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

1. Introduction

In this paper, we consider 3D Navier-Stokes-Voigt (NSV) equation with a distributed delay external force depending on the velocity of the fluid:(1)ut-α2Δut-νΔu+u·∇u+∇p=∫-h0Gs,ut+sds,divut,x=0,t,x∈τ,+∞×Ω,ut,x=0,t,x∈τ,+∞×∂Ω,ut,x=ϕt-τ,x,t∈τ-h,τ,x∈Ω,where u=(u1,u2,u3) is the velocity field of the fluid, p is the pressure, ν>0 is the kinematic viscosity, α>0 is the length scale parameter of the elasticity of the fluid, the external force G and initial velocity field ϕ are defined in the interval of time [-h,0], where h is a fixed positive number and Ω is a bounded smooth domain of R3.

The NSV equation was introduced by Oskolkov [1] to give an approximate description of the Kelvin-Voigt fluid and was proposed as a regularization of 3D Navier-Stokes equation for the purpose of direct numerical simulations in [2]. Since the term -α2Δut changes the parabolic character of the equation, the NSV equation being well posed in 3D, many authors have studied the long time dynamics of this model. Kalantarov and Titi [3] investigated the existence of the global attractor, the estimation of the upper bounds for the number of determining modes, and the dimension of global attractor of the semigroup generated by the equations. By a useful decomposition method, Yue and Zhong [4] proved the asymptotic regularity of solution of NSV equation and obtained the existence of the uniform attractor; they also described the structure of the uniform attractor and its regularity. García-Luengo et al. [5] investigated the existence and relationship between minimal pullback attractor for the universe of fixed bounded sets and universe given by a tempered condition.

Partial differential equations with delays arise from various fields, like physics, control theory, and so on (see, e.g., [6–10]); the unknown functions depend on not only present stage but also some past stage. The existence and stability of solution and global attractor for Navier-Stokes equation with discrete delay were established in [11–13]. The existence of pullback attractors in CH01 and CH01∩H2 was proved for the processes associated with nonclassical diffusion equations with variable bounded delay in [14, 15]. Delay effect has been considered on an unbounded domain in [16]. The existence of pullback attractor for a Navier-Stokes equation with infinite discrete delay effect was studied in [17].

The aim of this paper is to investigate the NSV equation with a distributed delay, instead of the discussions with finite delays in the references. Our purpose is twofold. We first show the existence and uniqueness of solution to NSV equation (1) with a distributed delay; then we prove the existence of pullback attractor for the process generated by the NSV equation (1).

This paper is organized as follows. In Section 2, we give some preliminary results and prove existence of solution to NSV equation with a distributed delay. In Section 3, we derive the existence of pullback attractor by prior estimates and contractive functions.

2. Existence of Solutions

In order to prove the existence of solutions to problem (1), we define the function spaces (2)V=u∈C0∞Ω3,divu=0.H is the closure of V in (L2(Ω))3 with the inner product (·,·) and associate norm |·|, V is the closure of V in (H01(Ω))3 with scalar product ((·,·)) and associate norm ·, where (3)u,v=∑i,j=13∫Ω∂uj∂xi∂vj∂xidx,∀u,v∈H01Ω3;it follows that V⊂H≡H′⊂V′, where the injections are dense and compact. We will use ·∗ for the norm in V′ and ·,· for the duality pairing between V and V′.

Define the linear continuous operator A:V→V′ as(4)Au,v=u,v,∀u,v∈V.We denote D(A)={u∈V,Au∈H}; one has that D(A)=(H2(Ω))3∩V and Au=-PΔu, for all u∈D(A) is the Stokes operator, where P is the orthoprojector from (L2(Ω))3 onto H; also denote CH=C0([-h,0];H) and CV=C0([-h,0];V).

Define the trilinear form b on V×V×V by (5)bu,v,w=∑i,j=13∫Ωui∂vj∂xiwjdx,∀u,v,w∈Vand the operator B:V×V→V′ as (6)Bu,v,w=bu,v,w,∀u,v,w∈V,and denote B(u)=B(u,u).

The trilinear form b satisfies that(7)bu,v,w=-bu,w,v,bu,v,v=0,∀u,v,w∈V.We also recall that there exists a constant C depending only on Ω such that(8)bu,v,w≤Cu1/2Au1/2vw,u∈DA,v∈V,w∈H,bu,v,w≤Cuvw1/2w1/2,u∈V,v∈V,w∈V,bu,v,w≤Cu1/2u1/2vw,u∈V,v∈V,w∈V.

For the term containing the time delay, G:R×H→H satisfies that

(H1)G(·,u):R→H is a measurable function,

(H2)G(t,0)=0 for all t∈R,

(H3) there exists a positive constant L, such that ∀R>0; if |u|<R and |v|<R, then(9)Gt,u-Gt,v2≤Lu-v2.

Remark 1.

Hypotheses H2-(H3) imply that |G(t,u)|2≤L|u|2, so we have |G(t,u)|2∈L∞(τ,T) for |u|<R.

Problem (1) can be rewritten as(10)∂∂tu+α2Au+νAu+Bu,u=∫-h0Gs,ut+sds,ut,x=ϕt-τ,x,t∈τ-h,τ,x∈Ω;then we get the existence of solution to problem (10).

Theorem 2.

Let ϕ∈CV, let G:R×H→H satisfy the hypotheses (H1)–(H3), and let τ∈R. Then, ∀T>τ; there exists a unique weak solution to (10) such that(11)ut,x∈Cτ-h,T;V,∂ut,x∂t∈L2τ,T;V.Moreover, if ϕ∈CD(A), then problem (10) admits a strong solution.

Proof.

Consider the Galerkin approximations for problem (10):(12)ddtum+α2Aum+νAum+Bum,um=∫-h0Gs,umt+sds,where um=∑j=1mujmej, Aum=∑j=1mλjujmej, and ej and λj are the corresponding orthonormal eigenfunctions and eigenvalues of operator A, respectively; then, (13)um2=∑j=1mλjujm2,Aum2=∑j=1mλj2ujm2,um2=∑j=1mujm2.

We now derive a prior estimate for the Galerkin approximate solution. Multiplying (12) by ujm, summing from j=1 to m and using the fact (14)Bum,um,um=bum,um,um=0,we obtain that, for a.e. t>τ,(15)ddtum2+α2um2+2νum2=2∫-h0Gs,umt+sds,um≤∫-h0Gs,umt+sds2+um2.Integrating (15) from τ to t, we obtain that (16)umt2+α2umt2+2ν∫τtums2ds≤umτ2+α2umτ2+∫τt∫-h0Gs,umr+sds2dr+∫τtumr2dr.Remark 1 implies that(17)∫τt∫-h0Gs,umr+sds2dr≤∫-h0∫τtGs,umr+s2drds≤∫-h0∫τtLumr+s2drds≤L∫-h0∫τ+st+sumr2drds≤L∫-h0∫τ-hτumr2dr+∫τtumr2drds≤Lh2ϕCH2+Lh∫τtumr2dr.Then, ∀t∈(τ,T) and(18)umt2+α2umt2+2ν∫τtums2ds≤Lh+2∫τtums2ds+Lh2ϕCH2+ϕCH2+α2ϕCV2.So, we have (19)umt2≤C∫τtums2ds+CϕCH2+ϕCH2+α2ϕCV2.The Gronwall inequality implies that(20)umt2≤C.Putting (20) into the right-hand side of (18), we have (21)α2umt2+2ν∫τtums2ds≤C,∀t∈τ,T.This implies that(22)um is bounded in L∞τ,T;V∩L2τ,T;V.

Now, multiplying (12) by ∂tum and integrating over Ω, we have (23)∂tum2+α2∂tum2+ν2ddtum2≤bum,um,∂tum+∫-h0Gs,umt+sds,∂tum≤cum3/2um1/2∂tum+12∫-h0Gs,umt+sds2+12∂tum2;since(24)um3/2um1/2∂tum≤cum3um+α22∂tum2;then(25)∂tum2+α2∂tum2+νddtum2≤cum3um+∫-h0Gs,umt+sds2,integrating the above inequality from τ to t, by (17), (20), and (22) we have(26)∫τt∂tum2+α2∂tum2ds+νumt2≤νumτ2+c∫τtum3umds+∫τt∫-h0Gs,umr+sds2dr≤νϕτ2+c∫τtum3umds+cϕCH2+c∫τtumr2dr≤νϕCV2+c∫τtum3umds+cϕCH2+c∫τtumr2dr.Since {um} is bounded in L∞(τ,T;V)∩L2(τ,T;V), we obtain that(27)∂tum is bounded in L2τ,T;V.

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

We next consider the uniqueness of solution. Let u,v be two solutions to problem (10) corresponding the initial data ϕ and ψ, respectively.

Denote w=u-v; then, we have(28)∂∂tw+α2Aw+νAw+Bu,u-Bv,v=∫-h0Gs,ut+sds-∫-h0Gs,vt+sds.Multiplying (28) by w and integrating over Ω, we obtain(29)12ddtw2+α2w2+νw2+Bu,u,w-Bv,v,w=∫-h0Gs,ut+sds-∫-h0Gs,vt+sds,w.Notice that(30)Bu,u,w-Bv,v,w=bu,u,u-v-bv,u,u-v+bv,u,u-v-bv,v,u-v=bu-v,u,u-v-bv,u-v,u-v=bw,u,w.

Substituting (30) into (29) and integrating from τ to t, we get(31)wt2+α2wt2+2ν∫τtws2ds-wτ2-α2wτ2≤∫τtbw,u,wds+∫τtws2ds+∫τt∫-h0Gs,ur+sds-∫-h0Gs,vr+sds2dr.(H3) implies that(32)∫τt∫-h0Gs,ur+sds-∫-h0Gs,vr+sds2dr≤∫-h0∫τtGs,ur+s-Gs,vr+s2drds≤∫-h0∫τtGs,ur+s-Gs,vr+s2drds≤L∫-h0∫τtur+s-vr+s2drds≤L∫-h0∫τ+st+sur-vr2drds≤L∫-h0∫τ-hτur-vr2dr+∫τtur-vr2drds≤Lh2ϕ-ψCH2+Lh∫τtws2ds.As the property of operator b and Poincaré, we have(33)∫τtbw,u,wds≤C∫τtw3/2w1/2uds≤C∫τtw2uds.Substituting (32) and (33) into (31), we get (34)wt2+α2wt2+2ν∫τtws2ds≤∫τtw2uds+Lh2ϕ-ψCH2+Lh+1∫τtws2ds+wτ2+α2wτ2≤∫τtw2uds+Lh2ϕ-ψCH2+Lh+1∫τtws2ds+ϕ-ψCH2+α2ϕ-ψCV2≤C∫τtws2ds+Cϕ-ψCV2.The last inequality comes from Poincaré inequality and the boundedness of u. Therefore, the Gronwall inequality implies the uniqueness of the solution. The proof is complete.

3. Existence of Pullback Attractor

In this section, we will prove the existence of pullback attractor to problem (10). First we give existence of pullback absorbing set for the process {U(t,τ)} generated by the global solution to problem (10).

Lemma 3.

Assume (H1)–(H3) hold and L≤ν2λ12ɛ/4eɛh; then, the process {U(t,τ)} is pullback dissipative, where 0<ɛ<min{ν/α2,νλ1/4}.

Proof.

Multiplying (10) by u and integrating over Ω, we obtain (35)∂∂tu2+α2u2+2νu2≤ηu2+1η∫-h0Gs,ut+sds2,where η is a constant determined later.

By Poincaré inequality, we have (36)∂∂tu2+α2u2+νu2+νλ1-ηu2≤1η∫-h0Gs,ut+sds2.Since(37)e-ɛt∂∂teɛtu2+α2u2=ɛu2+α2u2+∂∂tu2+α2u2,then(38)e-ɛt∂∂teɛtu2+α2u2≤ɛ+η-νλ1u2+ɛα2-νu2+1η∫-h0Gs,ut+sds2.Integrating (38) from τ to t, we get(39)eɛtut2+α2ut2-eɛτuτ2+α2uτ2≤∫τteɛsɛ+η-νλ1u2+ɛα2-νu2ds+1η∫τteɛr∫-h0Gs,ur+sds2dr.Assumptions (H1)–(H3) imply that(40)∫τteɛr∫-h0Gs,ur+sds2dr≤∫τteɛr∫-h0Gs,ur+s2dsdr≤L∫-h0∫τteɛrur+s2drds≤L∫-h0∫τ+st+seɛr-sur2drds≤L∫-h0∫τ-hteɛr-sur2drds≤L∫-h0e-ɛs∫τ-hteɛrur2drds≤L∫-h0e-ɛs∫τ-hτeɛrur2dr+∫τteɛrur2drds≤Leɛhɛ1ɛϕCH2eɛτ+∫τteɛrur2dr≤Leɛτ+hɛ2ϕCH2+Leɛhɛ·∫τteɛrur2dr.Substituting (40) into (39), we have (41)eɛtut2+α2ut2-eɛτuτ2+α2uτ2≤∫τteɛsɛ+η-νλ1+Leɛh2ηɛu2+ɛα2-νu2ds+Leɛτ+h2ηɛ2ϕCH2.Let η=(1/2)νλ1, choosing 0<ɛ<min{ν/α2,νλ1/4}. L≤ν2λ12ɛ/4eɛh implies that (42)maxɛ+η-νλ1+η-1Leɛh2ɛ,ɛα2-ν<0;then,(43)eɛtut2+α2ut2≤Leɛτ+hɛ2νλ1ϕCH2+eɛτuτ2+α2uτ2,which implies (44)ut2+α2ut2≤Leɛτ+h-tɛ2νλ1ϕCH2+eɛτ-tϕτ2+α2ϕτ2.Now, if we take t≥τ+h, then, for θ∈[-h,0], we have (45)ut+θ2+α2ut+θ2≤Leɛτ+h-t-θɛ2νλ1ϕCH2+eɛτ-t-θϕCH2+α2ϕCV2.

We denote by R the set of all functions r:(-∞,+∞)→(0,+∞) such that (46)limτ→-∞eατr2τ=0.Then, the closed ball in CV defined by(47)B=φ∈CV:φCV2≤1is pullback absorbing set for {U(t,τ)}. The proof is complete.

We next prove the asymptotic compactness of solution to problem (10) by contractive functions; see [19, 20].

Let X be a Banach space and let B be a bounded subset of X. We call a function Φ(·,·) which, defined on X×X, is a contractive function on B×B if for any sequence {xn}n=1∞⊂B there is a subsequence {xnk}k=1∞⊂{xn}n=1∞ such that (48)liml→∞limk→∞Φxnk,xnl=0.Denote all such contractive functions on B×B by C(B).

Theorem 4 (see [<xref ref-type="bibr" rid="B9">19</xref>]).

Let {S(t)}t≥τ be a semigroup on a Banach space (X,·) and have a bounded absorbing set B, ∀ϵ>0; there exist T=T(B,ϵ) and Φ∈C(B), such that (49)STx-STy≤ϵ+ΦTx,y,∀x,y∈B,where ΦT depends on T. Then, {S(t)}t≥τ is asymptotically compact in X.

Lemma 5.

Assume that (H1)–(H3) hold; the process {U(t,τ)}t≥τ generated by the global solution to problem (10) is asymptotically compact.

Proof.

Let ui(t) be the solution to problem (10) with initial data ϕi∈B (i=1,2), respectively. Denote v(t)=u1(t)-u2(t); then, v(t) satisfies the equivalent abstract equation(50)∂∂tv+α2Av+νAv+Bu1,u1-Bu2,u2=∫-h0Gs,u1t+s-Gs,u2t+sdswith the initial condition v(t)=ϕ1(t-τ)-ϕ2(t-τ), t∈[τ-h,τ].

Set an energy function(51)Evt=12∫Ωv2dx+α22∫Ω∇v2dx.Multiplying (50) by v and integrating over [s,T]×Ω with T>t+τ, s≥τ, we have(52)EvT-Evs+ν∫sT∫Ω∇v2dxdr+∫sT∫ΩBu1-Bu2vrdxdr=∫sT∫Ω∫-h0Gσ,u1r+σ-Gσ,u2r+σvrdσdxdr;then we have(53)ν∫sT∫Ω∇v2dxdr≤Evs-∫sT∫ΩBu1-Bu2vrdxdr+∫sT∫Ω∫-h0Gσ,u1r+σ-Gσ,u2r+σvrdσdxdr.Using Poincaré inequality and (51) and (53), we have(54)∫τTEvsds=12∫τT∫Ωv2dxdr+α22∫τT∫Ω∇v2dxdr≤C∫τT∫Ω∇v2dxdr≤CEvτ-C∫τT∫ΩBu1-Bu2vrdxdr+C∫τT∫Ω∫-h0Gσ,u1r+σ-Gσ,u2r+σvrdσdxdr.Integrating (52) from τ to T with respect to s, we obtain(55)TEvT+ν∫τT∫sT∫Ω∇v2dxdrds=∫τTEvsds-∫τT∫sT∫ΩBu1-Bu2vrdxdrds+∫τT∫sT∫Ω∫-h0Gσ,u1r+σ-Gσ,u2r+σvrdσdxdrds.Substituting (54) into (55), we get(56)TEvT+ν∫τT∫sT∫Ω∇v2dxdrds≤CEvτ-C∫τT∫ΩBu1-Bu2vrdxdr+C∫τT∫Ω∫-h0Gσ,u1r+σ-Gσ,u2r+σvrdσdxdr-∫τT∫sT∫ΩBu1-Bu2vrdxdrds+∫τT∫sT∫Ω∫-h0Gσ,u1r+σ-Gσ,u2r+σvrdσdxdrds.Set(57)C0=CEvτ=C2∫Ωϕ10-ϕ202dx+Cα22∫Ω∇ϕ10-∇ϕ202dx,(58)Φu1,u2=-C∫τT∫ΩBu1-Bu2vrdxdr+C∫τT∫Ω∫-h0Gσ,u1r+σ-Gσ,u2r+σvrdσdxdr-∫τT∫sT∫ΩBu1-Bu2vrdxdrds+∫τT∫sT∫Ω∫-h0Gσ,u1r+σ-Gσ,u2r+σvrdσdxdrds;then, by (56), we have(59)EvT≤C0T+1TΦu1,u2.One has ∀ϵ>0; we choose T>0 large enough such that C0/T<ϵ. By Theorem 4, it suffices to prove that Φ(u1,u2) is a contractive function, since B is a bounded positive invariant set.

If um→u (m→∞), we have the limits(60)limn→∞limm→∞∫τT∫ΩBun-Bumun-umdxdr=limn→∞limm→∞∫τT∫Ωun-um·∇un-um·∇un-umun-umdxdr=0,limn→∞limm→∞∫τT∫sT∫ΩBun-Bumun-umdxdrds=limn→∞limm→∞∫τT∫sT∫Ωun-um·∇un-um·∇un-umun-umdxdrds=0.By (H1)–(H3), we have(61)limn→∞limm→∞∫τT∫Ω∫-h0Gσ,unr+σ-Gσ,umr+σvrdσdxdr≤limn→∞limm→∞∫τT∫Ω∫-h0Lunr+σ-umr+σunr-umrdσdxdr=0,limn→∞limm→∞∫τT∫sT∫Ω∫-h0Gσ,unr+σ-Gσ,umr+σvrdσdxdrds≤limn→∞limm→∞∫τT∫sT∫Ω∫-h0Lun-umr+σun-umrdσdxdrds=0.Combining (60)-(61) with (58), we have that Φ(u1,u2) is a contractive function.

The proof is complete.

Theorem 6.

Assume that (H1)–(H3) hold and L≤ν2λ12ɛ/4eɛh; then, the process {U(t,τ)} generated by the global solution for problem (10) has a pullback attractor.

Proof.

Lemma 3 implies {U(t,τ)} has a pullback absorbing set B and Lemma 5 implies {U(t,τ)} is asymptotically compact; we obtain the conclusion immediately.

According to Theorem 6, 0<ɛ<min{ν/α2,νλ1/4} and L≤ν2λ12ɛ/4eɛh. Under the assumptions (H1)–(H3), the NSV equation (1) with a distributed delay has a pullback attractor.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

Yantao Guo carried out the long time behavior of solutions. Shuilin Cheng carried out the pullback attractor. Yanbin Tang carried out the distributed delay. All authors read and approved the final paper.

Acknowledgment

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

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