We consider a nonautonomous 2D Leray-α model of fluid turbulence. We prove the existence of the uniform attractor Aα. We also study the convergence of Aα as α goes to zero. More precisely, we prove that the uniform attractor Aα converges to the uniform attractor of the 2D Navier-Stokes system as α tends to zero.

1. Introduction

In the past decades, the study of nonautonomous dynamical systems has been paid much attention as evidenced by the references cited in [1–8]. In [9], the author considers some special classes of nonautonomous dynamical systems and studies the existence and uniqueness of uniform attractors. In [10], the authors present a general approach that is well suited to construct the uniform attractor of some equations arising in mathematical physics (see also [11, 12]). In this approach, instead of considering a single process associated with the dynamical system, the authors consider a family of processes depending on a parameter (symbol) σ in some Banach space. The approach preserves the leading concept of invariance, which implies the structure of the uniform attractors.

In this article, we study the following nonautonomous 2D Leray-α model:(1)∂v∂t-νΔv+u·∇v+∇p=g0x,t,v=u-α2Δu,∇·u=0,∇·v=0,vτ=vτ,where u is the velocity vector field, p is the pressure, and ν is the viscosity coefficient. The spatial variable x belongs to the two-dimensional torus T2=0,2πL2 and α is a parameter. Precise assumptions on the external force g0 are given below. Formally, the above system is the 2D Navier-Stokes system when α=0.

The 2D Leray-α model has received much attention over the past years (see [13] and the references therein) because of its importance in the description of fluid motion and turbulence. The 3D version of (1), namely, the 3D Leray-α model, was considered in [14] as a large eddy simulation subgrid scale model of 3D turbulence. In [15], the authors studied the relations between the long-time dynamics of the 3D Leray-alpha model and the 3D Navier-Stokes system. They found that bounded sets of solutions of the 3D Leray-α model converge to the trajectory attractor of the 3D Navier-Stokes system as time tends to infinity and α approaches zero. In particular, they showed that the trajectory attractor of the 3D Leray-α model converges to the trajectory attractor of the 3D Navier-Stokes system. In [16], analogous results were proven for the 3D Navier-Stokes-α model. In [17], the authors studied the convergence of the solution of the 2D stochastic Leray-α model to the solution of the stochastic 2D Navier-Stokes equations as α approaches 0. In particular, they proved the convergence in probability with the rate of convergence at most O(α).

The 2D Leray-α model has been studied analytically in [18] and computationally in [13]. In [18], the authors investigated the rate of convergence of four alpha models (2D Navier-Stokes-α model, 2D Leray-α model, 2D modified Leray-α model, and 2D simplified Bardina model) in the 2D case subject to periodic boundary conditions. In particular, they showed upper bounds in terms of α for the difference between solutions of the 2D α-models and solutions of the 2D Navier-Stokes system. They found that all the four α-models have the same order of convergence and error estimates. We also note that the autonomous and nonautonomous 2D Navier-Stokes-α models were considered in [6, 19]. In [19], they proved that the global attractors of the 2D Navier-Stokes-α model converge to a subset of the global attractor of the 2D Navier-Stokes system when α approaches 0. In [6], the authors studied the convergence of the uniform attractors of the 2D Navier-Stokes-α model when α tends to zero. They found that the uniform attractors of the 2D Navier-Stokes-α model converge to the uniform attractor of the 2D Navier-Stokes system when α approaches zero.

The purpose of this paper is to prove analogous results for the nonautonomous 2D Leray-α model. More precisely, we prove that the uniform attractors for the 2D Leray-α model converge to the uniform attractor of the 2D Navier-Stokes system when α approaches zero (see Theorem 13). Uniform attractors are not invariant under the family of processes; this brings about some difficulties in proving upper semicontinuous property. The proof of the convergence of the uniform attractors of the 2D Leray-α model uses the structure of uniform attractors which says that each uniform attractor is a union of kernels.

The article is structured as follows. In Section 2, we recall some properties of the uniform attractor for the 2D Navier-Stokes equations. In Section 3, we prove the existence and the structure of the uniform attractor of the 2D Leray-α model. In Section 4, we prove the convergence of the uniform attractors of the 2D Leray-α model to the uniform attractor of the 2D Navier-Stokes system as α approaches zero.

2. The 2D Navier-Stokes System and Its Uniform Attractor

We consider the nonautonomous 2D Navier-Stokes system with periodic boundary conditions:(2)∂u∂t-νΔu+u·∇u+∇p=g0t,x,∇·u=0.In (2), u=u(x,t)=(u1(x,t),u2(x,t)) is the unknown vector field in T2 describing the motion of the fluid. The scalar function p(x,t) is the unknown pressure and g0(x,t) is a given field of external force. Let F be the set of trigonometric polynomials of two variables with periodic domain T2 and spatial average zero; that is, for every Φ∈F, ∫T2Φ(x)dx=0. We then set(3)V=Φ∈F2: ∇·Φ=0.We denote by H and V the closure of V in L2T22 and H1T22, respectively. The norms in H and V are denoted, respectively, by · and ·.

We denote by P:L2T22→H the Helmholtz-Leray orthogonal projection operator and by A=-PΔ the Stokes operator, subject to periodic boundary conditions, with domain D(A)=H2T22∩V. We note that in the space periodic case(4)A=-PΔ=-Δ.The operator A-1 is a self-adjoint positive definite compact operator from H into H. By 0<2π/L2=λ1≤λ2≤⋯, we denote the eigenvalues of A in the 2D case. It is well known that, in two dimensions, the eigenvalues of operator A satisfy Weyl’s type formula (see, e.g., [13, 15]); namely, there exists a constant c0>0 such that(5)jc0≤λjλ1≤c0jfor j=1,2,….By(6)u,v=A1/2u,A1/2v=∇u,∇v,u=A1/2uforu,v∈V,we denote the scalar product and the norm in V, respectively. Let V′ be the dual space of V. For every v∈V′, we denote by v,u the value of the functional v from V′ on a vector u∈V. The operator A is an isomorphism from V to V′. In particular ((w,u))=Aw,u for all w,u∈V.

The Poincaré inequalities read(7)u2≤λ1-1u2,∀u∈V,(8)uV′2≤λ1-1u2,∀u∈H.For every w1,w2∈V, we define the bilinear operator(9)Bw1,w2=Pw1·∇w2.In the following lemma, we list certain relevant inequalities and properties of B (see, e.g., [11]).

Lemma 1.

The bilinear operator B defined in (9) satisfies the following.

B can be extended as a continuous bilinear map B:V×V→V′. In particular, B satisfies the following inequalities:(10)Bu,v,wV′≤cu1/2u1/2vw1/2w1/2∀u,v,w∈V,(11)Bu,v,wV′≤cu1/2u1/2v1/2v1/2w∀u,v,w∈V,(12)Bu,v,w≤cu∞vw,∀u∈DA,v∈V,w∈H,(13)Bu,v,w≤cu∇vw,∀u∈H,v∈DA3/2,w∈H,(14)Bu,v,wDA′≤cuvw∞,∀u∈H,v∈V,w∈DA.Moreover, for every w1,w2,w3∈V, we have(15)Bw1,w2,w3V′=-Bw1,w3,w2V′,and in particular(16)Bw1,w2,w2V′=0.We apply the operator P to both sides of (2) and obtain an equivalent system:(17)∂u∂t+νAu+Bu,u=g0x,t.The initial condition is posed at t=τ,τ∈R:(18)uτ=uτ∈H.

In order to clarify the assumptions on the external force g0, we introduce the following notation. Given a Banach space X, we denote by Lb2(R;X) the subspace of Lloc2(R;X) of translation bounded functions; that is, for Ψ(s)∈Lb2(R;X), we have(19)ΨLb2R;X2=supt∈R∫tt+1ΨsX2ds<∞.We now give from [10] the definition and some properties of translation compact functions.

Definition 2.

A function Ψ∈Lloc2(R;X) is said to be translation compact in Lloc2(R;X) if the set of its translations {Ψ(t+h),h∈R} is precompact in Lloc2(R;X) for the local convergence topology.

The set(20)HΨ=Ψt+h,h∈RLloc2R;Xis called the hull of the function Ψ in the space Lloc2(R;X), where ·X denotes the closure in the space X. Note that if Ψ is translation compact in Lloc2(R;X), then its hull H(Ψ) is compact in Lloc2(R;X). The hull H(g) of g(x,t) in the space Lloc2(R;H) is(21)Hg=g·,t+h,h∈RLloc2R;H.The following proposition gives the existence and uniqueness of weak solutions of problems (17)-(18) (see [10] for the proof).

Proposition 3.

Let g0∈Lb2(R;H) and let uτ∈H. Problems (17)-(18) have unique solutions u∈C(Rτ;H)∩Lloc2(Rτ;V) and ∂u/∂t∈Lloc2(Rτ;V′), where Rτ=[τ,+∞). The following estimates hold:(22)ut2≤uτ2e-λt-τ+λ-11+λ-1g0Lb22,ut2+ν∫τtus2ds≤uτ+λ-1∫τtg0s2ds,where λ=νλ1.

From Proposition 3, we can define a process {Ug0(t,τ)}:Ug0(t,τ)uτ=u(t),t≥τ, where u(t) is a solution of (17)-(18).

Now, we are given a field external force g0 that is translation compact function in L2loc(R;H). In particular, g0 is translation bounded in Lloc2(R;H).

Let H(g0) be the hull of g0∈Lloc2(R;H). Consider the family of Cauchy problems(23)∂u∂t+νAu+Bu,u=gx,t,uτ=uτ,g∈Hg0.For all g∈H(g0), problem (23) has a unique solution u(t) and estimates in (22) hold. Thus the family of processes {Ug(t,τ)},g∈H(g0) acting on H corresponds to problem (23).

We denote by Kg the kernel of the process {Ugα(t,τ)} with the external force g∈H(g0). Let us recall that Kg is the family of all complete solutions u(t),t∈R, of (23) which are bounded in the norm of H. The set Kg(s)={u(s),u∈Kg}⊂H is called the kernel section at t=s.

The following result gives the existence and the structure of the uniform attractor of the process {Ug0(t,τ)} (see [10] for the proof).

Proposition 4.

If g0 is translation compact function in Lloc2(R;H), then the process {Ug0(t,τ)} corresponding to (17) with external force g0(x,s) has the uniform (withrespecttoτ∈R) attractor A0 that coincides with the uniform (w.r.tg∈H(g0)) attractor AH(g0) of the family of processes {Ug(t,τ)},g∈H(g0) and(24)A0=AHg0=⋃g∈Hg0Kg0,where Kg is the kernel of the process {Ug(t,τ)}. The kernel Kg is nonempty for all g∈H(g0).

3. The 2D Leray-<bold><italic>α</italic></bold> Model and Its Uniform Attractor3.1. The 2D Leray-<italic>α</italic> Model

We consider the following system with periodic boundary conditions:(25)∂v∂t-νΔv+u·∇v+∇p=g0x,t,x∈T2,v=u-α2Δu,∇·u=0,∇·v=0.This system is an approximation of the 2D Navier-Stokes system discussed in the previous section. The unknown functions are the vector fields v=v(x,t)=(v1,v2) or u=u(x,t)=(u1,u2) and the scalar function p=p(x,t). In (25), α is a fixed positive parameter which is called the subgrid length scale of the model. For α=0, the function v=u and we obtain exactly the 2D Navier-Stokes system.

We can rewrite system (25) in an equivalent form using the standard projector P in H and excluding the pressure as in the previous section, where all the necessary notations were defined. We obtain the system(26)∂v∂t+νAv+Bu,v=g0x,t,v=u+α2Au.We supplement system (26) with the initial data(27)vτ=vτ∈H.It follows from the embedding theorem in R2 that H2(T2)⊂L∞(T2). In particular, we have the energy inequality(28)uL∞T22≤cαu+α2Au≤cαv,∀u∈H2∩V, where v=u+α2Au and c(α) is a constant that depends on α. We obtain from inequality (28) that(29)Bu,v≤cuL∞T22v≤c1αvv,where v=u+α2Au.

Consider an arbitrary function v(·)∈Lloc2(Rτ;V)∩L∞(Rτ;H). Then, from (29), we conclude that(30)Bu·,v·∈Lloc2Rτ;H.We study weak solutions v(x,t) of system (25) belonging to the space Lloc2(Rτ;V)∩L∞(Rτ;H). Then(31)Av∈Lloc2Rτ;V′,∂tv∈Lloc2Rτ;V′.We now formulate the theorem on the existence and uniqueness of weak solutions of problems (26)-(27).

Theorem 5.

Let α>0, let g0∈Lb2(R;H), and let vτ∈H. Systems (26)-(27) have unique weak solutions v∈C(Rτ;H)∩Lloc2(Rτ;V) and ∂tv∈Lloc2(Rτ;V′). The following estimates hold:(32)ut2≤vt2≤vτ2e-λt-τ+λ-11+λ-1g0Lb2R;H2,(33)vt2+ν∫τtvs2ds≤vτ2+λ-1∫τtg0s2ds,(34)t-τvt2≤Ct-τ,vτ2,∫τtg0s2ds,where λ=νλ1 and C(z,R,R1) is a monotone continuous function of z=t-τ,R and R1.

To prove the estimates in (32)-(34), we will need the following lemma whose proof is given in [10].

Lemma 6.

Let a real function z(t),t≥0, be uniformly continuous and satisfy the inequality(35)dzdt+λzt≤ft,t≥0,where λ>0,f(t)≥0 for all t≥0, and f∈Lloc1(R+). Suppose also that(36)∫tt+1fsds≤M,∀t≥0.Then z(t)≤z(0)e-λt+M(1+λ-1),∀t≥0.

Proof of Theorem <xref ref-type="statement" rid="thm1">5</xref>.

The existence and uniqueness of weak solutions are quite analogous to the proof of the existence and uniqueness theorem for the 2D Navier-Stokes system [10]. Let us prove the estimate in (32). We take the scalar product of (26) with v and use relation (16); we obtain(37)12ddtvt2+νvt2=g0t,vt≤ν2vt2+12νg0tV′2≤ν2vt2+12νλ1g0t2.Using Poincaré inequality (7), we arrive at(38)ddtvt2+λvt2≤λ-1g0t2,where λ=νλ1. Applying Lemma 6 with(39)zt=vt+τ2;ft=λ-1g0t2;∫tt+1fsds≤λ-1∫tt+1g0s2ds≤λ-1g0Lb2R;H2=M,we get(40)vt+τ2≤vτ2e-λt+λ-11+λ-1g0Lb2R;H2;that is,(41)vt2≤vτ2e-λt-τ+λ-11+λ-1g0Lb2R;H2.This proves (32). Multiplying (26) by tAv, we have(42)12ddttvt2-12vt2+νtAvt2+tBu,v,Av=tg0t,Av.Recall that(43)g0t,Av≤ν4Avt2+1νg0t2.From (29), we have(44)Bu,v,Av≤Bu,vAv≤c1αvvAv≤ν4Avt2+c12ανv2v2.Replacing (43) and (44) in (42), we get(45)ddttvt2+νtAvt2≤vt2+2tνg0t2+2c12ανtvt2vt2.Let us set y(t)=tvt2 and obtain(46)dydt≤2c12ανvt2y+vt2+2tνg0t2.Using Gronwall’s lemma, we obtain(47)tvt2≤∫0tvs2+s2νg0s2dsexp∫0t2c12ανvs2ds.From the estimate in (33), we deduce from (47) that(48)tvt2≤1νv02+λ-1+2t∫0tg0s2dsexp2c12αν2v02+2c12αλ-1ν2∫0tg0s2ds≤Ct,v02,∫0tg0s2ds,where(49)Cz,R,R1=1νR+λ-1+2zR1exp2c12αν2R+2c12αλ-1ν2R1.This ends the proof of Theorem 5.

Remark 7.

We note that the estimates in (32) and (33) are independent of α. This fact plays the key role in the proof of the convergence of solutions of the 2D Leray-αmodel to the solution of the 2D Navier-Stokes system as α→0+.

3.2. The Uniform Attractor <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M238"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="script">A</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> of the 2D Leray-<italic>α</italic> Model

In this subsection, we prove the existence of the uniform attractor for the 2D Leray-α model. We consider the process {Ug0α(t,τ)},t≥τ,τ∈R corresponding to problems (26)-(27). More precisely, the mapping Ug0α(t,τ):H→H is defined by(50)Ug0αt,τvτ=vt,for all vτ∈H,t≥τ,τ∈R, where v is solution of (26)-(27). It follows from (32) that the process {Ug0α(t,τ)} has the uniform (w.r.t.τ∈R) absorbing set(51)B0=v∈H:v2≤2R02,where R02=λ-1(1+λ-1)g0Lb2(R;H)2 and the set B0 is bounded in H. Therefore, for any bounded (in H) set O, there exists a time t(O) such that(52)Ug0αt+τ,τO⊂B0,for all t>t(O) and τ∈R.

Proposition 8.

The process {Ug0α(t,τ)} associated with (26)-(27) is uniformly compact in H and has a uniformly absorbing set B1 (bounded in V) defined by(53)B1=⋃τ∈RUg0ατ+1,τB0,where B0 is given by (51). Moreover, the process {Ug0α(t,τ)} has a uniform attractor Aα which satisfies(54)Aα⊂B0∪B1.

Proof.

From (34) and (51), it is clear that B1 is bounded in V and hence is relatively compact in H. From (34), it is also clear that B1 is uniform (with respect to τ∈R) absorbing set for the process {Ug0α(t,τ)}. The rest of the proof of the proposition follows the general theory on uniform global attractors [10]. This ends the proof of the proposition.

From the general theory on uniform global attractors in [10], the global attractor Aα given in Proposition 8 satisfies the following:

For any bounded (in H) set O, supτ∈RdistHUg0α(t+τ,τ)O,Aα→0 as t→∞.

Aα is the minimal set that satisfies (i).

3.3. The Structure of the Uniform Attractor of the 2D Leray-<italic>α</italic> Model

We consider the system(55)∂v∂t+νAv+Bu,v=g0,vτ=vτ,v=u+α2Au.We assume that g0 is translation compact in the space Lloc2(R;H). Let H(g0) be the hull of g0 in Lloc2(R;H). For all g∈H(g0), the problem(56)∂v∂t+νAv+Bu,v=gt,x,v=u+α2Au,vτ=vτhas a unique solution v(t) and the estimates in (32)–(34) hold. For g∈H(g0), system (56) generates a process {Ugα(t,τ)} that satisfies the same properties as the process {Ug0α(t,τ)}. The family of processes {Ugα(t,τ)},g∈H(g), acting on H corresponds to (56).

Proposition 9.

The family of processes {Ugα(t,τ)},g∈H(g0), corresponding to (56) is uniformly (with respect to g∈H(g0)) bounded, uniformly compact, and H×H(g0),H-continuous.

Proof.

The uniform boundedness of the family of processes {Ugα(t,τ)},g∈H(g0), follows from (32) and the fact that(57)gLb2R;H2≤g0Lb2R;H2,∀g∈Hg0.This estimate also implies that the set B0={v∈H;v2≤2R02}, where R02=λ-1(1+λ-1)g0Lb2R;H2, is uniformly (with respect to g∈H(g0) absorbing. The set(58)B1=⋃g∈Hg0⋃τ∈RUgτ+1,τB0is also uniformly absorbing. By (34), the set B1 is bounded in V and therefore, by the compactness of the embedding V↪H, B1 is precompact in H. Hence the family {Ugα(t,τ)},g∈H(g0), is uniformly compact.

Let us verify the H×H(g0),H-continuity of the processes {Ugα(t,τ)},g∈H(g0). We consider two symbols g1 and g2 and the corresponding solutions v1 and v2 of problem (56) with initial data v1τ and v2τ, respectively. Denote(59)wt=v1t-v2t=Ug1t,τv1τ-Ug2t,τv2τ,q=g1-g2.The function w satisfies the equation(60)∂w∂t+νAw+Bu1,v1-Bu2,v2=q.We take the inner product of (60) with w; we obtain(61)12ddtw2+νw2+Bu1-u2,v2,w=q,w.Using the estimate in (10), we arrive at(62)Bu1-u2,v2,w≤cu1-u21/2u1-u21/2v2w1/2w1/2≤cw1/2w1/2w1/2w1/2v2≤cwwv2≤ν4w2+cw2v22.Also we have(63)q,w≤qw≤λ-1qw≤ν4w2+c1q2.Using (62) and (63) in (61), we get (64)ddtw2+νw2≤cw2v22+c1q2.Let us set y(t)=wt2 and we obtain(65)ddtyt≤cv22yt+c1q2.Using Gronwall’s lemma, we obtain(66)wt2≤wτ2+∫τtc1qs2dsexp∫τtcv2s2ds.With the estimate in (33), we get(67)∫τtv2s2ds≤1νv2τ2+λ-1∫τtg2s2ds.The estimate in (67) proves that ∫τtv2s2ds is bounded, and (66) implies the H×H(g0),H-continuity of the family of processes {Ugα(t,τ)},g∈H(g0). This ends the proof of the proposition.

Theorem 10.

If g0 is translation compact in L2loc(R;H), then the process {Ug0(t,τ)} corresponding to (55) with external force g0(x,t) has the uniform (with respect to τ∈R) attractor Aα that coincides with the uniform (with respect to g∈H(g0)) attractor AH(g0)α of the family of processes {Ugα(t,τ)},g∈H(g0).

Moreover,(68)Aα=AHg0α=⋃g∈Hg0Kgα0,where Kgα is the kernel of the process {Ugα(t,τ)}. The kernel Kgα is nonempty for all g∈H(g0).

In the next section, we study the asymptotic behavior of the uniform attractor of the 2D Leray-α model.

4. Convergence of the Uniform Attractors of the 2D Leray-<bold><italic>α</italic></bold> Model

In the previous sections, we have proven the existence and the structure of the uniform attractor:

Aα of the process {Ug0α(t,τ)} generated by the solutions of the 2D Leray-α model.

A0 of the process {Ug0(t,τ)} generated by the solutions of the 2D Navier-Stokes system.

Our aim in this section is to prove the convergence of the uniform attractors Aα to the uniform attractor A0 as α approaches 0; that is,(69)limn→∞distHAαn,A0=0,

if αn→0+.

The following proposition is the key.
Proposition 11.

Let {gn},g∈H(g0), and a sequence of functions vαn(t)∈Kgnαn(t) satisfy the following conditions:

αn→0+ as n→∞.

gn⇀g in Hg0 as n→∞.

vαn(t)⇀v(t) in H as n→∞.

Then v is a weak solution of the 2D Navier-Stokes system with external force g; that is, v∈Kg.

For the proof of this proposition, we need an estimate for the derivative ∂tv in which constants are independent of α similar to that proven for v in (32)-(33).

Proposition 12.

Let g0∈Lb2(R;H) and let vτ∈H. Then any solution v(t) of (26)-(27) satisfies the following inequalities:(70)∫τT∂tvsV∗4/3ds3/4≤cvτ2+R22,(71)∫τT∂tvsV∗2ds1/2≤cvτ2+R22,where c depends on λ1,ν. R2 depends on λ1,ν and g0Lb2R;H. The numbers c and R2 are independent of α.

Proof.

Consider the operator B(u(t),v(t)), where v=u+α2Au. We note that(72)u≤v,u≤v.From inequalities (10) and (72), we get(73)Bu,vV∗≤cu1/2u1/2v≤cv1/2v3/2.We deduce that(74)∫τTBus,vsV∗4/3ds3/4≤c∫τTvs2/3vs2ds3/4≤cesssups∈τ,Tvs1/2∫τTvs2ds3/4≤cvτ2e-λT+λ-11+λ-1g0Lb2R;H21/41νvτ2+λ-1ν∫τTg0s2ds3/4≤cvτ2e-λT+λ-11+λ-1g0Lb2R;H21/41νvτ2+λ-1νT+1g0Lb2R;H23/4≤cvτ2+λ-11+λ-1g0Lb2R;H2+λ-1T+1g0Lb2R;H2≤cvτ2+R2′2,where R2′2=cλ-1(1+λ-1)g0Lb2R;H2+λ-1T+1g0Lb2(R;H)2. Using the triangle inequality, it follows from (26) that(75)∫τT∂tvsV∗4/3ds3/4≤ν∫τTAvsV∗4/3ds3/4+∫τTBus,vsV∗4/3ds3/4+∫τTg0sV∗4/3ds3/4≤ν∫τTvs4/3ds3/4+∫τTBus,vsV∗4/3ds3/4+λ-1/2∫τTg0s4/3ds3/4≤ν∫τTvs2ds1/2+∫τTBus,vsV∗4/3ds3/4+λ-1/2∫τTg0s2ds1/2≤ν1νvτ2+λ-1ν∫τTg0s2ds1/2+cvτ2+R2′2+T+1λ-frac12g0Lb2R;H≤cvτ2+λ-1T+1g0Lb2R;H2+R2′2+T+1λ-1/2g0Lb2R;H+1≤cvτ2+R22,where R22=λ-1(T+1)g0Lb2(R;H)2+R2′2+(T+1)λ-1/2g0Lb2(R;H)+1. This proves (70).

For the proof of (71), we use inequalities (11) and (72) and we get(76)Bu,vV∗≤cu1/2u1/2v1/2v1/2≤v1/2v1/2v1/2v1/2≤cvv.We then have(77)∫τTBus,vsV∗2ds1/2≤c∫τTvs2vs2ds1/2≤cesssups∈τ,Tvs∫τTvs2ds1/2≤cvτ2e-λT+λ-11+λ-1g0L2bR;H21/21νvτ2+λ-1ν∫τTg0s2ds1/2≤cvτ2e-λT+λ-11+λ-1g0L2bR;H21/21νvτ2+λ-1νT+1g0Lb2R;H21/2≤cvτ2+λ-11+λ-1g0Lb2R;H2+λ-1T+1g0Lb2R;H2≤cvτ2+R2′2.It follows from (26) that(78)∫τT∂tvsV∗2ds1/2≤ν∫τTAvsV∗2ds1/2+∫τTBus,vsV∗2ds1/2+∫τTg0sV∗2ds1/2≤ν∫τTvs2ds1/2+∫τTBus,vsV∗2ds1/2+λ-1/2∫τTg0s2ds1/2≤ν∫τTvs2ds1/2+∫τTBus,vsV∗2ds1/2+λ-1/2∫τTg0s2ds1/2≤ν1νvτ2+λ-1ν∫τTg0s2ds1/2+cvτ2+R2′2+T+1λ-1/2g0Lb2R;H≤cvτ2+λ-1T+1g0Lb2R;H2+R2′2+T+1λ-1/2g0Lb2R;H+1≤cvτ2+R22.This ends the proof of the proposition.

Proof of Proposition <xref ref-type="statement" rid="prop5">11</xref>.

We prove that v is a weak solution of the 2D Navier-Stokes system on every interval (τ,T). The function vαn satisfies the equation(79)∂tvαn+νAvαn+Buαn,vαn=gn.From the estimates in (32)-(33) and (71), we have(80)vαnt2≤vτ2e-λt-τ+λ-11+λ-1gnLb2R;H2,ν∫τtvαns2ds≤vτ2+λ-1∫τtgns2ds,∫τT∂tvαnsV∗2ds1/2≤cvτ2+2λ-1T+1gnLb2R;H2+cλ-11+λ-1gnLb2R;H2+T+1λ-1/2gnLb2R;H+1.Since each bounded sequence in a reflexive Banach space has a weakly convergent subsequence (see [20], Theorem 21.D, p. 255), we can choose a subsequence {vαn(t)} of {vαn(t)} such that(81)vαnt⇀vtinH,(82)∂vαn∂t⇀v′tinL2τ,T;V′,(83)vαn⇀vinL2τ,T;V,as n→∞. The convergence (82) uses the fact that the generalized derivatives are compatible with the weak limits (see [20], Proposition 23.19, p. 419). From (83), we obtain(84)Avαn⇀AvinL2τ,T;V′.In order to establish the equality, it is sufficient to prove that the sequence B(uαn,vαn) converges to B(v(·),v(·)) in D(τ,T;V′) as n→∞. Notice that(85)uαn⇀vweakly in L2τ,T;V.Indeed, the function uαn satisfies the equation(86)uαn+αn2Auαn=vαn.Since uαn is bounded in L2(τ,T;V), then, passing to a subsequence, we may assume that uαn converges to a function w(·) weakly in L2(τ,T;V); that is,(87)uαn⇀win L2τ,T;V.Then the sequence Auαn⇀Aw weakly in L2(τ,T;V′) and(88)αnAuαn⇀0weakly in L2τ,T:V′.Therefore, in equality (86), we may pass to the limit in the space L2(τ,T:V′) and obtain that(89)w=limn→∞uαn=limn→∞vαn=v.Then, (87) and (89) imply (85).

From (71), the sequences ∂tvn and ∂tun are bounded in L2(τ,T;V′). Then the Aubin compactness theorem [21] implies that, passing to a subsequence, we may assume that vαn and uαn converge to v(·) strongly in L2(τ,T;H). Therefore, we may assume that(90)vαnx,t⟶vx,tfora.e.x,t∈T2×τ,T,uαnx,t⟶vx,tfora.e.x,t∈T2×τ,T.We recall that(91)Buαn,vαn=P∑i=12∂iuαnivαn.It follows from (90) that(92)uαnix,tvαnx,t⟶vix,tvx,tfora.e.x,t∈T2×τ,T.Using the estimate in (11), we deduce that(93)uαnivαn is bounded in L2τ,T;H,L2T2×τ,T2.Applying the known lemma on weak convergence from [21], we conclude from (92) and (93) that(94)uαnivαn⇀vivweakly in L2T2×τ,T2 and weakly in L2(τ,T;H). We then deduce from (91) that(95)Buαn,vαn⇀Bv,vweakly in L2τ,T;V′.We have then proven that v(·) is a weak solution of the 2D Navier-Stokes equations with external force g. This completes the proof of the proposition.

Now we present and prove the main result of this paper.

Theorem 13.

Let Aαn be the uniform attractor of the 2D Leray-α model and let A0 be the uniform attractor of the 2D Navier-Stokes system. Then one has(96)AαnconvergestoA0asnapproaches∞;that is,(97)limn→∞distHAαn,A0=0.

Remark 14.

In (97), distH denotes the Hausdorff semidistance defined by(98)distHX,Y=supx∈Xinfy∈Yx-y.

Proof of Theorem <xref ref-type="statement" rid="thm3">13</xref>.

Assume that distHAαn,A0↛0. Hence, by the compactness of A0, we can choose a positive constant δ>0 and a subsequence {m} of {n} and ψm∈Aαm satisfying(99)distHψm,A0≥δ,∀m≥1.We recall that(100)Aαm=⋃g∈Hg0Kgαm0.Therefore, since ψm∈Aαm, there exist σm∈H(g0) and vm∈Kσmαm such that ψm=vm(0).

Since (t↦vm(t+h))∈Kσm(·+h)αm∀h∈R, it follows that vm(t)∈Aαm⊂B0∀t∈R. Since B0 is an absorbing set for the process Uσmαm(t,τ) (see (51)), we have(101)vmt2≤2R02,where R0 is independent of m and α(σmLb2(R;H)2≤g0Lb2(R;H)2). Also, since H(g0) is compact in Lloc2(R;H) and {σm}⊂H(g0), there exists a subsequence of vm and g∈H(g0) such that(102)σm⇀gin Hg0.Using the fact that each bounded sequence in a reflexive Banach space has a weakly convergent subsequence (see [20], Theorem 21.D, p. 255) and the boundedness (101), we deduce that(103)vmt converges weakly in H.Then, using the standard Cantor diagonal procedure as in [8, 15, 16], we can deduce a function ϕ(s),s∈R, and a sequence {mj} such that(104)vmjt⇀ϕtweakly in H as j⟶∞.From Proposition 11, we have that ϕ is a weak solution of the 2D Navier-Stokes equations. For t=0, we have(105)ψmj⇀ϕ0in H.Using the fact that Aαm⊂B1, where B1 is given by (53) (B1 is uniformly absorbing set), we have(106)ψmj⟶ϕ0in H,since ψmj is bounded in V. Also, since A0=⋃g∈H(g0)Kg(0), we get ϕ(0)∈Kg(0)⊂A0. Passing to the limit in (99), we obtain δ=0; and this contradicts the fact that δ>0. This ends the proof of the theorem.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

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