120591

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

minusint

119879

120591

int

119879

119904

int

Ω

[119861 (1199061) minus 119861 (119906

2)] V (119903) 119889119909 119889119903 119889119904

+int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903 119889119904

(56)

Set

1198620 = 119862119864V (120591) =119862

2int

Ω

100381610038161003816100381610038161206011(0) minus 1206012 (0)1003816100381610038161003816

1003816

2119889119909+

1198621205722

2int

Ω

10038161003816100381610038161003816nabla120601

1(0) minus nabla1206012 (0)1003816100381610038161003816

1003816

2119889119909 (57)

Φ(1199061 119906

2) = minus119862int

119879

120591

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

+119862int

119879

120591

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

minusint

119879

120591

int

119879

119904

int

Ω

[119861 (1199061) minus 119861 (119906

2)] V (119903) 119889119909 119889119903 119889119904

+int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903 119889119904

(58)

8 Discrete Dynamics in Nature and Society

then by (56) we have

119864V (119879) le1198620119879

+

1119879

Φ(1199061 119906

2) (59)

One has forall120598 gt 0 we choose 119879 gt 0 large enough such that1198620119879 lt 120598 By Theorem 4 it suffices to prove that Φ(1199061 1199062) is

a contractive function since119861 is a bounded positive invariantset

If 119906119898 rarr 119906 (119898 rarr infin) we have the limits

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

(119861 (119906119899) minus 119861 (119906

119898)) (119906119899minus 119906119898) 119889119909 119889119903

= lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

[((119906119899minus 119906119898) sdot nabla) 119906

119899minus (119906119898sdot nabla) (119906

119899minus 119906119898)] (119906119899minus 119906119898) 119889119909 119889119903 = 0

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

(119861 (119906119899) minus 119861 (119906

119898)) (119906119899minus 119906119898) 119889119909 119889119903 119889119904

= lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

[((119906119899minus 119906119898) sdot nabla) 119906

119899minus (119906119898sdot nabla) (119906

119899minus 119906119898)] (119906119899minus 119906119898) 119889119909 119889119903 119889119904 = 0

(60)

By (1198671)ndash(1198673) we have

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

int

0

minusℎ

(119866 (120590 119906119899(119903 + 120590)) minus 119866 (120590 119906

119898(119903 + 120590))) V (119903) 119889120590 119889119909 119889119903

le lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

int

0

minusℎ

1198711003816100381610038161003816119906119899(119903 + 120590) minus 119906

119898(119903 + 120590)

1003816100381610038161003816

1003816100381610038161003816119906119899(119903) minus 119906

119898(119903)1003816100381610038161003816119889120590 119889119909 119889119903 = 0

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

(119866 (120590 119906119899(119903 + 120590)) minus 119866 (120590 119906

119898(119903 + 120590))) V (119903) 119889120590 119889119909 119889119903 119889119904

le lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

1198711003816100381610038161003816(119906119899minus 119906119898) (119903 + 120590)

1003816100381610038161003816

1003816100381610038161003816(119906119899minus 119906119898) (119903)

1003816100381610038161003816119889120590 119889119909 119889119903 119889119904 = 0

(61)

Combining (60)-(61) with (58) we have that Φ(1199061 1199062) is acontractive function

The proof is complete

Theorem 6 Assume that (1198671)ndash(1198673) hold and 119871 le ]2120582211205764119890120576ℎ then the process 119880(119905 120591) generated by the global solutionfor problem (10) has a pullback attractor

Proof Lemma 3 implies 119880(119905 120591) has a pullback absorbing set119861 and Lemma 5 implies 119880(119905 120591) is asymptotically compactwe obtain the conclusion immediately

According to Theorem 6 0 lt 120576 lt min]1205722 ]12058214 and119871 le ]2120582211205764119890

120576ℎ Under the assumptions (1198671)ndash(1198673) the NSVequation (1) with a distributed delay has a pullback attractor

Conflict of Interests

The authors declare that they have no competing interests

Authorsrsquo Contribution

Yantao Guo carried out the long time behavior of solutionsShuilin Cheng carried out the pullback attractor YanbinTang carried out the distributed delay All authors read andapproved the final paper

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China (Grant nos 11471129 and 11272277)

References

[1] A P Oskolkov ldquoThe uniqueness and solvability in the largeof boundary value problems for the equations of motion ofaqueous solutions of polymersrdquo Zapiski Nauchnykh SeminarovLeningrad Otdel Mathematics Institute Stekov (LOMI) vol 38pp 98ndash136 1973

Discrete Dynamics in Nature and Society 9

[2] Y Cao E M Lunasin and E S Titi ldquoGlobal well-posedness ofthe three-dimensional viscous and inviscid simplified Bardinaturbulence modelsrdquo Communications in Mathematical Sciencesvol 4 no 4 pp 823ndash848 2006

[3] V K Kalantarov and E S Titi ldquoGlobal attractors and determin-ing modes for the 3D Navier-Stokes-Voight equationsrdquo ChineseAnnals ofMathematics Series B vol 30 no 6 pp 697ndash714 2009

[4] G-C Yue and C-K Zhong ldquoAttractors for autonomous andnonautonomous 3D Navier-Stokes-Voight equationsrdquo Discreteand Continuous Dynamical Systems Series B vol 16 no 3 pp985ndash1002 2011

[5] J Garcıa-Luengo P Marın-Rubio and J Real ldquoPullback attrac-tors for three-dimensional non-autonomous Navier-Stokes-Voigt equationsrdquo Nonlinearity vol 25 no 4 pp 905ndash930 2012

[6] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977

[7] Y Tang and M Wang ldquoA remark on exponential stabilityof time-delayed Burgers equationrdquo Discrete and ContinuousDynamical Systems Series B vol 12 no 1 pp 219ndash225 2009

[8] Y Tang and L Zhou ldquoStability switch and Hopf bifurcationfor a diffusive prey-predator system with delayrdquo Journal ofMathematical Analysis and Applications vol 334 no 2 pp1290ndash1307 2007

[9] J H Wu Theory and Applications of Partial Differential Equa-tions Springer New York NY USA 1996

[10] L Zhou Y B Tang and S Hussein ldquoStability and Hopf bifur-cation for a delay competition diffusion systemrdquoChaos Solitonsamp Fractals vol 14 no 8 pp 1201ndash1225 2002

[11] T Caraballo and J Real ldquoNavier-Stokes equations with delaysrdquoThe Royal Society of London Proceedings Series A Mathemat-ical Physical and Engineering Sciences vol 457 no 2014 pp2441ndash2453 2001

[12] T Caraballo and J Real ldquoAsymptotic behaviour of two-dimen-sional Navier-Stokes equations with delaysrdquo The Royal Societyof London Proceedings Series A Mathematical Physical andEngineering Sciences vol 459 no 2040 pp 3181ndash3194 2003

[13] T Caraballo and J Real ldquoAttractors for 2D-Navier-Stokesmodels with delaysrdquo Journal of Differential Equations vol 205no 2 pp 271ndash297 2004

[14] Z Hu and Y Wang ldquoPullback attractors for a nonautonomousnonclassical diffusion equation with variable delayrdquo Journal ofMathematical Physics vol 53 no 7 Article ID 072702 17 pages2012

[15] H Y Li and Y M Qin ldquoPullback attractors for three-dimen-sional navier-stokes-voigt equations with delaysrdquo BoundaryValue Problems vol 2013 article 191 2013

[16] P Marın-Rubio and J Real ldquoAttractors for 2D-Navier-Stokesequations with delays on some unbounded domainsrdquoNonlinearAnalysis Theory Methods amp Applications vol 67 no 10 pp2784ndash2799 2007

[17] P Marın-Rubio J Real and J Valero ldquoPullback attractors for atwo-dimensional Navier-Stokesmodel in an infinite delay caserdquoNonlinear Analysis Theory Methods amp Applications vol 74 no5 pp 2012ndash2030 2011

[18] P Marın-Rubio and J Real ldquoPullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linearoperatorsrdquo Discrete and Continuous Dynamical Systems SeriesA vol 26 no 3 pp 989ndash1006 2010

[19] AKKhanmamedov ldquoGlobal attractors forwave equationswithnonlinear interior damping and critical exponentsrdquo Journal ofDifferential Equations vol 230 no 2 pp 702ndash719 2006

[20] L Yang and C-K Zhong ldquoGlobal attractor for plate equationwith nonlinear dampingrdquo Nonlinear Analysis Theory Methodsamp Applications vol 69 no 11 pp 3802ndash3810 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 3: Research Article Approximate Kelvin-Voigt Fluid …downloads.hindawi.com/journals/ddns/2015/721673.pdfWe consider the approximate D Kelvin-Voigt uid driven by an external force depending

Discrete Dynamics in Nature and Society 3

where 119906119898 = sum119898

119895=1 119906119898

119895119890119895119860119906119898 = sum

119898

119895=1 120582119895119906119898

119895119890119895 and 119890

119895and120582

119895are

the corresponding orthonormal eigenfunctions and eigen-values of operator 119860 respectively then

10038171003817100381710038171199061198981003817100381710038171003817

2=

119898

sum

119895=1120582119895(119906119898

119895)

2

10038161003816100381610038161198601199061198981003816100381610038161003816

2=

119898

sum

119895=11205822119895(119906119898

119895)

2

10038161003816100381610038161199061198981003816100381610038161003816

2=

119898

sum

119895=1(119906119898

119895)

2

(13)

We now derive a prior estimate for the Galerkin approx-imate solution Multiplying (12) by 119906119898

119895 summing from 119895 = 1

to119898 and using the fact

(119861 (119906119898 119906119898) 119906119898) = 119887 (119906

119898 119906119898 119906119898) = 0 (14)

we obtain that for ae 119905 gt 120591119889

119889119905

(10038161003816100381610038161199061198981003816100381610038161003816

2+120572

2 10038171003817100381710038171199061198981003817100381710038171003817

2) + 2] 100381710038171003817

10038171199061198981003817100381710038171003817

2

= 2(int0

minusℎ

119866 (119904 119906119898(119905 + 119904)) 119889119904 119906

119898)

le

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906119898(119905 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

+10038161003816100381610038161199061198981003816100381610038161003816

2

(15)

Integrating (15) from 120591 to 119905 we obtain that

1003816100381610038161003816119906119898(119905)1003816100381610038161003816

2+120572

2 1003817100381710038171003817119906119898(119905)1003817100381710038171003817

2+ 2]int

119905

120591

1003817100381710038171003817119906119898(119904)1003817100381710038171003817

2119889119904

le1003816100381610038161003816119906119898(120591)

1003816100381610038161003816

2+120572

2 1003817100381710038171003817119906119898(120591)

1003817100381710038171003817

2

+int

119905

120591

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906119898(119903 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

119889119903

+int

119905

120591

1003816100381610038161003816119906119898(119903)1003816100381610038161003816

2119889119903

(16)

Remark 1 implies that

int

119905

120591

10038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906119898(119903 + 119904)) 119889119904

10038161003816100381610038161003816100381610038161003816

2

119889119903

le int

0

minusℎ

int

119905

120591

1003816100381610038161003816119866 (119904 119906

119898(119903 + 119904))

1003816100381610038161003816

2119889119903 119889119904

le int

0

minusℎ

int

119905

120591

1198711003816100381610038161003816119906119898(119903 + 119904)

1003816100381610038161003816

2119889119903 119889119904

le 119871int

0

minusℎ

int

119905+119904

120591+119904

1003816100381610038161003816119906119898(119903)1003816100381610038161003816

2119889119903 119889119904

le 119871int

0

minusℎ

(int

120591

120591minusℎ

1003816100381610038161003816119906119898(119903)1003816100381610038161003816

2119889119903 +int

119905

120591

1003816100381610038161003816119906119898(119903)1003816100381610038161003816

2119889119903) 119889119904

le 119871ℎ2 10038171003817100381710038171206011003817100381710038171003817

2119862119867

+119871ℎint

119905

120591

1003816100381610038161003816119906119898(119903)1003816100381610038161003816

2119889119903

(17)

Then forall119905 isin (120591 119879) and

1003816100381610038161003816119906119898(119905)1003816100381610038161003816

2+120572

2 1003817100381710038171003817119906119898(119905)1003817100381710038171003817

2+ 2]int

119905

120591

1003817100381710038171003817119906119898(119904)1003817100381710038171003817

2119889119904

le (119871ℎ + 2) int119905

120591

1003816100381610038161003816119906119898(119904)1003816100381610038161003816

2119889119904 + 119871ℎ

2 10038171003817100381710038171206011003817100381710038171003817

2119862119867

+10038161003816100381610038161206011003816100381610038161003816

2119862119867

+1205722 10038171003817100381710038171206011003817100381710038171003817

2119862119881

(18)

So we have

1003816100381610038161003816119906119898(119905)1003816100381610038161003816

2le 119862int

119905

120591

1003816100381610038161003816119906119898(119904)1003816100381610038161003816

2119889119904 +119862

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+10038161003816100381610038161206011003816100381610038161003816

2119862119867

+1205722 10038171003817100381710038171206011003817100381710038171003817

2119862119881

(19)

The Gronwall inequality implies that

1003816100381610038161003816119906119898(119905)1003816100381610038161003816

2le 119862 (20)

Putting (20) into the right-hand side of (18) we have

1205722 1003817100381710038171003817119906119898(119905)1003817100381710038171003817

2+ 2]int

119905

120591

1003817100381710038171003817119906119898(119904)1003817100381710038171003817

2119889119904 le 119862 forall119905 isin (120591 119879) (21)

This implies that

119906119898 is bounded in 119871

infin(120591 119879 119881) cap 119871

2(120591 119879 119881) (22)

Now multiplying (12) by 120597119905119906119898 and integrating overΩ we

have

10038161003816100381610038161205971199051199061198981003816100381610038161003816

2+120572

2 10038171003817100381710038171205971199051199061198981003817100381710038171003817

2+

]2119889

119889119905

10038171003817100381710038171199061198981003817100381710038171003817

2

le1003816100381610038161003816119887 (119906119898 119906119898 120597119905119906119898)1003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

(int

0

minusℎ

119866 (119904 119906119898(119905 + 119904)) 119889119904 120597

119905119906119898)

10038161003816100381610038161003816100381610038161003816

le 11988810038171003817100381710038171199061198981003817100381710038171003817

32 10038161003816100381610038161199061198981003816100381610038161003816

12 10038171003817100381710038171205971199051199061198981003817100381710038171003817

+

12

10038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906119898(119905 + 119904)) 119889119904

10038161003816100381610038161003816100381610038161003816

2

+

1210038161003816100381610038161205971199051199061198981003816100381610038161003816

2

(23)

since

10038171003817100381710038171199061198981003817100381710038171003817

32 10038161003816100381610038161199061198981003816100381610038161003816

12 10038171003817100381710038171205971199051199061198981003817100381710038171003817le 119888

10038171003817100381710038171199061198981003817100381710038171003817

3 10038161003816100381610038161199061198981003816100381610038161003816+

1205722

210038171003817100381710038171205971199051199061198981003817100381710038171003817

2 (24)

then

10038161003816100381610038161205971199051199061198981003816100381610038161003816

2+120572

2 10038171003817100381710038171205971199051199061198981003817100381710038171003817

2+ ]

119889

119889119905

10038171003817100381710038171199061198981003817100381710038171003817

2

le 11988810038171003817100381710038171199061198981003817100381710038171003817

3 10038161003816100381610038161199061198981003816100381610038161003816+

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906119898(119905 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

(25)

4 Discrete Dynamics in Nature and Society

integrating the above inequality from 120591 to 119905 by (17) (20) and(22) we have

int

119905

120591

(10038161003816100381610038161205971199051199061198981003816100381610038161003816

2+120572

2 10038171003817100381710038171205971199051199061198981003817100381710038171003817

2) 119889119904 + ] 100381710038171003817

1003817119906119898(119905)1003817100381710038171003817

2

le ] 1003817100381710038171003817119906119898(120591)

1003817100381710038171003817

2+ 119888int

119905

120591

10038171003817100381710038171199061198981003817100381710038171003817

3 10038161003816100381610038161199061198981003816100381610038161003816119889119904

+int

119905

120591

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906119898(119903 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

119889119903

le ] 1003817100381710038171003817120601 (120591)

1003817100381710038171003817

2+ 119888int

119905

120591

10038171003817100381710038171199061198981003817100381710038171003817

3 10038161003816100381610038161199061198981003816100381610038161003816119889119904 + 119888

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+ 119888int

119905

120591

1003816100381610038161003816119906119898(119903)1003816100381610038161003816

2119889119903

le ] 10038171003817100381710038171206011003817100381710038171003817

2119862119881+ 119888int

119905

120591

10038171003817100381710038171199061198981003817100381710038171003817

3 10038161003816100381610038161199061198981003816100381610038161003816119889119904 + 119888

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+ 119888int

119905

120591

1003816100381610038161003816119906119898(119903)1003816100381610038161003816

2119889119903

(26)

Since 119906119898 is bounded in 119871infin(120591 119879 119881) cap 1198712(120591 119879 119881) we obtainthat

120597119905119906119898 is bounded in 119871

2(120591 119879 119881) (27)

By the Faedo-Galerkin scheme for example see [14 18]according to the estimates (22) and (27) we can get existenceof the weak solution here we omit the details

We next consider the uniqueness of solution Let 119906 V betwo solutions to problem (10) corresponding the initial data120601 and 120595 respectively

Denote 119908 = 119906 minus V then we have

120597

120597119905

(119908+1205722119860119908)+ ]119860119908+119861 (119906 119906) minus 119861 (V V)

= int

0

minusℎ

119866 (119904 119906 (119905 + 119904)) 119889119904 minusint

0

minusℎ

119866 (119904 V (119905 + 119904)) 119889119904(28)

Multiplying (28) by 119908 and integrating overΩ we obtain

12119889

119889119905

(|119908|2+120572

2119908

2) + ] 1199082 + (119861 (119906 119906) 119908)

minus (119861 (V V) 119908)

= (int

0

minusℎ

119866 (119904 119906 (119905 + 119904)) 119889119904 minusint

0

minusℎ

119866 (119904 V (119905 + 119904)) 119889119904 119908)

(29)

Notice that

(119861 (119906 119906) 119908) minus (119861 (V V) 119908)

= 119887 (119906 119906 119906 minus V) minus 119887 (V 119906 119906 minus V) + 119887 (V 119906 119906 minus V)

minus 119887 (V V 119906 minus V)

= 119887 (119906 minus V 119906 119906 minus V) minus 119887 (V 119906 minus V 119906 minus V)

= 119887 (119908 119906 119908)

(30)

Substituting (30) into (29) and integrating from 120591 to 119905 weget

|119908 (119905)|2+120572

2119908 (119905)

2+ 2]int

119905

120591

119908 (119904)2119889119904 minus |119908 (120591)|

2

minus1205722119908 (120591)

2le int

119905

120591

|119887 (119908 119906 119908)| 119889119904 +int

119905

120591

|119908 (119904)|2119889119904

+int

119905

120591

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119903 + 119904)) 119889119904

minusint

0

minusℎ

119866 (119904 V (119903 + 119904)) 119889119904100381610038161003816100381610038161003816100381610038161003816

2

119889119903

(31)

(1198673) implies that

int

119905

120591

10038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119903 + 119904)) 119889119904 minusint

0

minusℎ

119866 (119904 V (119903 + 119904)) 11988911990410038161003816100381610038161003816100381610038161003816

2

119889119903

le int

0

minusℎ

int

119905

120591

|119866 (119904 119906 (119903 + 119904)) minus119866 (119904 V (119903 + 119904))|2 119889119903 119889119904

le int

0

minusℎ

int

119905

120591

|119866 (119904 119906 (119903 + 119904)) minus119866 (119904 V (119903 + 119904))|2 119889119903 119889119904

le 119871int

0

minusℎ

int

119905

120591

|119906 (119903 + 119904) minus V (119903 + 119904)|2 119889119903 119889119904

le 119871int

0

minusℎ

int

119905+119904

120591+119904

|119906 (119903) minus V (119903)|2 119889119903 119889119904

le 119871int

0

minusℎ

(int

120591

120591minusℎ

|119906 (119903) minus V (119903)|2 119889119903

+int

119905

120591

|119906 (119903) minus V (119903)|2 119889119903) 119889119904 le 119871ℎ2 1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119867

+119871ℎint

119905

120591

|119908 (119904)|2119889119904

(32)

As the property of operator 119887 and Poincare we have

int

119905

120591

|119887 (119908 119906 119908)| 119889119904 le 119862int

119905

120591

11990832

|119908|12

119906 119889119904

le 119862int

119905

120591

1199082119906 119889119904

(33)

Discrete Dynamics in Nature and Society 5

Substituting (32) and (33) into (31) we get

|119908 (119905)|2+120572

2119908 (119905)

2+ 2]int

119905

120591

119908 (119904)2119889119904

le int

119905

120591

1199082119906 119889119904 + 119871ℎ

2 1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119867

+ (119871ℎ + 1) int119905

120591

|119908 (119904)|2119889119904 + |119908 (120591)|

2

+1205722119908 (120591)

2

le int

119905

120591

1199082119906 119889119904 + 119871ℎ

2 1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119867

+ (119871ℎ + 1) int119905

120591

|119908 (119904)|2119889119904 +

1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119867

+1205722 1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119881

le 119862int

119905

120591

119908 (119904)2119889119904 +119862

1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119881

(34)

The last inequality comes from Poincare inequality andthe boundedness of 119906 Therefore the Gronwall inequalityimplies the uniqueness of the solution The proof is com-plete

3 Existence of Pullback Attractor

In this section we will prove the existence of pullbackattractor to problem (10) First we give existence of pullbackabsorbing set for the process 119880(119905 120591) generated by the globalsolution to problem (10)

Lemma 3 Assume (1198671)ndash(1198673) hold and 119871 le ]2120582211205764119890120576ℎ then

the process 119880(119905 120591) is pullback dissipative where 0 lt 120576 lt

min]1205722 ]12058214

Proof Multiplying (10) by119906 and integrating overΩ we obtain

120597

120597119905

(|119906|2+120572

2119906

2) + 2] 1199062

le 120578 |119906|2+

1120578

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119905 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

(35)

where 120578 is a constant determined laterBy Poincare inequality we have

120597

120597119905

(|119906|2+120572

2119906

2) + ] 1199062 + (]1205821 minus 120578) |119906|

2

le

1120578

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119905 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

(36)

Since

119890minus120576119905 120597

120597119905

[119890120576119905(|119906|

2+120572

2119906

2)]

= 120576 (|119906|2+120572

2119906

2) +

120597

120597119905

(|119906|2+120572

2119906

2)

(37)

then

119890minus120576119905 120597

120597119905

[119890120576119905(|119906|

2+120572

2119906

2)]

le (120576 + 120578 minus ]1205821) |119906|2+ (120576120572

2minus ]) 1199062

+

1120578

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119905 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

(38)

Integrating (38) from 120591 to 119905 we get

119890120576119905(|119906 (119905)|

2+120572

2119906 (119905)

2)

minus 119890120576120591(|119906 (120591)|

2+120572

2119906 (120591)

2)

le int

119905

120591

119890120576119904[(120576 + 120578 minus ]1205821) |119906|

2+ (120576120572

2minus ]) 1199062] 119889119904

+

1120578

int

119905

120591

119890120576119903

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119903 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

119889119903

(39)

Assumptions (1198671)ndash(1198673) imply that

int

119905

120591

119890120576119903

10038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119903 + 119904)) 119889119904

10038161003816100381610038161003816100381610038161003816

2

119889119903

le int

119905

120591

119890120576119903int

0

minusℎ

|119866 (119904 119906 (119903 + 119904))|2119889119904 119889119903

le 119871int

0

minusℎ

int

119905

120591

119890120576119903|119906 (119903 + 119904)|

2119889119903 119889119904

le 119871int

0

minusℎ

int

119905+119904

120591+119904

119890120576(119903minus119904)

|119906 (119903)|2119889119903 119889119904

le 119871int

0

minusℎ

int

119905

120591minusℎ

119890120576(119903minus119904)

|119906 (119903)|2119889119903 119889119904

le 119871int

0

minusℎ

119890minus120576119904

int

119905

120591minusℎ

119890120576119903|119906 (119903)|

2119889119903 119889119904

le 119871int

0

minusℎ

119890minus120576119904

(int

120591

120591minusℎ

119890120576119903|119906 (119903)|

2119889119903

+int

119905

120591

119890120576119903|119906 (119903)|

2119889119903) 119889119904 le

119871119890120576ℎ

120576

(

1120576

10038171003817100381710038171206011003817100381710038171003817

2119862119867

119890120576120591

+int

119905

120591

119890120576119903|119906 (119903)|

2119889119903) le

119871119890120576(120591+ℎ)

1205762

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+

119871119890120576ℎ

120576

sdot int

119905

120591

119890120576119903|119906 (119903)|

2119889119903

(40)

Substituting (40) into (39) we have

119890120576119905(|119906 (119905)|

2+120572

2119906 (119905)

2) minus 119890120576120591(|119906 (120591)|

2+120572

2119906 (120591)

2)

le int

119905

120591

119890120576119904[(120576 + 120578 minus ]1205821 +

119871119890120576ℎ

2120578120576) |119906|

2

+ (1205761205722minus ]) 1199062]119889119904 +

119871119890120576(120591+ℎ)

2120578120576210038171003817100381710038171206011003817100381710038171003817

2119862119867

(41)

6 Discrete Dynamics in Nature and Society

Let 120578 = (12)]1205821 choosing 0 lt 120576 lt min]1205722 ]12058214 119871 le

]2120582211205764119890120576ℎ implies that

max120576+ 120578minus ]1205821 +120578minus1119871119890120576ℎ

2120576 120576120572

2minus ] lt 0 (42)

then

119890120576119905(|119906 (119905)|

2+120572

2119906 (119905)

2)

le

119871119890120576(120591+ℎ)

1205762]1205821

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+ 119890120576120591(|119906 (120591)|

2+120572

2119906 (120591)

2)

(43)

which implies

|119906 (119905)|2+120572

2119906 (119905)

2

le

119871119890120576(120591+ℎminus119905)

1205762]1205821

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+ 119890120576(120591minus119905)

(1003816100381610038161003816120601 (120591)

1003816100381610038161003816

2+120572

2 1003817100381710038171003817120601 (120591)

1003817100381710038171003817

2)

(44)

Now if we take 119905 ge 120591 + ℎ then for 120579 isin [minusℎ 0] we have

|119906 (119905 + 120579)|2+120572

2119906 (119905 + 120579)

2

le

119871119890120576(120591+ℎminus119905minus120579)

1205762]1205821

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+ 119890120576(120591minus119905minus120579)

(10038171003817100381710038171206011003817100381710038171003817

2119862119867

+1205722 10038171003817100381710038171206011003817100381710038171003817

2119862119881)

(45)

We denote byR the set of all functions 119903 (minusinfin +infin) rarr

(0 +infin) such that

lim120591rarrminusinfin

1198901205721205911199032(120591) = 0 (46)

Then the closed ball in 119862119881defined by

119861 = 120593 isin11986211988110038171003817100381710038171205931003817100381710038171003817

2119862119881le 1 (47)

is pullback absorbing set for 119880(119905 120591) The proof is complete

We next prove the asymptotic compactness of solution toproblem (10) by contractive functions see [19 20]

Let 119883 be a Banach space and let 119861 be a bounded subsetof 119883 We call a function Φ(sdot sdot) which defined on 119883 times 119883 is acontractive function on 119861times119861 if for any sequence 119909

119899infin

119899=1sub 119861

there is a subsequence 119909119899119896infin

119896=1 sub 119909119899infin

119899=1 such that

lim119897rarrinfin

lim119896rarrinfin

Φ(119909119899119896 119909119899119897) = 0 (48)

Denote all such contractive functions on 119861 times 119861 by 119862(119861)

Theorem 4 (see [19]) Let 119878(119905)119905ge120591

be a semigroup on aBanach space (119883 sdot ) and have a bounded absorbing set 119861forall120598 gt 0 there exist 119879 = 119879(119861 120598) and Φ isin 119862(119861) such that

1003817100381710038171003817119878 (119879) 119909 minus 119878 (119879) 119910

1003817100381710038171003817le 120598 +Φ

119879(119909 119910) forall119909 119910 isin 119861 (49)

where Φ119879depends on 119879 Then 119878(119905)

119905ge120591is asymptotically

compact in119883

Lemma 5 Assume that (1198671)ndash(1198673) hold the process 119880(119905 120591)119905ge120591generated by the global solution to problem (10) is asymptoti-cally compact

Proof Let 119906119894(119905) be the solution to problem (10) with initialdata 120601119894 isin 119861 (119894 = 1 2) respectively Denote V(119905) = 119906

1(119905)minus119906

2(119905)

then V(119905) satisfies the equivalent abstract equation

120597

120597119905

(V+1205722119860V) + ]119860V+119861 (1199061 1199061) minus119861 (1199062 1199062)

= int

0

minusℎ

[119866 (119904 1199061(119905 + 119904)) minus119866 (s 1199062 (119905 + 119904))] 119889119904

(50)

with the initial condition V(119905) = 1206011(119905 minus 120591) minus 120601

2(119905 minus 120591) 119905 isin

[120591 minus ℎ 120591]Set an energy function

119864V (119905) =12int

Ω

|V|2 119889119909+1205722

2int

Ω

|nablaV|2 119889119909 (51)

Multiplying (50) by V and integrating over [119904 119879]timesΩwith 119879 gt

119905 + 120591 119904 ge 120591 we have

119864V (119879) minus119864V (119904) + ]int119879

119904

int

Ω

|nablaV|2 119889119909 119889119903

+int

119879

119904

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

= int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

(52)

then we have

]int119879

119904

int

Ω

|nablaV|2 119889119909 119889119903 le 119864V (119904) minusint119879

119904

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

+int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

(53)

Discrete Dynamics in Nature and Society 7

Using Poincare inequality and (51) and (53) we have

int

119879

120591

119864V (119904) 119889119904 =12int

119879

120591

int

Ω

|V|2 119889119909 119889119903 +1205722

2int

119879

120591

int

Ω

|nablaV|2 119889119909 119889119903 le 119862int

119879

120591

int

Ω

|nablaV|2 119889119909 119889119903

le 119862119864V (120591) minus119862int119879

120591

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

+119862int

119879

120591

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

(54)

Integrating (52) from 120591 to 119879 with respect to 119904 we obtain

119879119864V (119879) + ]int119879

120591

int

119879

119904

int

Ω

|nablaV|2 119889119909 119889119903 119889119904 = int

119879

120591

119864V (119904) 119889119904 minusint119879

120591

int

119879

119904

int

Ω

[119861 (1199061) minus 119861 (119906

2)] V (119903) 119889119909 119889119903 119889119904

+int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903 119889119904

(55)

Substituting (54) into (55) we get

119879119864V (119879) + ]int119879

120591

int

119879

119904

int

Ω

|nablaV|2 119889119909 119889119903 119889119904 le 119862119864V (120591) minus119862int119879

120591

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

+119862int

119879

120591

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

minusint

119879

120591

int

119879

119904

int

Ω

[119861 (1199061) minus 119861 (119906

2)] V (119903) 119889119909 119889119903 119889119904

+int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903 119889119904

(56)

Set

1198620 = 119862119864V (120591) =119862

2int

Ω

100381610038161003816100381610038161206011(0) minus 1206012 (0)1003816100381610038161003816

1003816

2119889119909+

1198621205722

2int

Ω

10038161003816100381610038161003816nabla120601

1(0) minus nabla1206012 (0)1003816100381610038161003816

1003816

2119889119909 (57)

Φ(1199061 119906

2) = minus119862int

119879

120591

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

+119862int

119879

120591

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

minusint

119879

120591

int

119879

119904

int

Ω

[119861 (1199061) minus 119861 (119906

2)] V (119903) 119889119909 119889119903 119889119904

+int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903 119889119904

(58)

8 Discrete Dynamics in Nature and Society

then by (56) we have

119864V (119879) le1198620119879

+

1119879

Φ(1199061 119906

2) (59)

One has forall120598 gt 0 we choose 119879 gt 0 large enough such that1198620119879 lt 120598 By Theorem 4 it suffices to prove that Φ(1199061 1199062) is

a contractive function since119861 is a bounded positive invariantset

If 119906119898 rarr 119906 (119898 rarr infin) we have the limits

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

(119861 (119906119899) minus 119861 (119906

119898)) (119906119899minus 119906119898) 119889119909 119889119903

= lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

[((119906119899minus 119906119898) sdot nabla) 119906

119899minus (119906119898sdot nabla) (119906

119899minus 119906119898)] (119906119899minus 119906119898) 119889119909 119889119903 = 0

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

(119861 (119906119899) minus 119861 (119906

119898)) (119906119899minus 119906119898) 119889119909 119889119903 119889119904

= lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

[((119906119899minus 119906119898) sdot nabla) 119906

119899minus (119906119898sdot nabla) (119906

119899minus 119906119898)] (119906119899minus 119906119898) 119889119909 119889119903 119889119904 = 0

(60)

By (1198671)ndash(1198673) we have

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

int

0

minusℎ

(119866 (120590 119906119899(119903 + 120590)) minus 119866 (120590 119906

119898(119903 + 120590))) V (119903) 119889120590 119889119909 119889119903

le lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

int

0

minusℎ

1198711003816100381610038161003816119906119899(119903 + 120590) minus 119906

119898(119903 + 120590)

1003816100381610038161003816

1003816100381610038161003816119906119899(119903) minus 119906

119898(119903)1003816100381610038161003816119889120590 119889119909 119889119903 = 0

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

(119866 (120590 119906119899(119903 + 120590)) minus 119866 (120590 119906

119898(119903 + 120590))) V (119903) 119889120590 119889119909 119889119903 119889119904

le lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

1198711003816100381610038161003816(119906119899minus 119906119898) (119903 + 120590)

1003816100381610038161003816

1003816100381610038161003816(119906119899minus 119906119898) (119903)

1003816100381610038161003816119889120590 119889119909 119889119903 119889119904 = 0

(61)

Combining (60)-(61) with (58) we have that Φ(1199061 1199062) is acontractive function

The proof is complete

Theorem 6 Assume that (1198671)ndash(1198673) hold and 119871 le ]2120582211205764119890120576ℎ then the process 119880(119905 120591) generated by the global solutionfor problem (10) has a pullback attractor

Proof Lemma 3 implies 119880(119905 120591) has a pullback absorbing set119861 and Lemma 5 implies 119880(119905 120591) is asymptotically compactwe obtain the conclusion immediately

According to Theorem 6 0 lt 120576 lt min]1205722 ]12058214 and119871 le ]2120582211205764119890

120576ℎ Under the assumptions (1198671)ndash(1198673) the NSVequation (1) with a distributed delay has a pullback attractor

Conflict of Interests

The authors declare that they have no competing interests

Authorsrsquo Contribution

Yantao Guo carried out the long time behavior of solutionsShuilin Cheng carried out the pullback attractor YanbinTang carried out the distributed delay All authors read andapproved the final paper

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China (Grant nos 11471129 and 11272277)

References

[1] A P Oskolkov ldquoThe uniqueness and solvability in the largeof boundary value problems for the equations of motion ofaqueous solutions of polymersrdquo Zapiski Nauchnykh SeminarovLeningrad Otdel Mathematics Institute Stekov (LOMI) vol 38pp 98ndash136 1973

Discrete Dynamics in Nature and Society 9

[2] Y Cao E M Lunasin and E S Titi ldquoGlobal well-posedness ofthe three-dimensional viscous and inviscid simplified Bardinaturbulence modelsrdquo Communications in Mathematical Sciencesvol 4 no 4 pp 823ndash848 2006

[3] V K Kalantarov and E S Titi ldquoGlobal attractors and determin-ing modes for the 3D Navier-Stokes-Voight equationsrdquo ChineseAnnals ofMathematics Series B vol 30 no 6 pp 697ndash714 2009

[4] G-C Yue and C-K Zhong ldquoAttractors for autonomous andnonautonomous 3D Navier-Stokes-Voight equationsrdquo Discreteand Continuous Dynamical Systems Series B vol 16 no 3 pp985ndash1002 2011

[5] J Garcıa-Luengo P Marın-Rubio and J Real ldquoPullback attrac-tors for three-dimensional non-autonomous Navier-Stokes-Voigt equationsrdquo Nonlinearity vol 25 no 4 pp 905ndash930 2012

[6] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977

[7] Y Tang and M Wang ldquoA remark on exponential stabilityof time-delayed Burgers equationrdquo Discrete and ContinuousDynamical Systems Series B vol 12 no 1 pp 219ndash225 2009

[8] Y Tang and L Zhou ldquoStability switch and Hopf bifurcationfor a diffusive prey-predator system with delayrdquo Journal ofMathematical Analysis and Applications vol 334 no 2 pp1290ndash1307 2007

[9] J H Wu Theory and Applications of Partial Differential Equa-tions Springer New York NY USA 1996

[10] L Zhou Y B Tang and S Hussein ldquoStability and Hopf bifur-cation for a delay competition diffusion systemrdquoChaos Solitonsamp Fractals vol 14 no 8 pp 1201ndash1225 2002

[11] T Caraballo and J Real ldquoNavier-Stokes equations with delaysrdquoThe Royal Society of London Proceedings Series A Mathemat-ical Physical and Engineering Sciences vol 457 no 2014 pp2441ndash2453 2001

[12] T Caraballo and J Real ldquoAsymptotic behaviour of two-dimen-sional Navier-Stokes equations with delaysrdquo The Royal Societyof London Proceedings Series A Mathematical Physical andEngineering Sciences vol 459 no 2040 pp 3181ndash3194 2003

[13] T Caraballo and J Real ldquoAttractors for 2D-Navier-Stokesmodels with delaysrdquo Journal of Differential Equations vol 205no 2 pp 271ndash297 2004

[14] Z Hu and Y Wang ldquoPullback attractors for a nonautonomousnonclassical diffusion equation with variable delayrdquo Journal ofMathematical Physics vol 53 no 7 Article ID 072702 17 pages2012

[15] H Y Li and Y M Qin ldquoPullback attractors for three-dimen-sional navier-stokes-voigt equations with delaysrdquo BoundaryValue Problems vol 2013 article 191 2013

[16] P Marın-Rubio and J Real ldquoAttractors for 2D-Navier-Stokesequations with delays on some unbounded domainsrdquoNonlinearAnalysis Theory Methods amp Applications vol 67 no 10 pp2784ndash2799 2007

[17] P Marın-Rubio J Real and J Valero ldquoPullback attractors for atwo-dimensional Navier-Stokesmodel in an infinite delay caserdquoNonlinear Analysis Theory Methods amp Applications vol 74 no5 pp 2012ndash2030 2011

[18] P Marın-Rubio and J Real ldquoPullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linearoperatorsrdquo Discrete and Continuous Dynamical Systems SeriesA vol 26 no 3 pp 989ndash1006 2010

[19] AKKhanmamedov ldquoGlobal attractors forwave equationswithnonlinear interior damping and critical exponentsrdquo Journal ofDifferential Equations vol 230 no 2 pp 702ndash719 2006

[20] L Yang and C-K Zhong ldquoGlobal attractor for plate equationwith nonlinear dampingrdquo Nonlinear Analysis Theory Methodsamp Applications vol 69 no 11 pp 3802ndash3810 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 4: Research Article Approximate Kelvin-Voigt Fluid …downloads.hindawi.com/journals/ddns/2015/721673.pdfWe consider the approximate D Kelvin-Voigt uid driven by an external force depending

4 Discrete Dynamics in Nature and Society

integrating the above inequality from 120591 to 119905 by (17) (20) and(22) we have

int

119905

120591

(10038161003816100381610038161205971199051199061198981003816100381610038161003816

2+120572

2 10038171003817100381710038171205971199051199061198981003817100381710038171003817

2) 119889119904 + ] 100381710038171003817

1003817119906119898(119905)1003817100381710038171003817

2

le ] 1003817100381710038171003817119906119898(120591)

1003817100381710038171003817

2+ 119888int

119905

120591

10038171003817100381710038171199061198981003817100381710038171003817

3 10038161003816100381610038161199061198981003816100381610038161003816119889119904

+int

119905

120591

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906119898(119903 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

119889119903

le ] 1003817100381710038171003817120601 (120591)

1003817100381710038171003817

2+ 119888int

119905

120591

10038171003817100381710038171199061198981003817100381710038171003817

3 10038161003816100381610038161199061198981003816100381610038161003816119889119904 + 119888

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+ 119888int

119905

120591

1003816100381610038161003816119906119898(119903)1003816100381610038161003816

2119889119903

le ] 10038171003817100381710038171206011003817100381710038171003817

2119862119881+ 119888int

119905

120591

10038171003817100381710038171199061198981003817100381710038171003817

3 10038161003816100381610038161199061198981003816100381610038161003816119889119904 + 119888

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+ 119888int

119905

120591

1003816100381610038161003816119906119898(119903)1003816100381610038161003816

2119889119903

(26)

Since 119906119898 is bounded in 119871infin(120591 119879 119881) cap 1198712(120591 119879 119881) we obtainthat

120597119905119906119898 is bounded in 119871

2(120591 119879 119881) (27)

By the Faedo-Galerkin scheme for example see [14 18]according to the estimates (22) and (27) we can get existenceof the weak solution here we omit the details

We next consider the uniqueness of solution Let 119906 V betwo solutions to problem (10) corresponding the initial data120601 and 120595 respectively

Denote 119908 = 119906 minus V then we have

120597

120597119905

(119908+1205722119860119908)+ ]119860119908+119861 (119906 119906) minus 119861 (V V)

= int

0

minusℎ

119866 (119904 119906 (119905 + 119904)) 119889119904 minusint

0

minusℎ

119866 (119904 V (119905 + 119904)) 119889119904(28)

Multiplying (28) by 119908 and integrating overΩ we obtain

12119889

119889119905

(|119908|2+120572

2119908

2) + ] 1199082 + (119861 (119906 119906) 119908)

minus (119861 (V V) 119908)

= (int

0

minusℎ

119866 (119904 119906 (119905 + 119904)) 119889119904 minusint

0

minusℎ

119866 (119904 V (119905 + 119904)) 119889119904 119908)

(29)

Notice that

(119861 (119906 119906) 119908) minus (119861 (V V) 119908)

= 119887 (119906 119906 119906 minus V) minus 119887 (V 119906 119906 minus V) + 119887 (V 119906 119906 minus V)

minus 119887 (V V 119906 minus V)

= 119887 (119906 minus V 119906 119906 minus V) minus 119887 (V 119906 minus V 119906 minus V)

= 119887 (119908 119906 119908)

(30)

Substituting (30) into (29) and integrating from 120591 to 119905 weget

|119908 (119905)|2+120572

2119908 (119905)

2+ 2]int

119905

120591

119908 (119904)2119889119904 minus |119908 (120591)|

2

minus1205722119908 (120591)

2le int

119905

120591

|119887 (119908 119906 119908)| 119889119904 +int

119905

120591

|119908 (119904)|2119889119904

+int

119905

120591

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119903 + 119904)) 119889119904

minusint

0

minusℎ

119866 (119904 V (119903 + 119904)) 119889119904100381610038161003816100381610038161003816100381610038161003816

2

119889119903

(31)

(1198673) implies that

int

119905

120591

10038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119903 + 119904)) 119889119904 minusint

0

minusℎ

119866 (119904 V (119903 + 119904)) 11988911990410038161003816100381610038161003816100381610038161003816

2

119889119903

le int

0

minusℎ

int

119905

120591

|119866 (119904 119906 (119903 + 119904)) minus119866 (119904 V (119903 + 119904))|2 119889119903 119889119904

le int

0

minusℎ

int

119905

120591

|119866 (119904 119906 (119903 + 119904)) minus119866 (119904 V (119903 + 119904))|2 119889119903 119889119904

le 119871int

0

minusℎ

int

119905

120591

|119906 (119903 + 119904) minus V (119903 + 119904)|2 119889119903 119889119904

le 119871int

0

minusℎ

int

119905+119904

120591+119904

|119906 (119903) minus V (119903)|2 119889119903 119889119904

le 119871int

0

minusℎ

(int

120591

120591minusℎ

|119906 (119903) minus V (119903)|2 119889119903

+int

119905

120591

|119906 (119903) minus V (119903)|2 119889119903) 119889119904 le 119871ℎ2 1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119867

+119871ℎint

119905

120591

|119908 (119904)|2119889119904

(32)

As the property of operator 119887 and Poincare we have

int

119905

120591

|119887 (119908 119906 119908)| 119889119904 le 119862int

119905

120591

11990832

|119908|12

119906 119889119904

le 119862int

119905

120591

1199082119906 119889119904

(33)

Discrete Dynamics in Nature and Society 5

Substituting (32) and (33) into (31) we get

|119908 (119905)|2+120572

2119908 (119905)

2+ 2]int

119905

120591

119908 (119904)2119889119904

le int

119905

120591

1199082119906 119889119904 + 119871ℎ

2 1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119867

+ (119871ℎ + 1) int119905

120591

|119908 (119904)|2119889119904 + |119908 (120591)|

2

+1205722119908 (120591)

2

le int

119905

120591

1199082119906 119889119904 + 119871ℎ

2 1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119867

+ (119871ℎ + 1) int119905

120591

|119908 (119904)|2119889119904 +

1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119867

+1205722 1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119881

le 119862int

119905

120591

119908 (119904)2119889119904 +119862

1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119881

(34)

The last inequality comes from Poincare inequality andthe boundedness of 119906 Therefore the Gronwall inequalityimplies the uniqueness of the solution The proof is com-plete

3 Existence of Pullback Attractor

In this section we will prove the existence of pullbackattractor to problem (10) First we give existence of pullbackabsorbing set for the process 119880(119905 120591) generated by the globalsolution to problem (10)

Lemma 3 Assume (1198671)ndash(1198673) hold and 119871 le ]2120582211205764119890120576ℎ then

the process 119880(119905 120591) is pullback dissipative where 0 lt 120576 lt

min]1205722 ]12058214

Proof Multiplying (10) by119906 and integrating overΩ we obtain

120597

120597119905

(|119906|2+120572

2119906

2) + 2] 1199062

le 120578 |119906|2+

1120578

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119905 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

(35)

where 120578 is a constant determined laterBy Poincare inequality we have

120597

120597119905

(|119906|2+120572

2119906

2) + ] 1199062 + (]1205821 minus 120578) |119906|

2

le

1120578

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119905 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

(36)

Since

119890minus120576119905 120597

120597119905

[119890120576119905(|119906|

2+120572

2119906

2)]

= 120576 (|119906|2+120572

2119906

2) +

120597

120597119905

(|119906|2+120572

2119906

2)

(37)

then

119890minus120576119905 120597

120597119905

[119890120576119905(|119906|

2+120572

2119906

2)]

le (120576 + 120578 minus ]1205821) |119906|2+ (120576120572

2minus ]) 1199062

+

1120578

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119905 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

(38)

Integrating (38) from 120591 to 119905 we get

119890120576119905(|119906 (119905)|

2+120572

2119906 (119905)

2)

minus 119890120576120591(|119906 (120591)|

2+120572

2119906 (120591)

2)

le int

119905

120591

119890120576119904[(120576 + 120578 minus ]1205821) |119906|

2+ (120576120572

2minus ]) 1199062] 119889119904

+

1120578

int

119905

120591

119890120576119903

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119903 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

119889119903

(39)

Assumptions (1198671)ndash(1198673) imply that

int

119905

120591

119890120576119903

10038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119903 + 119904)) 119889119904

10038161003816100381610038161003816100381610038161003816

2

119889119903

le int

119905

120591

119890120576119903int

0

minusℎ

|119866 (119904 119906 (119903 + 119904))|2119889119904 119889119903

le 119871int

0

minusℎ

int

119905

120591

119890120576119903|119906 (119903 + 119904)|

2119889119903 119889119904

le 119871int

0

minusℎ

int

119905+119904

120591+119904

119890120576(119903minus119904)

|119906 (119903)|2119889119903 119889119904

le 119871int

0

minusℎ

int

119905

120591minusℎ

119890120576(119903minus119904)

|119906 (119903)|2119889119903 119889119904

le 119871int

0

minusℎ

119890minus120576119904

int

119905

120591minusℎ

119890120576119903|119906 (119903)|

2119889119903 119889119904

le 119871int

0

minusℎ

119890minus120576119904

(int

120591

120591minusℎ

119890120576119903|119906 (119903)|

2119889119903

+int

119905

120591

119890120576119903|119906 (119903)|

2119889119903) 119889119904 le

119871119890120576ℎ

120576

(

1120576

10038171003817100381710038171206011003817100381710038171003817

2119862119867

119890120576120591

+int

119905

120591

119890120576119903|119906 (119903)|

2119889119903) le

119871119890120576(120591+ℎ)

1205762

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+

119871119890120576ℎ

120576

sdot int

119905

120591

119890120576119903|119906 (119903)|

2119889119903

(40)

Substituting (40) into (39) we have

119890120576119905(|119906 (119905)|

2+120572

2119906 (119905)

2) minus 119890120576120591(|119906 (120591)|

2+120572

2119906 (120591)

2)

le int

119905

120591

119890120576119904[(120576 + 120578 minus ]1205821 +

119871119890120576ℎ

2120578120576) |119906|

2

+ (1205761205722minus ]) 1199062]119889119904 +

119871119890120576(120591+ℎ)

2120578120576210038171003817100381710038171206011003817100381710038171003817

2119862119867

(41)

6 Discrete Dynamics in Nature and Society

Let 120578 = (12)]1205821 choosing 0 lt 120576 lt min]1205722 ]12058214 119871 le

]2120582211205764119890120576ℎ implies that

max120576+ 120578minus ]1205821 +120578minus1119871119890120576ℎ

2120576 120576120572

2minus ] lt 0 (42)

then

119890120576119905(|119906 (119905)|

2+120572

2119906 (119905)

2)

le

119871119890120576(120591+ℎ)

1205762]1205821

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+ 119890120576120591(|119906 (120591)|

2+120572

2119906 (120591)

2)

(43)

which implies

|119906 (119905)|2+120572

2119906 (119905)

2

le

119871119890120576(120591+ℎminus119905)

1205762]1205821

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+ 119890120576(120591minus119905)

(1003816100381610038161003816120601 (120591)

1003816100381610038161003816

2+120572

2 1003817100381710038171003817120601 (120591)

1003817100381710038171003817

2)

(44)

Now if we take 119905 ge 120591 + ℎ then for 120579 isin [minusℎ 0] we have

|119906 (119905 + 120579)|2+120572

2119906 (119905 + 120579)

2

le

119871119890120576(120591+ℎminus119905minus120579)

1205762]1205821

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+ 119890120576(120591minus119905minus120579)

(10038171003817100381710038171206011003817100381710038171003817

2119862119867

+1205722 10038171003817100381710038171206011003817100381710038171003817

2119862119881)

(45)

We denote byR the set of all functions 119903 (minusinfin +infin) rarr

(0 +infin) such that

lim120591rarrminusinfin

1198901205721205911199032(120591) = 0 (46)

Then the closed ball in 119862119881defined by

119861 = 120593 isin11986211988110038171003817100381710038171205931003817100381710038171003817

2119862119881le 1 (47)

is pullback absorbing set for 119880(119905 120591) The proof is complete

We next prove the asymptotic compactness of solution toproblem (10) by contractive functions see [19 20]

Let 119883 be a Banach space and let 119861 be a bounded subsetof 119883 We call a function Φ(sdot sdot) which defined on 119883 times 119883 is acontractive function on 119861times119861 if for any sequence 119909

119899infin

119899=1sub 119861

there is a subsequence 119909119899119896infin

119896=1 sub 119909119899infin

119899=1 such that

lim119897rarrinfin

lim119896rarrinfin

Φ(119909119899119896 119909119899119897) = 0 (48)

Denote all such contractive functions on 119861 times 119861 by 119862(119861)

Theorem 4 (see [19]) Let 119878(119905)119905ge120591

be a semigroup on aBanach space (119883 sdot ) and have a bounded absorbing set 119861forall120598 gt 0 there exist 119879 = 119879(119861 120598) and Φ isin 119862(119861) such that

1003817100381710038171003817119878 (119879) 119909 minus 119878 (119879) 119910

1003817100381710038171003817le 120598 +Φ

119879(119909 119910) forall119909 119910 isin 119861 (49)

where Φ119879depends on 119879 Then 119878(119905)

119905ge120591is asymptotically

compact in119883

Lemma 5 Assume that (1198671)ndash(1198673) hold the process 119880(119905 120591)119905ge120591generated by the global solution to problem (10) is asymptoti-cally compact

Proof Let 119906119894(119905) be the solution to problem (10) with initialdata 120601119894 isin 119861 (119894 = 1 2) respectively Denote V(119905) = 119906

1(119905)minus119906

2(119905)

then V(119905) satisfies the equivalent abstract equation

120597

120597119905

(V+1205722119860V) + ]119860V+119861 (1199061 1199061) minus119861 (1199062 1199062)

= int

0

minusℎ

[119866 (119904 1199061(119905 + 119904)) minus119866 (s 1199062 (119905 + 119904))] 119889119904

(50)

with the initial condition V(119905) = 1206011(119905 minus 120591) minus 120601

2(119905 minus 120591) 119905 isin

[120591 minus ℎ 120591]Set an energy function

119864V (119905) =12int

Ω

|V|2 119889119909+1205722

2int

Ω

|nablaV|2 119889119909 (51)

Multiplying (50) by V and integrating over [119904 119879]timesΩwith 119879 gt

119905 + 120591 119904 ge 120591 we have

119864V (119879) minus119864V (119904) + ]int119879

119904

int

Ω

|nablaV|2 119889119909 119889119903

+int

119879

119904

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

= int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

(52)

then we have

]int119879

119904

int

Ω

|nablaV|2 119889119909 119889119903 le 119864V (119904) minusint119879

119904

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

+int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

(53)

Discrete Dynamics in Nature and Society 7

Using Poincare inequality and (51) and (53) we have

int

119879

120591

119864V (119904) 119889119904 =12int

119879

120591

int

Ω

|V|2 119889119909 119889119903 +1205722

2int

119879

120591

int

Ω

|nablaV|2 119889119909 119889119903 le 119862int

119879

120591

int

Ω

|nablaV|2 119889119909 119889119903

le 119862119864V (120591) minus119862int119879

120591

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

+119862int

119879

120591

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

(54)

Integrating (52) from 120591 to 119879 with respect to 119904 we obtain

119879119864V (119879) + ]int119879

120591

int

119879

119904

int

Ω

|nablaV|2 119889119909 119889119903 119889119904 = int

119879

120591

119864V (119904) 119889119904 minusint119879

120591

int

119879

119904

int

Ω

[119861 (1199061) minus 119861 (119906

2)] V (119903) 119889119909 119889119903 119889119904

+int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903 119889119904

(55)

Substituting (54) into (55) we get

119879119864V (119879) + ]int119879

120591

int

119879

119904

int

Ω

|nablaV|2 119889119909 119889119903 119889119904 le 119862119864V (120591) minus119862int119879

120591

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

+119862int

119879

120591

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

minusint

119879

120591

int

119879

119904

int

Ω

[119861 (1199061) minus 119861 (119906

2)] V (119903) 119889119909 119889119903 119889119904

+int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903 119889119904

(56)

Set

1198620 = 119862119864V (120591) =119862

2int

Ω

100381610038161003816100381610038161206011(0) minus 1206012 (0)1003816100381610038161003816

1003816

2119889119909+

1198621205722

2int

Ω

10038161003816100381610038161003816nabla120601

1(0) minus nabla1206012 (0)1003816100381610038161003816

1003816

2119889119909 (57)

Φ(1199061 119906

2) = minus119862int

119879

120591

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

+119862int

119879

120591

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

minusint

119879

120591

int

119879

119904

int

Ω

[119861 (1199061) minus 119861 (119906

2)] V (119903) 119889119909 119889119903 119889119904

+int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903 119889119904

(58)

8 Discrete Dynamics in Nature and Society

then by (56) we have

119864V (119879) le1198620119879

+

1119879

Φ(1199061 119906

2) (59)

One has forall120598 gt 0 we choose 119879 gt 0 large enough such that1198620119879 lt 120598 By Theorem 4 it suffices to prove that Φ(1199061 1199062) is

a contractive function since119861 is a bounded positive invariantset

If 119906119898 rarr 119906 (119898 rarr infin) we have the limits

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

(119861 (119906119899) minus 119861 (119906

119898)) (119906119899minus 119906119898) 119889119909 119889119903

= lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

[((119906119899minus 119906119898) sdot nabla) 119906

119899minus (119906119898sdot nabla) (119906

119899minus 119906119898)] (119906119899minus 119906119898) 119889119909 119889119903 = 0

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

(119861 (119906119899) minus 119861 (119906

119898)) (119906119899minus 119906119898) 119889119909 119889119903 119889119904

= lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

[((119906119899minus 119906119898) sdot nabla) 119906

119899minus (119906119898sdot nabla) (119906

119899minus 119906119898)] (119906119899minus 119906119898) 119889119909 119889119903 119889119904 = 0

(60)

By (1198671)ndash(1198673) we have

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

int

0

minusℎ

(119866 (120590 119906119899(119903 + 120590)) minus 119866 (120590 119906

119898(119903 + 120590))) V (119903) 119889120590 119889119909 119889119903

le lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

int

0

minusℎ

1198711003816100381610038161003816119906119899(119903 + 120590) minus 119906

119898(119903 + 120590)

1003816100381610038161003816

1003816100381610038161003816119906119899(119903) minus 119906

119898(119903)1003816100381610038161003816119889120590 119889119909 119889119903 = 0

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

(119866 (120590 119906119899(119903 + 120590)) minus 119866 (120590 119906

119898(119903 + 120590))) V (119903) 119889120590 119889119909 119889119903 119889119904

le lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

1198711003816100381610038161003816(119906119899minus 119906119898) (119903 + 120590)

1003816100381610038161003816

1003816100381610038161003816(119906119899minus 119906119898) (119903)

1003816100381610038161003816119889120590 119889119909 119889119903 119889119904 = 0

(61)

Combining (60)-(61) with (58) we have that Φ(1199061 1199062) is acontractive function

The proof is complete

Theorem 6 Assume that (1198671)ndash(1198673) hold and 119871 le ]2120582211205764119890120576ℎ then the process 119880(119905 120591) generated by the global solutionfor problem (10) has a pullback attractor

Proof Lemma 3 implies 119880(119905 120591) has a pullback absorbing set119861 and Lemma 5 implies 119880(119905 120591) is asymptotically compactwe obtain the conclusion immediately

According to Theorem 6 0 lt 120576 lt min]1205722 ]12058214 and119871 le ]2120582211205764119890

120576ℎ Under the assumptions (1198671)ndash(1198673) the NSVequation (1) with a distributed delay has a pullback attractor

Conflict of Interests

The authors declare that they have no competing interests

Authorsrsquo Contribution

Yantao Guo carried out the long time behavior of solutionsShuilin Cheng carried out the pullback attractor YanbinTang carried out the distributed delay All authors read andapproved the final paper

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China (Grant nos 11471129 and 11272277)

References

[1] A P Oskolkov ldquoThe uniqueness and solvability in the largeof boundary value problems for the equations of motion ofaqueous solutions of polymersrdquo Zapiski Nauchnykh SeminarovLeningrad Otdel Mathematics Institute Stekov (LOMI) vol 38pp 98ndash136 1973

Discrete Dynamics in Nature and Society 9

[2] Y Cao E M Lunasin and E S Titi ldquoGlobal well-posedness ofthe three-dimensional viscous and inviscid simplified Bardinaturbulence modelsrdquo Communications in Mathematical Sciencesvol 4 no 4 pp 823ndash848 2006

[3] V K Kalantarov and E S Titi ldquoGlobal attractors and determin-ing modes for the 3D Navier-Stokes-Voight equationsrdquo ChineseAnnals ofMathematics Series B vol 30 no 6 pp 697ndash714 2009

[4] G-C Yue and C-K Zhong ldquoAttractors for autonomous andnonautonomous 3D Navier-Stokes-Voight equationsrdquo Discreteand Continuous Dynamical Systems Series B vol 16 no 3 pp985ndash1002 2011

[5] J Garcıa-Luengo P Marın-Rubio and J Real ldquoPullback attrac-tors for three-dimensional non-autonomous Navier-Stokes-Voigt equationsrdquo Nonlinearity vol 25 no 4 pp 905ndash930 2012

[6] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977

[7] Y Tang and M Wang ldquoA remark on exponential stabilityof time-delayed Burgers equationrdquo Discrete and ContinuousDynamical Systems Series B vol 12 no 1 pp 219ndash225 2009

[8] Y Tang and L Zhou ldquoStability switch and Hopf bifurcationfor a diffusive prey-predator system with delayrdquo Journal ofMathematical Analysis and Applications vol 334 no 2 pp1290ndash1307 2007

[9] J H Wu Theory and Applications of Partial Differential Equa-tions Springer New York NY USA 1996

[10] L Zhou Y B Tang and S Hussein ldquoStability and Hopf bifur-cation for a delay competition diffusion systemrdquoChaos Solitonsamp Fractals vol 14 no 8 pp 1201ndash1225 2002

[11] T Caraballo and J Real ldquoNavier-Stokes equations with delaysrdquoThe Royal Society of London Proceedings Series A Mathemat-ical Physical and Engineering Sciences vol 457 no 2014 pp2441ndash2453 2001

[12] T Caraballo and J Real ldquoAsymptotic behaviour of two-dimen-sional Navier-Stokes equations with delaysrdquo The Royal Societyof London Proceedings Series A Mathematical Physical andEngineering Sciences vol 459 no 2040 pp 3181ndash3194 2003

[13] T Caraballo and J Real ldquoAttractors for 2D-Navier-Stokesmodels with delaysrdquo Journal of Differential Equations vol 205no 2 pp 271ndash297 2004

[14] Z Hu and Y Wang ldquoPullback attractors for a nonautonomousnonclassical diffusion equation with variable delayrdquo Journal ofMathematical Physics vol 53 no 7 Article ID 072702 17 pages2012

[15] H Y Li and Y M Qin ldquoPullback attractors for three-dimen-sional navier-stokes-voigt equations with delaysrdquo BoundaryValue Problems vol 2013 article 191 2013

[16] P Marın-Rubio and J Real ldquoAttractors for 2D-Navier-Stokesequations with delays on some unbounded domainsrdquoNonlinearAnalysis Theory Methods amp Applications vol 67 no 10 pp2784ndash2799 2007

[17] P Marın-Rubio J Real and J Valero ldquoPullback attractors for atwo-dimensional Navier-Stokesmodel in an infinite delay caserdquoNonlinear Analysis Theory Methods amp Applications vol 74 no5 pp 2012ndash2030 2011

[18] P Marın-Rubio and J Real ldquoPullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linearoperatorsrdquo Discrete and Continuous Dynamical Systems SeriesA vol 26 no 3 pp 989ndash1006 2010

[19] AKKhanmamedov ldquoGlobal attractors forwave equationswithnonlinear interior damping and critical exponentsrdquo Journal ofDifferential Equations vol 230 no 2 pp 702ndash719 2006

[20] L Yang and C-K Zhong ldquoGlobal attractor for plate equationwith nonlinear dampingrdquo Nonlinear Analysis Theory Methodsamp Applications vol 69 no 11 pp 3802ndash3810 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 5: Research Article Approximate Kelvin-Voigt Fluid …downloads.hindawi.com/journals/ddns/2015/721673.pdfWe consider the approximate D Kelvin-Voigt uid driven by an external force depending

Discrete Dynamics in Nature and Society 5

Substituting (32) and (33) into (31) we get

|119908 (119905)|2+120572

2119908 (119905)

2+ 2]int

119905

120591

119908 (119904)2119889119904

le int

119905

120591

1199082119906 119889119904 + 119871ℎ

2 1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119867

+ (119871ℎ + 1) int119905

120591

|119908 (119904)|2119889119904 + |119908 (120591)|

2

+1205722119908 (120591)

2

le int

119905

120591

1199082119906 119889119904 + 119871ℎ

2 1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119867

+ (119871ℎ + 1) int119905

120591

|119908 (119904)|2119889119904 +

1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119867

+1205722 1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119881

le 119862int

119905

120591

119908 (119904)2119889119904 +119862

1003817100381710038171003817120601 minus120595

1003817100381710038171003817

2119862119881

(34)

The last inequality comes from Poincare inequality andthe boundedness of 119906 Therefore the Gronwall inequalityimplies the uniqueness of the solution The proof is com-plete

3 Existence of Pullback Attractor

In this section we will prove the existence of pullbackattractor to problem (10) First we give existence of pullbackabsorbing set for the process 119880(119905 120591) generated by the globalsolution to problem (10)

Lemma 3 Assume (1198671)ndash(1198673) hold and 119871 le ]2120582211205764119890120576ℎ then

the process 119880(119905 120591) is pullback dissipative where 0 lt 120576 lt

min]1205722 ]12058214

Proof Multiplying (10) by119906 and integrating overΩ we obtain

120597

120597119905

(|119906|2+120572

2119906

2) + 2] 1199062

le 120578 |119906|2+

1120578

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119905 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

(35)

where 120578 is a constant determined laterBy Poincare inequality we have

120597

120597119905

(|119906|2+120572

2119906

2) + ] 1199062 + (]1205821 minus 120578) |119906|

2

le

1120578

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119905 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

(36)

Since

119890minus120576119905 120597

120597119905

[119890120576119905(|119906|

2+120572

2119906

2)]

= 120576 (|119906|2+120572

2119906

2) +

120597

120597119905

(|119906|2+120572

2119906

2)

(37)

then

119890minus120576119905 120597

120597119905

[119890120576119905(|119906|

2+120572

2119906

2)]

le (120576 + 120578 minus ]1205821) |119906|2+ (120576120572

2minus ]) 1199062

+

1120578

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119905 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

(38)

Integrating (38) from 120591 to 119905 we get

119890120576119905(|119906 (119905)|

2+120572

2119906 (119905)

2)

minus 119890120576120591(|119906 (120591)|

2+120572

2119906 (120591)

2)

le int

119905

120591

119890120576119904[(120576 + 120578 minus ]1205821) |119906|

2+ (120576120572

2minus ]) 1199062] 119889119904

+

1120578

int

119905

120591

119890120576119903

100381610038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119903 + 119904)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

119889119903

(39)

Assumptions (1198671)ndash(1198673) imply that

int

119905

120591

119890120576119903

10038161003816100381610038161003816100381610038161003816

int

0

minusℎ

119866 (119904 119906 (119903 + 119904)) 119889119904

10038161003816100381610038161003816100381610038161003816

2

119889119903

le int

119905

120591

119890120576119903int

0

minusℎ

|119866 (119904 119906 (119903 + 119904))|2119889119904 119889119903

le 119871int

0

minusℎ

int

119905

120591

119890120576119903|119906 (119903 + 119904)|

2119889119903 119889119904

le 119871int

0

minusℎ

int

119905+119904

120591+119904

119890120576(119903minus119904)

|119906 (119903)|2119889119903 119889119904

le 119871int

0

minusℎ

int

119905

120591minusℎ

119890120576(119903minus119904)

|119906 (119903)|2119889119903 119889119904

le 119871int

0

minusℎ

119890minus120576119904

int

119905

120591minusℎ

119890120576119903|119906 (119903)|

2119889119903 119889119904

le 119871int

0

minusℎ

119890minus120576119904

(int

120591

120591minusℎ

119890120576119903|119906 (119903)|

2119889119903

+int

119905

120591

119890120576119903|119906 (119903)|

2119889119903) 119889119904 le

119871119890120576ℎ

120576

(

1120576

10038171003817100381710038171206011003817100381710038171003817

2119862119867

119890120576120591

+int

119905

120591

119890120576119903|119906 (119903)|

2119889119903) le

119871119890120576(120591+ℎ)

1205762

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+

119871119890120576ℎ

120576

sdot int

119905

120591

119890120576119903|119906 (119903)|

2119889119903

(40)

Substituting (40) into (39) we have

119890120576119905(|119906 (119905)|

2+120572

2119906 (119905)

2) minus 119890120576120591(|119906 (120591)|

2+120572

2119906 (120591)

2)

le int

119905

120591

119890120576119904[(120576 + 120578 minus ]1205821 +

119871119890120576ℎ

2120578120576) |119906|

2

+ (1205761205722minus ]) 1199062]119889119904 +

119871119890120576(120591+ℎ)

2120578120576210038171003817100381710038171206011003817100381710038171003817

2119862119867

(41)

6 Discrete Dynamics in Nature and Society

Let 120578 = (12)]1205821 choosing 0 lt 120576 lt min]1205722 ]12058214 119871 le

]2120582211205764119890120576ℎ implies that

max120576+ 120578minus ]1205821 +120578minus1119871119890120576ℎ

2120576 120576120572

2minus ] lt 0 (42)

then

119890120576119905(|119906 (119905)|

2+120572

2119906 (119905)

2)

le

119871119890120576(120591+ℎ)

1205762]1205821

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+ 119890120576120591(|119906 (120591)|

2+120572

2119906 (120591)

2)

(43)

which implies

|119906 (119905)|2+120572

2119906 (119905)

2

le

119871119890120576(120591+ℎminus119905)

1205762]1205821

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+ 119890120576(120591minus119905)

(1003816100381610038161003816120601 (120591)

1003816100381610038161003816

2+120572

2 1003817100381710038171003817120601 (120591)

1003817100381710038171003817

2)

(44)

Now if we take 119905 ge 120591 + ℎ then for 120579 isin [minusℎ 0] we have

|119906 (119905 + 120579)|2+120572

2119906 (119905 + 120579)

2

le

119871119890120576(120591+ℎminus119905minus120579)

1205762]1205821

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+ 119890120576(120591minus119905minus120579)

(10038171003817100381710038171206011003817100381710038171003817

2119862119867

+1205722 10038171003817100381710038171206011003817100381710038171003817

2119862119881)

(45)

We denote byR the set of all functions 119903 (minusinfin +infin) rarr

(0 +infin) such that

lim120591rarrminusinfin

1198901205721205911199032(120591) = 0 (46)

Then the closed ball in 119862119881defined by

119861 = 120593 isin11986211988110038171003817100381710038171205931003817100381710038171003817

2119862119881le 1 (47)

is pullback absorbing set for 119880(119905 120591) The proof is complete

We next prove the asymptotic compactness of solution toproblem (10) by contractive functions see [19 20]

Let 119883 be a Banach space and let 119861 be a bounded subsetof 119883 We call a function Φ(sdot sdot) which defined on 119883 times 119883 is acontractive function on 119861times119861 if for any sequence 119909

119899infin

119899=1sub 119861

there is a subsequence 119909119899119896infin

119896=1 sub 119909119899infin

119899=1 such that

lim119897rarrinfin

lim119896rarrinfin

Φ(119909119899119896 119909119899119897) = 0 (48)

Denote all such contractive functions on 119861 times 119861 by 119862(119861)

Theorem 4 (see [19]) Let 119878(119905)119905ge120591

be a semigroup on aBanach space (119883 sdot ) and have a bounded absorbing set 119861forall120598 gt 0 there exist 119879 = 119879(119861 120598) and Φ isin 119862(119861) such that

1003817100381710038171003817119878 (119879) 119909 minus 119878 (119879) 119910

1003817100381710038171003817le 120598 +Φ

119879(119909 119910) forall119909 119910 isin 119861 (49)

where Φ119879depends on 119879 Then 119878(119905)

119905ge120591is asymptotically

compact in119883

Lemma 5 Assume that (1198671)ndash(1198673) hold the process 119880(119905 120591)119905ge120591generated by the global solution to problem (10) is asymptoti-cally compact

Proof Let 119906119894(119905) be the solution to problem (10) with initialdata 120601119894 isin 119861 (119894 = 1 2) respectively Denote V(119905) = 119906

1(119905)minus119906

2(119905)

then V(119905) satisfies the equivalent abstract equation

120597

120597119905

(V+1205722119860V) + ]119860V+119861 (1199061 1199061) minus119861 (1199062 1199062)

= int

0

minusℎ

[119866 (119904 1199061(119905 + 119904)) minus119866 (s 1199062 (119905 + 119904))] 119889119904

(50)

with the initial condition V(119905) = 1206011(119905 minus 120591) minus 120601

2(119905 minus 120591) 119905 isin

[120591 minus ℎ 120591]Set an energy function

119864V (119905) =12int

Ω

|V|2 119889119909+1205722

2int

Ω

|nablaV|2 119889119909 (51)

Multiplying (50) by V and integrating over [119904 119879]timesΩwith 119879 gt

119905 + 120591 119904 ge 120591 we have

119864V (119879) minus119864V (119904) + ]int119879

119904

int

Ω

|nablaV|2 119889119909 119889119903

+int

119879

119904

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

= int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

(52)

then we have

]int119879

119904

int

Ω

|nablaV|2 119889119909 119889119903 le 119864V (119904) minusint119879

119904

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

+int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

(53)

Discrete Dynamics in Nature and Society 7

Using Poincare inequality and (51) and (53) we have

int

119879

120591

119864V (119904) 119889119904 =12int

119879

120591

int

Ω

|V|2 119889119909 119889119903 +1205722

2int

119879

120591

int

Ω

|nablaV|2 119889119909 119889119903 le 119862int

119879

120591

int

Ω

|nablaV|2 119889119909 119889119903

le 119862119864V (120591) minus119862int119879

120591

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

+119862int

119879

120591

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

(54)

Integrating (52) from 120591 to 119879 with respect to 119904 we obtain

119879119864V (119879) + ]int119879

120591

int

119879

119904

int

Ω

|nablaV|2 119889119909 119889119903 119889119904 = int

119879

120591

119864V (119904) 119889119904 minusint119879

120591

int

119879

119904

int

Ω

[119861 (1199061) minus 119861 (119906

2)] V (119903) 119889119909 119889119903 119889119904

+int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903 119889119904

(55)

Substituting (54) into (55) we get

119879119864V (119879) + ]int119879

120591

int

119879

119904

int

Ω

|nablaV|2 119889119909 119889119903 119889119904 le 119862119864V (120591) minus119862int119879

120591

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

+119862int

119879

120591

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

minusint

119879

120591

int

119879

119904

int

Ω

[119861 (1199061) minus 119861 (119906

2)] V (119903) 119889119909 119889119903 119889119904

+int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903 119889119904

(56)

Set

1198620 = 119862119864V (120591) =119862

2int

Ω

100381610038161003816100381610038161206011(0) minus 1206012 (0)1003816100381610038161003816

1003816

2119889119909+

1198621205722

2int

Ω

10038161003816100381610038161003816nabla120601

1(0) minus nabla1206012 (0)1003816100381610038161003816

1003816

2119889119909 (57)

Φ(1199061 119906

2) = minus119862int

119879

120591

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

+119862int

119879

120591

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

minusint

119879

120591

int

119879

119904

int

Ω

[119861 (1199061) minus 119861 (119906

2)] V (119903) 119889119909 119889119903 119889119904

+int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903 119889119904

(58)

8 Discrete Dynamics in Nature and Society

then by (56) we have

119864V (119879) le1198620119879

+

1119879

Φ(1199061 119906

2) (59)

One has forall120598 gt 0 we choose 119879 gt 0 large enough such that1198620119879 lt 120598 By Theorem 4 it suffices to prove that Φ(1199061 1199062) is

a contractive function since119861 is a bounded positive invariantset

If 119906119898 rarr 119906 (119898 rarr infin) we have the limits

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

(119861 (119906119899) minus 119861 (119906

119898)) (119906119899minus 119906119898) 119889119909 119889119903

= lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

[((119906119899minus 119906119898) sdot nabla) 119906

119899minus (119906119898sdot nabla) (119906

119899minus 119906119898)] (119906119899minus 119906119898) 119889119909 119889119903 = 0

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

(119861 (119906119899) minus 119861 (119906

119898)) (119906119899minus 119906119898) 119889119909 119889119903 119889119904

= lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

[((119906119899minus 119906119898) sdot nabla) 119906

119899minus (119906119898sdot nabla) (119906

119899minus 119906119898)] (119906119899minus 119906119898) 119889119909 119889119903 119889119904 = 0

(60)

By (1198671)ndash(1198673) we have

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

int

0

minusℎ

(119866 (120590 119906119899(119903 + 120590)) minus 119866 (120590 119906

119898(119903 + 120590))) V (119903) 119889120590 119889119909 119889119903

le lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

int

0

minusℎ

1198711003816100381610038161003816119906119899(119903 + 120590) minus 119906

119898(119903 + 120590)

1003816100381610038161003816

1003816100381610038161003816119906119899(119903) minus 119906

119898(119903)1003816100381610038161003816119889120590 119889119909 119889119903 = 0

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

(119866 (120590 119906119899(119903 + 120590)) minus 119866 (120590 119906

119898(119903 + 120590))) V (119903) 119889120590 119889119909 119889119903 119889119904

le lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

1198711003816100381610038161003816(119906119899minus 119906119898) (119903 + 120590)

1003816100381610038161003816

1003816100381610038161003816(119906119899minus 119906119898) (119903)

1003816100381610038161003816119889120590 119889119909 119889119903 119889119904 = 0

(61)

Combining (60)-(61) with (58) we have that Φ(1199061 1199062) is acontractive function

The proof is complete

Theorem 6 Assume that (1198671)ndash(1198673) hold and 119871 le ]2120582211205764119890120576ℎ then the process 119880(119905 120591) generated by the global solutionfor problem (10) has a pullback attractor

Proof Lemma 3 implies 119880(119905 120591) has a pullback absorbing set119861 and Lemma 5 implies 119880(119905 120591) is asymptotically compactwe obtain the conclusion immediately

According to Theorem 6 0 lt 120576 lt min]1205722 ]12058214 and119871 le ]2120582211205764119890

120576ℎ Under the assumptions (1198671)ndash(1198673) the NSVequation (1) with a distributed delay has a pullback attractor

Conflict of Interests

The authors declare that they have no competing interests

Authorsrsquo Contribution

Yantao Guo carried out the long time behavior of solutionsShuilin Cheng carried out the pullback attractor YanbinTang carried out the distributed delay All authors read andapproved the final paper

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China (Grant nos 11471129 and 11272277)

References

[1] A P Oskolkov ldquoThe uniqueness and solvability in the largeof boundary value problems for the equations of motion ofaqueous solutions of polymersrdquo Zapiski Nauchnykh SeminarovLeningrad Otdel Mathematics Institute Stekov (LOMI) vol 38pp 98ndash136 1973

Discrete Dynamics in Nature and Society 9

[2] Y Cao E M Lunasin and E S Titi ldquoGlobal well-posedness ofthe three-dimensional viscous and inviscid simplified Bardinaturbulence modelsrdquo Communications in Mathematical Sciencesvol 4 no 4 pp 823ndash848 2006

[3] V K Kalantarov and E S Titi ldquoGlobal attractors and determin-ing modes for the 3D Navier-Stokes-Voight equationsrdquo ChineseAnnals ofMathematics Series B vol 30 no 6 pp 697ndash714 2009

[4] G-C Yue and C-K Zhong ldquoAttractors for autonomous andnonautonomous 3D Navier-Stokes-Voight equationsrdquo Discreteand Continuous Dynamical Systems Series B vol 16 no 3 pp985ndash1002 2011

[5] J Garcıa-Luengo P Marın-Rubio and J Real ldquoPullback attrac-tors for three-dimensional non-autonomous Navier-Stokes-Voigt equationsrdquo Nonlinearity vol 25 no 4 pp 905ndash930 2012

[6] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977

[7] Y Tang and M Wang ldquoA remark on exponential stabilityof time-delayed Burgers equationrdquo Discrete and ContinuousDynamical Systems Series B vol 12 no 1 pp 219ndash225 2009

[8] Y Tang and L Zhou ldquoStability switch and Hopf bifurcationfor a diffusive prey-predator system with delayrdquo Journal ofMathematical Analysis and Applications vol 334 no 2 pp1290ndash1307 2007

[9] J H Wu Theory and Applications of Partial Differential Equa-tions Springer New York NY USA 1996

[10] L Zhou Y B Tang and S Hussein ldquoStability and Hopf bifur-cation for a delay competition diffusion systemrdquoChaos Solitonsamp Fractals vol 14 no 8 pp 1201ndash1225 2002

[11] T Caraballo and J Real ldquoNavier-Stokes equations with delaysrdquoThe Royal Society of London Proceedings Series A Mathemat-ical Physical and Engineering Sciences vol 457 no 2014 pp2441ndash2453 2001

[12] T Caraballo and J Real ldquoAsymptotic behaviour of two-dimen-sional Navier-Stokes equations with delaysrdquo The Royal Societyof London Proceedings Series A Mathematical Physical andEngineering Sciences vol 459 no 2040 pp 3181ndash3194 2003

[13] T Caraballo and J Real ldquoAttractors for 2D-Navier-Stokesmodels with delaysrdquo Journal of Differential Equations vol 205no 2 pp 271ndash297 2004

[14] Z Hu and Y Wang ldquoPullback attractors for a nonautonomousnonclassical diffusion equation with variable delayrdquo Journal ofMathematical Physics vol 53 no 7 Article ID 072702 17 pages2012

[15] H Y Li and Y M Qin ldquoPullback attractors for three-dimen-sional navier-stokes-voigt equations with delaysrdquo BoundaryValue Problems vol 2013 article 191 2013

[16] P Marın-Rubio and J Real ldquoAttractors for 2D-Navier-Stokesequations with delays on some unbounded domainsrdquoNonlinearAnalysis Theory Methods amp Applications vol 67 no 10 pp2784ndash2799 2007

[17] P Marın-Rubio J Real and J Valero ldquoPullback attractors for atwo-dimensional Navier-Stokesmodel in an infinite delay caserdquoNonlinear Analysis Theory Methods amp Applications vol 74 no5 pp 2012ndash2030 2011

[18] P Marın-Rubio and J Real ldquoPullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linearoperatorsrdquo Discrete and Continuous Dynamical Systems SeriesA vol 26 no 3 pp 989ndash1006 2010

[19] AKKhanmamedov ldquoGlobal attractors forwave equationswithnonlinear interior damping and critical exponentsrdquo Journal ofDifferential Equations vol 230 no 2 pp 702ndash719 2006

[20] L Yang and C-K Zhong ldquoGlobal attractor for plate equationwith nonlinear dampingrdquo Nonlinear Analysis Theory Methodsamp Applications vol 69 no 11 pp 3802ndash3810 2008

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Research Article Approximate Kelvin-Voigt Fluid …downloads.hindawi.com/journals/ddns/2015/721673.pdfWe consider the approximate D Kelvin-Voigt uid driven by an external force depending

6 Discrete Dynamics in Nature and Society

Let 120578 = (12)]1205821 choosing 0 lt 120576 lt min]1205722 ]12058214 119871 le

]2120582211205764119890120576ℎ implies that

max120576+ 120578minus ]1205821 +120578minus1119871119890120576ℎ

2120576 120576120572

2minus ] lt 0 (42)

then

119890120576119905(|119906 (119905)|

2+120572

2119906 (119905)

2)

le

119871119890120576(120591+ℎ)

1205762]1205821

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+ 119890120576120591(|119906 (120591)|

2+120572

2119906 (120591)

2)

(43)

which implies

|119906 (119905)|2+120572

2119906 (119905)

2

le

119871119890120576(120591+ℎminus119905)

1205762]1205821

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+ 119890120576(120591minus119905)

(1003816100381610038161003816120601 (120591)

1003816100381610038161003816

2+120572

2 1003817100381710038171003817120601 (120591)

1003817100381710038171003817

2)

(44)

Now if we take 119905 ge 120591 + ℎ then for 120579 isin [minusℎ 0] we have

|119906 (119905 + 120579)|2+120572

2119906 (119905 + 120579)

2

le

119871119890120576(120591+ℎminus119905minus120579)

1205762]1205821

10038171003817100381710038171206011003817100381710038171003817

2119862119867

+ 119890120576(120591minus119905minus120579)

(10038171003817100381710038171206011003817100381710038171003817

2119862119867

+1205722 10038171003817100381710038171206011003817100381710038171003817

2119862119881)

(45)

We denote byR the set of all functions 119903 (minusinfin +infin) rarr

(0 +infin) such that

lim120591rarrminusinfin

1198901205721205911199032(120591) = 0 (46)

Then the closed ball in 119862119881defined by

119861 = 120593 isin11986211988110038171003817100381710038171205931003817100381710038171003817

2119862119881le 1 (47)

is pullback absorbing set for 119880(119905 120591) The proof is complete

We next prove the asymptotic compactness of solution toproblem (10) by contractive functions see [19 20]

Let 119883 be a Banach space and let 119861 be a bounded subsetof 119883 We call a function Φ(sdot sdot) which defined on 119883 times 119883 is acontractive function on 119861times119861 if for any sequence 119909

119899infin

119899=1sub 119861

there is a subsequence 119909119899119896infin

119896=1 sub 119909119899infin

119899=1 such that

lim119897rarrinfin

lim119896rarrinfin

Φ(119909119899119896 119909119899119897) = 0 (48)

Denote all such contractive functions on 119861 times 119861 by 119862(119861)

Theorem 4 (see [19]) Let 119878(119905)119905ge120591

be a semigroup on aBanach space (119883 sdot ) and have a bounded absorbing set 119861forall120598 gt 0 there exist 119879 = 119879(119861 120598) and Φ isin 119862(119861) such that

1003817100381710038171003817119878 (119879) 119909 minus 119878 (119879) 119910

1003817100381710038171003817le 120598 +Φ

119879(119909 119910) forall119909 119910 isin 119861 (49)

where Φ119879depends on 119879 Then 119878(119905)

119905ge120591is asymptotically

compact in119883

Lemma 5 Assume that (1198671)ndash(1198673) hold the process 119880(119905 120591)119905ge120591generated by the global solution to problem (10) is asymptoti-cally compact

Proof Let 119906119894(119905) be the solution to problem (10) with initialdata 120601119894 isin 119861 (119894 = 1 2) respectively Denote V(119905) = 119906

1(119905)minus119906

2(119905)

then V(119905) satisfies the equivalent abstract equation

120597

120597119905

(V+1205722119860V) + ]119860V+119861 (1199061 1199061) minus119861 (1199062 1199062)

= int

0

minusℎ

[119866 (119904 1199061(119905 + 119904)) minus119866 (s 1199062 (119905 + 119904))] 119889119904

(50)

with the initial condition V(119905) = 1206011(119905 minus 120591) minus 120601

2(119905 minus 120591) 119905 isin

[120591 minus ℎ 120591]Set an energy function

119864V (119905) =12int

Ω

|V|2 119889119909+1205722

2int

Ω

|nablaV|2 119889119909 (51)

Multiplying (50) by V and integrating over [119904 119879]timesΩwith 119879 gt

119905 + 120591 119904 ge 120591 we have

119864V (119879) minus119864V (119904) + ]int119879

119904

int

Ω

|nablaV|2 119889119909 119889119903

+int

119879

119904

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

= int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

(52)

then we have

]int119879

119904

int

Ω

|nablaV|2 119889119909 119889119903 le 119864V (119904) minusint119879

119904

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

+int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

(53)

Discrete Dynamics in Nature and Society 7

Using Poincare inequality and (51) and (53) we have

int

119879

120591

119864V (119904) 119889119904 =12int

119879

120591

int

Ω

|V|2 119889119909 119889119903 +1205722

2int

119879

120591

int

Ω

|nablaV|2 119889119909 119889119903 le 119862int

119879

120591

int

Ω

|nablaV|2 119889119909 119889119903

le 119862119864V (120591) minus119862int119879

120591

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

+119862int

119879

120591

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

(54)

Integrating (52) from 120591 to 119879 with respect to 119904 we obtain

119879119864V (119879) + ]int119879

120591

int

119879

119904

int

Ω

|nablaV|2 119889119909 119889119903 119889119904 = int

119879

120591

119864V (119904) 119889119904 minusint119879

120591

int

119879

119904

int

Ω

[119861 (1199061) minus 119861 (119906

2)] V (119903) 119889119909 119889119903 119889119904

+int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903 119889119904

(55)

Substituting (54) into (55) we get

119879119864V (119879) + ]int119879

120591

int

119879

119904

int

Ω

|nablaV|2 119889119909 119889119903 119889119904 le 119862119864V (120591) minus119862int119879

120591

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

+119862int

119879

120591

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

minusint

119879

120591

int

119879

119904

int

Ω

[119861 (1199061) minus 119861 (119906

2)] V (119903) 119889119909 119889119903 119889119904

+int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903 119889119904

(56)

Set

1198620 = 119862119864V (120591) =119862

2int

Ω

100381610038161003816100381610038161206011(0) minus 1206012 (0)1003816100381610038161003816

1003816

2119889119909+

1198621205722

2int

Ω

10038161003816100381610038161003816nabla120601

1(0) minus nabla1206012 (0)1003816100381610038161003816

1003816

2119889119909 (57)

Φ(1199061 119906

2) = minus119862int

119879

120591

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

+119862int

119879

120591

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

minusint

119879

120591

int

119879

119904

int

Ω

[119861 (1199061) minus 119861 (119906

2)] V (119903) 119889119909 119889119903 119889119904

+int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903 119889119904

(58)

8 Discrete Dynamics in Nature and Society

then by (56) we have

119864V (119879) le1198620119879

+

1119879

Φ(1199061 119906

2) (59)

One has forall120598 gt 0 we choose 119879 gt 0 large enough such that1198620119879 lt 120598 By Theorem 4 it suffices to prove that Φ(1199061 1199062) is

a contractive function since119861 is a bounded positive invariantset

If 119906119898 rarr 119906 (119898 rarr infin) we have the limits

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

(119861 (119906119899) minus 119861 (119906

119898)) (119906119899minus 119906119898) 119889119909 119889119903

= lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

[((119906119899minus 119906119898) sdot nabla) 119906

119899minus (119906119898sdot nabla) (119906

119899minus 119906119898)] (119906119899minus 119906119898) 119889119909 119889119903 = 0

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

(119861 (119906119899) minus 119861 (119906

119898)) (119906119899minus 119906119898) 119889119909 119889119903 119889119904

= lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

[((119906119899minus 119906119898) sdot nabla) 119906

119899minus (119906119898sdot nabla) (119906

119899minus 119906119898)] (119906119899minus 119906119898) 119889119909 119889119903 119889119904 = 0

(60)

By (1198671)ndash(1198673) we have

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

int

0

minusℎ

(119866 (120590 119906119899(119903 + 120590)) minus 119866 (120590 119906

119898(119903 + 120590))) V (119903) 119889120590 119889119909 119889119903

le lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

int

0

minusℎ

1198711003816100381610038161003816119906119899(119903 + 120590) minus 119906

119898(119903 + 120590)

1003816100381610038161003816

1003816100381610038161003816119906119899(119903) minus 119906

119898(119903)1003816100381610038161003816119889120590 119889119909 119889119903 = 0

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

(119866 (120590 119906119899(119903 + 120590)) minus 119866 (120590 119906

119898(119903 + 120590))) V (119903) 119889120590 119889119909 119889119903 119889119904

le lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

1198711003816100381610038161003816(119906119899minus 119906119898) (119903 + 120590)

1003816100381610038161003816

1003816100381610038161003816(119906119899minus 119906119898) (119903)

1003816100381610038161003816119889120590 119889119909 119889119903 119889119904 = 0

(61)

Combining (60)-(61) with (58) we have that Φ(1199061 1199062) is acontractive function

The proof is complete

Theorem 6 Assume that (1198671)ndash(1198673) hold and 119871 le ]2120582211205764119890120576ℎ then the process 119880(119905 120591) generated by the global solutionfor problem (10) has a pullback attractor

Proof Lemma 3 implies 119880(119905 120591) has a pullback absorbing set119861 and Lemma 5 implies 119880(119905 120591) is asymptotically compactwe obtain the conclusion immediately

According to Theorem 6 0 lt 120576 lt min]1205722 ]12058214 and119871 le ]2120582211205764119890

120576ℎ Under the assumptions (1198671)ndash(1198673) the NSVequation (1) with a distributed delay has a pullback attractor

Conflict of Interests

The authors declare that they have no competing interests

Authorsrsquo Contribution

Yantao Guo carried out the long time behavior of solutionsShuilin Cheng carried out the pullback attractor YanbinTang carried out the distributed delay All authors read andapproved the final paper

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China (Grant nos 11471129 and 11272277)

References

[1] A P Oskolkov ldquoThe uniqueness and solvability in the largeof boundary value problems for the equations of motion ofaqueous solutions of polymersrdquo Zapiski Nauchnykh SeminarovLeningrad Otdel Mathematics Institute Stekov (LOMI) vol 38pp 98ndash136 1973

Discrete Dynamics in Nature and Society 9

[2] Y Cao E M Lunasin and E S Titi ldquoGlobal well-posedness ofthe three-dimensional viscous and inviscid simplified Bardinaturbulence modelsrdquo Communications in Mathematical Sciencesvol 4 no 4 pp 823ndash848 2006

[3] V K Kalantarov and E S Titi ldquoGlobal attractors and determin-ing modes for the 3D Navier-Stokes-Voight equationsrdquo ChineseAnnals ofMathematics Series B vol 30 no 6 pp 697ndash714 2009

[4] G-C Yue and C-K Zhong ldquoAttractors for autonomous andnonautonomous 3D Navier-Stokes-Voight equationsrdquo Discreteand Continuous Dynamical Systems Series B vol 16 no 3 pp985ndash1002 2011

[5] J Garcıa-Luengo P Marın-Rubio and J Real ldquoPullback attrac-tors for three-dimensional non-autonomous Navier-Stokes-Voigt equationsrdquo Nonlinearity vol 25 no 4 pp 905ndash930 2012

[6] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977

[7] Y Tang and M Wang ldquoA remark on exponential stabilityof time-delayed Burgers equationrdquo Discrete and ContinuousDynamical Systems Series B vol 12 no 1 pp 219ndash225 2009

[8] Y Tang and L Zhou ldquoStability switch and Hopf bifurcationfor a diffusive prey-predator system with delayrdquo Journal ofMathematical Analysis and Applications vol 334 no 2 pp1290ndash1307 2007

[9] J H Wu Theory and Applications of Partial Differential Equa-tions Springer New York NY USA 1996

[10] L Zhou Y B Tang and S Hussein ldquoStability and Hopf bifur-cation for a delay competition diffusion systemrdquoChaos Solitonsamp Fractals vol 14 no 8 pp 1201ndash1225 2002

[11] T Caraballo and J Real ldquoNavier-Stokes equations with delaysrdquoThe Royal Society of London Proceedings Series A Mathemat-ical Physical and Engineering Sciences vol 457 no 2014 pp2441ndash2453 2001

[12] T Caraballo and J Real ldquoAsymptotic behaviour of two-dimen-sional Navier-Stokes equations with delaysrdquo The Royal Societyof London Proceedings Series A Mathematical Physical andEngineering Sciences vol 459 no 2040 pp 3181ndash3194 2003

[13] T Caraballo and J Real ldquoAttractors for 2D-Navier-Stokesmodels with delaysrdquo Journal of Differential Equations vol 205no 2 pp 271ndash297 2004

[14] Z Hu and Y Wang ldquoPullback attractors for a nonautonomousnonclassical diffusion equation with variable delayrdquo Journal ofMathematical Physics vol 53 no 7 Article ID 072702 17 pages2012

[15] H Y Li and Y M Qin ldquoPullback attractors for three-dimen-sional navier-stokes-voigt equations with delaysrdquo BoundaryValue Problems vol 2013 article 191 2013

[16] P Marın-Rubio and J Real ldquoAttractors for 2D-Navier-Stokesequations with delays on some unbounded domainsrdquoNonlinearAnalysis Theory Methods amp Applications vol 67 no 10 pp2784ndash2799 2007

[17] P Marın-Rubio J Real and J Valero ldquoPullback attractors for atwo-dimensional Navier-Stokesmodel in an infinite delay caserdquoNonlinear Analysis Theory Methods amp Applications vol 74 no5 pp 2012ndash2030 2011

[18] P Marın-Rubio and J Real ldquoPullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linearoperatorsrdquo Discrete and Continuous Dynamical Systems SeriesA vol 26 no 3 pp 989ndash1006 2010

[19] AKKhanmamedov ldquoGlobal attractors forwave equationswithnonlinear interior damping and critical exponentsrdquo Journal ofDifferential Equations vol 230 no 2 pp 702ndash719 2006

[20] L Yang and C-K Zhong ldquoGlobal attractor for plate equationwith nonlinear dampingrdquo Nonlinear Analysis Theory Methodsamp Applications vol 69 no 11 pp 3802ndash3810 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Research Article Approximate Kelvin-Voigt Fluid …downloads.hindawi.com/journals/ddns/2015/721673.pdfWe consider the approximate D Kelvin-Voigt uid driven by an external force depending

Discrete Dynamics in Nature and Society 7

Using Poincare inequality and (51) and (53) we have

int

119879

120591

119864V (119904) 119889119904 =12int

119879

120591

int

Ω

|V|2 119889119909 119889119903 +1205722

2int

119879

120591

int

Ω

|nablaV|2 119889119909 119889119903 le 119862int

119879

120591

int

Ω

|nablaV|2 119889119909 119889119903

le 119862119864V (120591) minus119862int119879

120591

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

+119862int

119879

120591

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

(54)

Integrating (52) from 120591 to 119879 with respect to 119904 we obtain

119879119864V (119879) + ]int119879

120591

int

119879

119904

int

Ω

|nablaV|2 119889119909 119889119903 119889119904 = int

119879

120591

119864V (119904) 119889119904 minusint119879

120591

int

119879

119904

int

Ω

[119861 (1199061) minus 119861 (119906

2)] V (119903) 119889119909 119889119903 119889119904

+int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903 119889119904

(55)

Substituting (54) into (55) we get

119879119864V (119879) + ]int119879

120591

int

119879

119904

int

Ω

|nablaV|2 119889119909 119889119903 119889119904 le 119862119864V (120591) minus119862int119879

120591

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

+119862int

119879

120591

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

minusint

119879

120591

int

119879

119904

int

Ω

[119861 (1199061) minus 119861 (119906

2)] V (119903) 119889119909 119889119903 119889119904

+int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903 119889119904

(56)

Set

1198620 = 119862119864V (120591) =119862

2int

Ω

100381610038161003816100381610038161206011(0) minus 1206012 (0)1003816100381610038161003816

1003816

2119889119909+

1198621205722

2int

Ω

10038161003816100381610038161003816nabla120601

1(0) minus nabla1206012 (0)1003816100381610038161003816

1003816

2119889119909 (57)

Φ(1199061 119906

2) = minus119862int

119879

120591

int

Ω

[119861 (1199061) minus119861 (119906

2)] V (119903) 119889119909 119889119903

+119862int

119879

120591

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903

minusint

119879

120591

int

119879

119904

int

Ω

[119861 (1199061) minus 119861 (119906

2)] V (119903) 119889119909 119889119903 119889119904

+int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

[119866 (120590 1199061(119903 + 120590)) minus 119866 (120590 119906

2(119903 + 120590))] V (119903) 119889120590 119889119909 119889119903 119889119904

(58)

8 Discrete Dynamics in Nature and Society

then by (56) we have

119864V (119879) le1198620119879

+

1119879

Φ(1199061 119906

2) (59)

One has forall120598 gt 0 we choose 119879 gt 0 large enough such that1198620119879 lt 120598 By Theorem 4 it suffices to prove that Φ(1199061 1199062) is

a contractive function since119861 is a bounded positive invariantset

If 119906119898 rarr 119906 (119898 rarr infin) we have the limits

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

(119861 (119906119899) minus 119861 (119906

119898)) (119906119899minus 119906119898) 119889119909 119889119903

= lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

[((119906119899minus 119906119898) sdot nabla) 119906

119899minus (119906119898sdot nabla) (119906

119899minus 119906119898)] (119906119899minus 119906119898) 119889119909 119889119903 = 0

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

(119861 (119906119899) minus 119861 (119906

119898)) (119906119899minus 119906119898) 119889119909 119889119903 119889119904

= lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

[((119906119899minus 119906119898) sdot nabla) 119906

119899minus (119906119898sdot nabla) (119906

119899minus 119906119898)] (119906119899minus 119906119898) 119889119909 119889119903 119889119904 = 0

(60)

By (1198671)ndash(1198673) we have

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

int

0

minusℎ

(119866 (120590 119906119899(119903 + 120590)) minus 119866 (120590 119906

119898(119903 + 120590))) V (119903) 119889120590 119889119909 119889119903

le lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

int

0

minusℎ

1198711003816100381610038161003816119906119899(119903 + 120590) minus 119906

119898(119903 + 120590)

1003816100381610038161003816

1003816100381610038161003816119906119899(119903) minus 119906

119898(119903)1003816100381610038161003816119889120590 119889119909 119889119903 = 0

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

(119866 (120590 119906119899(119903 + 120590)) minus 119866 (120590 119906

119898(119903 + 120590))) V (119903) 119889120590 119889119909 119889119903 119889119904

le lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

1198711003816100381610038161003816(119906119899minus 119906119898) (119903 + 120590)

1003816100381610038161003816

1003816100381610038161003816(119906119899minus 119906119898) (119903)

1003816100381610038161003816119889120590 119889119909 119889119903 119889119904 = 0

(61)

Combining (60)-(61) with (58) we have that Φ(1199061 1199062) is acontractive function

The proof is complete

Theorem 6 Assume that (1198671)ndash(1198673) hold and 119871 le ]2120582211205764119890120576ℎ then the process 119880(119905 120591) generated by the global solutionfor problem (10) has a pullback attractor

Proof Lemma 3 implies 119880(119905 120591) has a pullback absorbing set119861 and Lemma 5 implies 119880(119905 120591) is asymptotically compactwe obtain the conclusion immediately

According to Theorem 6 0 lt 120576 lt min]1205722 ]12058214 and119871 le ]2120582211205764119890

120576ℎ Under the assumptions (1198671)ndash(1198673) the NSVequation (1) with a distributed delay has a pullback attractor

Conflict of Interests

The authors declare that they have no competing interests

Authorsrsquo Contribution

Yantao Guo carried out the long time behavior of solutionsShuilin Cheng carried out the pullback attractor YanbinTang carried out the distributed delay All authors read andapproved the final paper

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China (Grant nos 11471129 and 11272277)

References

[1] A P Oskolkov ldquoThe uniqueness and solvability in the largeof boundary value problems for the equations of motion ofaqueous solutions of polymersrdquo Zapiski Nauchnykh SeminarovLeningrad Otdel Mathematics Institute Stekov (LOMI) vol 38pp 98ndash136 1973

Discrete Dynamics in Nature and Society 9

[2] Y Cao E M Lunasin and E S Titi ldquoGlobal well-posedness ofthe three-dimensional viscous and inviscid simplified Bardinaturbulence modelsrdquo Communications in Mathematical Sciencesvol 4 no 4 pp 823ndash848 2006

[3] V K Kalantarov and E S Titi ldquoGlobal attractors and determin-ing modes for the 3D Navier-Stokes-Voight equationsrdquo ChineseAnnals ofMathematics Series B vol 30 no 6 pp 697ndash714 2009

[4] G-C Yue and C-K Zhong ldquoAttractors for autonomous andnonautonomous 3D Navier-Stokes-Voight equationsrdquo Discreteand Continuous Dynamical Systems Series B vol 16 no 3 pp985ndash1002 2011

[5] J Garcıa-Luengo P Marın-Rubio and J Real ldquoPullback attrac-tors for three-dimensional non-autonomous Navier-Stokes-Voigt equationsrdquo Nonlinearity vol 25 no 4 pp 905ndash930 2012

[6] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977

[7] Y Tang and M Wang ldquoA remark on exponential stabilityof time-delayed Burgers equationrdquo Discrete and ContinuousDynamical Systems Series B vol 12 no 1 pp 219ndash225 2009

[8] Y Tang and L Zhou ldquoStability switch and Hopf bifurcationfor a diffusive prey-predator system with delayrdquo Journal ofMathematical Analysis and Applications vol 334 no 2 pp1290ndash1307 2007

[9] J H Wu Theory and Applications of Partial Differential Equa-tions Springer New York NY USA 1996

[10] L Zhou Y B Tang and S Hussein ldquoStability and Hopf bifur-cation for a delay competition diffusion systemrdquoChaos Solitonsamp Fractals vol 14 no 8 pp 1201ndash1225 2002

[11] T Caraballo and J Real ldquoNavier-Stokes equations with delaysrdquoThe Royal Society of London Proceedings Series A Mathemat-ical Physical and Engineering Sciences vol 457 no 2014 pp2441ndash2453 2001

[12] T Caraballo and J Real ldquoAsymptotic behaviour of two-dimen-sional Navier-Stokes equations with delaysrdquo The Royal Societyof London Proceedings Series A Mathematical Physical andEngineering Sciences vol 459 no 2040 pp 3181ndash3194 2003

[13] T Caraballo and J Real ldquoAttractors for 2D-Navier-Stokesmodels with delaysrdquo Journal of Differential Equations vol 205no 2 pp 271ndash297 2004

[14] Z Hu and Y Wang ldquoPullback attractors for a nonautonomousnonclassical diffusion equation with variable delayrdquo Journal ofMathematical Physics vol 53 no 7 Article ID 072702 17 pages2012

[15] H Y Li and Y M Qin ldquoPullback attractors for three-dimen-sional navier-stokes-voigt equations with delaysrdquo BoundaryValue Problems vol 2013 article 191 2013

[16] P Marın-Rubio and J Real ldquoAttractors for 2D-Navier-Stokesequations with delays on some unbounded domainsrdquoNonlinearAnalysis Theory Methods amp Applications vol 67 no 10 pp2784ndash2799 2007

[17] P Marın-Rubio J Real and J Valero ldquoPullback attractors for atwo-dimensional Navier-Stokesmodel in an infinite delay caserdquoNonlinear Analysis Theory Methods amp Applications vol 74 no5 pp 2012ndash2030 2011

[18] P Marın-Rubio and J Real ldquoPullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linearoperatorsrdquo Discrete and Continuous Dynamical Systems SeriesA vol 26 no 3 pp 989ndash1006 2010

[19] AKKhanmamedov ldquoGlobal attractors forwave equationswithnonlinear interior damping and critical exponentsrdquo Journal ofDifferential Equations vol 230 no 2 pp 702ndash719 2006

[20] L Yang and C-K Zhong ldquoGlobal attractor for plate equationwith nonlinear dampingrdquo Nonlinear Analysis Theory Methodsamp Applications vol 69 no 11 pp 3802ndash3810 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: Research Article Approximate Kelvin-Voigt Fluid …downloads.hindawi.com/journals/ddns/2015/721673.pdfWe consider the approximate D Kelvin-Voigt uid driven by an external force depending

8 Discrete Dynamics in Nature and Society

then by (56) we have

119864V (119879) le1198620119879

+

1119879

Φ(1199061 119906

2) (59)

One has forall120598 gt 0 we choose 119879 gt 0 large enough such that1198620119879 lt 120598 By Theorem 4 it suffices to prove that Φ(1199061 1199062) is

a contractive function since119861 is a bounded positive invariantset

If 119906119898 rarr 119906 (119898 rarr infin) we have the limits

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

(119861 (119906119899) minus 119861 (119906

119898)) (119906119899minus 119906119898) 119889119909 119889119903

= lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

[((119906119899minus 119906119898) sdot nabla) 119906

119899minus (119906119898sdot nabla) (119906

119899minus 119906119898)] (119906119899minus 119906119898) 119889119909 119889119903 = 0

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

(119861 (119906119899) minus 119861 (119906

119898)) (119906119899minus 119906119898) 119889119909 119889119903 119889119904

= lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

[((119906119899minus 119906119898) sdot nabla) 119906

119899minus (119906119898sdot nabla) (119906

119899minus 119906119898)] (119906119899minus 119906119898) 119889119909 119889119903 119889119904 = 0

(60)

By (1198671)ndash(1198673) we have

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

int

0

minusℎ

(119866 (120590 119906119899(119903 + 120590)) minus 119866 (120590 119906

119898(119903 + 120590))) V (119903) 119889120590 119889119909 119889119903

le lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

Ω

int

0

minusℎ

1198711003816100381610038161003816119906119899(119903 + 120590) minus 119906

119898(119903 + 120590)

1003816100381610038161003816

1003816100381610038161003816119906119899(119903) minus 119906

119898(119903)1003816100381610038161003816119889120590 119889119909 119889119903 = 0

lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

(119866 (120590 119906119899(119903 + 120590)) minus 119866 (120590 119906

119898(119903 + 120590))) V (119903) 119889120590 119889119909 119889119903 119889119904

le lim119899rarrinfin

lim119898rarrinfin

int

119879

120591

int

119879

119904

int

Ω

int

0

minusℎ

1198711003816100381610038161003816(119906119899minus 119906119898) (119903 + 120590)

1003816100381610038161003816

1003816100381610038161003816(119906119899minus 119906119898) (119903)

1003816100381610038161003816119889120590 119889119909 119889119903 119889119904 = 0

(61)

Combining (60)-(61) with (58) we have that Φ(1199061 1199062) is acontractive function

The proof is complete

Theorem 6 Assume that (1198671)ndash(1198673) hold and 119871 le ]2120582211205764119890120576ℎ then the process 119880(119905 120591) generated by the global solutionfor problem (10) has a pullback attractor

Proof Lemma 3 implies 119880(119905 120591) has a pullback absorbing set119861 and Lemma 5 implies 119880(119905 120591) is asymptotically compactwe obtain the conclusion immediately

According to Theorem 6 0 lt 120576 lt min]1205722 ]12058214 and119871 le ]2120582211205764119890

120576ℎ Under the assumptions (1198671)ndash(1198673) the NSVequation (1) with a distributed delay has a pullback attractor

Conflict of Interests

The authors declare that they have no competing interests

Authorsrsquo Contribution

Yantao Guo carried out the long time behavior of solutionsShuilin Cheng carried out the pullback attractor YanbinTang carried out the distributed delay All authors read andapproved the final paper

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China (Grant nos 11471129 and 11272277)

References

[1] A P Oskolkov ldquoThe uniqueness and solvability in the largeof boundary value problems for the equations of motion ofaqueous solutions of polymersrdquo Zapiski Nauchnykh SeminarovLeningrad Otdel Mathematics Institute Stekov (LOMI) vol 38pp 98ndash136 1973

Discrete Dynamics in Nature and Society 9

[2] Y Cao E M Lunasin and E S Titi ldquoGlobal well-posedness ofthe three-dimensional viscous and inviscid simplified Bardinaturbulence modelsrdquo Communications in Mathematical Sciencesvol 4 no 4 pp 823ndash848 2006

[3] V K Kalantarov and E S Titi ldquoGlobal attractors and determin-ing modes for the 3D Navier-Stokes-Voight equationsrdquo ChineseAnnals ofMathematics Series B vol 30 no 6 pp 697ndash714 2009

[4] G-C Yue and C-K Zhong ldquoAttractors for autonomous andnonautonomous 3D Navier-Stokes-Voight equationsrdquo Discreteand Continuous Dynamical Systems Series B vol 16 no 3 pp985ndash1002 2011

[5] J Garcıa-Luengo P Marın-Rubio and J Real ldquoPullback attrac-tors for three-dimensional non-autonomous Navier-Stokes-Voigt equationsrdquo Nonlinearity vol 25 no 4 pp 905ndash930 2012

[6] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977

[7] Y Tang and M Wang ldquoA remark on exponential stabilityof time-delayed Burgers equationrdquo Discrete and ContinuousDynamical Systems Series B vol 12 no 1 pp 219ndash225 2009

[8] Y Tang and L Zhou ldquoStability switch and Hopf bifurcationfor a diffusive prey-predator system with delayrdquo Journal ofMathematical Analysis and Applications vol 334 no 2 pp1290ndash1307 2007

[9] J H Wu Theory and Applications of Partial Differential Equa-tions Springer New York NY USA 1996

[10] L Zhou Y B Tang and S Hussein ldquoStability and Hopf bifur-cation for a delay competition diffusion systemrdquoChaos Solitonsamp Fractals vol 14 no 8 pp 1201ndash1225 2002

[11] T Caraballo and J Real ldquoNavier-Stokes equations with delaysrdquoThe Royal Society of London Proceedings Series A Mathemat-ical Physical and Engineering Sciences vol 457 no 2014 pp2441ndash2453 2001

[12] T Caraballo and J Real ldquoAsymptotic behaviour of two-dimen-sional Navier-Stokes equations with delaysrdquo The Royal Societyof London Proceedings Series A Mathematical Physical andEngineering Sciences vol 459 no 2040 pp 3181ndash3194 2003

[13] T Caraballo and J Real ldquoAttractors for 2D-Navier-Stokesmodels with delaysrdquo Journal of Differential Equations vol 205no 2 pp 271ndash297 2004

[14] Z Hu and Y Wang ldquoPullback attractors for a nonautonomousnonclassical diffusion equation with variable delayrdquo Journal ofMathematical Physics vol 53 no 7 Article ID 072702 17 pages2012

[15] H Y Li and Y M Qin ldquoPullback attractors for three-dimen-sional navier-stokes-voigt equations with delaysrdquo BoundaryValue Problems vol 2013 article 191 2013

[16] P Marın-Rubio and J Real ldquoAttractors for 2D-Navier-Stokesequations with delays on some unbounded domainsrdquoNonlinearAnalysis Theory Methods amp Applications vol 67 no 10 pp2784ndash2799 2007

[17] P Marın-Rubio J Real and J Valero ldquoPullback attractors for atwo-dimensional Navier-Stokesmodel in an infinite delay caserdquoNonlinear Analysis Theory Methods amp Applications vol 74 no5 pp 2012ndash2030 2011

[18] P Marın-Rubio and J Real ldquoPullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linearoperatorsrdquo Discrete and Continuous Dynamical Systems SeriesA vol 26 no 3 pp 989ndash1006 2010

[19] AKKhanmamedov ldquoGlobal attractors forwave equationswithnonlinear interior damping and critical exponentsrdquo Journal ofDifferential Equations vol 230 no 2 pp 702ndash719 2006

[20] L Yang and C-K Zhong ldquoGlobal attractor for plate equationwith nonlinear dampingrdquo Nonlinear Analysis Theory Methodsamp Applications vol 69 no 11 pp 3802ndash3810 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 9: Research Article Approximate Kelvin-Voigt Fluid …downloads.hindawi.com/journals/ddns/2015/721673.pdfWe consider the approximate D Kelvin-Voigt uid driven by an external force depending

Discrete Dynamics in Nature and Society 9

[2] Y Cao E M Lunasin and E S Titi ldquoGlobal well-posedness ofthe three-dimensional viscous and inviscid simplified Bardinaturbulence modelsrdquo Communications in Mathematical Sciencesvol 4 no 4 pp 823ndash848 2006

[3] V K Kalantarov and E S Titi ldquoGlobal attractors and determin-ing modes for the 3D Navier-Stokes-Voight equationsrdquo ChineseAnnals ofMathematics Series B vol 30 no 6 pp 697ndash714 2009

[4] G-C Yue and C-K Zhong ldquoAttractors for autonomous andnonautonomous 3D Navier-Stokes-Voight equationsrdquo Discreteand Continuous Dynamical Systems Series B vol 16 no 3 pp985ndash1002 2011

[5] J Garcıa-Luengo P Marın-Rubio and J Real ldquoPullback attrac-tors for three-dimensional non-autonomous Navier-Stokes-Voigt equationsrdquo Nonlinearity vol 25 no 4 pp 905ndash930 2012

[6] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977

[7] Y Tang and M Wang ldquoA remark on exponential stabilityof time-delayed Burgers equationrdquo Discrete and ContinuousDynamical Systems Series B vol 12 no 1 pp 219ndash225 2009

[8] Y Tang and L Zhou ldquoStability switch and Hopf bifurcationfor a diffusive prey-predator system with delayrdquo Journal ofMathematical Analysis and Applications vol 334 no 2 pp1290ndash1307 2007

[9] J H Wu Theory and Applications of Partial Differential Equa-tions Springer New York NY USA 1996

[10] L Zhou Y B Tang and S Hussein ldquoStability and Hopf bifur-cation for a delay competition diffusion systemrdquoChaos Solitonsamp Fractals vol 14 no 8 pp 1201ndash1225 2002

[11] T Caraballo and J Real ldquoNavier-Stokes equations with delaysrdquoThe Royal Society of London Proceedings Series A Mathemat-ical Physical and Engineering Sciences vol 457 no 2014 pp2441ndash2453 2001

[12] T Caraballo and J Real ldquoAsymptotic behaviour of two-dimen-sional Navier-Stokes equations with delaysrdquo The Royal Societyof London Proceedings Series A Mathematical Physical andEngineering Sciences vol 459 no 2040 pp 3181ndash3194 2003

[13] T Caraballo and J Real ldquoAttractors for 2D-Navier-Stokesmodels with delaysrdquo Journal of Differential Equations vol 205no 2 pp 271ndash297 2004

[14] Z Hu and Y Wang ldquoPullback attractors for a nonautonomousnonclassical diffusion equation with variable delayrdquo Journal ofMathematical Physics vol 53 no 7 Article ID 072702 17 pages2012

[15] H Y Li and Y M Qin ldquoPullback attractors for three-dimen-sional navier-stokes-voigt equations with delaysrdquo BoundaryValue Problems vol 2013 article 191 2013

[16] P Marın-Rubio and J Real ldquoAttractors for 2D-Navier-Stokesequations with delays on some unbounded domainsrdquoNonlinearAnalysis Theory Methods amp Applications vol 67 no 10 pp2784ndash2799 2007

[17] P Marın-Rubio J Real and J Valero ldquoPullback attractors for atwo-dimensional Navier-Stokesmodel in an infinite delay caserdquoNonlinear Analysis Theory Methods amp Applications vol 74 no5 pp 2012ndash2030 2011

[18] P Marın-Rubio and J Real ldquoPullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linearoperatorsrdquo Discrete and Continuous Dynamical Systems SeriesA vol 26 no 3 pp 989ndash1006 2010

[19] AKKhanmamedov ldquoGlobal attractors forwave equationswithnonlinear interior damping and critical exponentsrdquo Journal ofDifferential Equations vol 230 no 2 pp 702ndash719 2006

[20] L Yang and C-K Zhong ldquoGlobal attractor for plate equationwith nonlinear dampingrdquo Nonlinear Analysis Theory Methodsamp Applications vol 69 no 11 pp 3802ndash3810 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 10: Research Article Approximate Kelvin-Voigt Fluid …downloads.hindawi.com/journals/ddns/2015/721673.pdfWe consider the approximate D Kelvin-Voigt uid driven by an external force depending

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


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