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
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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
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
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)
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
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
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
[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
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
[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
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
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)
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
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
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
[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
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
[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
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)
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
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
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
[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
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
[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
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)
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
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
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
[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
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
[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
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
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
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
[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
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
[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
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
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
[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
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
[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
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
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
[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
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
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
[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
[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
[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