On the Convergence of the Uniform Attractor for the 2 D Leray-α Model

and Applied Analysis 3 |(B (u, V) , w)| ≤ c |u| ‖∇V‖ |w| , ∀u ∈ H, V ∈ D (A3/2) , w ∈ H, (13) 󵄨󵄨󵄨󵄨⟨B (u, V) , w⟩D(A)󸀠 󵄨󵄨󵄨󵄨 ≤ c |u| ‖V‖ ‖w‖∞ , ∀u ∈ H, V ∈ V, w ∈ D (A) . (14) Moreover, for every w1, w2, w3 ∈ V, we have ⟨B (w1, w2) , w3⟩V󸀠 = − ⟨B (w1, w3) , w2⟩V󸀠 , (15) and in particular ⟨B (w1, w2) , w2⟩V󸀠 = 0. (16) We apply the operator P to both sides of (2) and obtain an equivalent system: ∂u ∂t + ]Au + B (u, u) = g0 (x, t) . (17) The initial condition is posed at t = τ, τ ∈ R: u (τ) = uτ ∈ H. (18) In order to clarify the assumptions on the external force g0, we introduce the following notation. Given a Banach spaceX, we denote by L2b(R; X) the subspace of L2loc(R; X) of translation bounded functions; that is, for Ψ(s) ∈ L2b(R; X), we have ‖Ψ‖2L2 b (R;X) = sup t∈R ∫t+1 t ‖Ψ (s)‖2X ds < ∞. (19) We now give from [10] the definition and some properties of translation compact functions. Definition 2. A function Ψ ∈ L2loc(R; X) is said to be translation compact in L2loc(R; X) if the set of its translations {Ψ(t + h), h ∈ R} is precompact in L2loc(R; X) for the local convergence topology.


Introduction
In the past decades, the study of nonautonomous dynamical systems has been paid much attention as evidenced by the references cited in [1][2][3][4][5][6][7][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)  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- model: where  is the velocity vector field,  is the pressure, and ] is the viscosity coefficient.The spatial variable  belongs to the two-dimensional torus T 2 = [0, 2] 2 and  is a parameter.Precise assumptions on the external force  0 are given below.Formally, the above system is the 2D Navier-Stokes system when  = 0.The 2D Leray- 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- 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- model converge to the trajectory attractor of the 3D Navier-Stokes system as time tends to infinity and  approaches zero.In particular, they showed that the trajectory attractor of the 3D Leray- model converges to the trajectory attractor of the 3D Navier-Stokes system.In [16], analogous results were proven for the 3D Navier-Stokes- model.In [17], the authors studied the convergence of the solution of the 2D stochastic Leray- model to the solution of the stochastic 2D Navier-Stokes equations as  approaches 0. In particular, they proved the convergence in probability with the rate of convergence at most ().

The 2D Navier-Stokes System and Its Uniform Attractor
We consider the nonautonomous 2D Navier-Stokes system with periodic boundary conditions: In (2),  = (,) = ( 1 (, ),  2 (, )) is the unknown vector field in T 2 describing the motion of the fluid.The scalar function (, ) is the unknown pressure and  0 (, ) is a given field of external force.Let F be the set of trigonometric polynomials of two variables with periodic domain T 2 and spatial average zero; that is, for every Φ ∈ F, ∫ T 2 Φ() = 0.
We then set We denote by  and  the closure of V in  2 (T 2 ) 2 and  1 (T 2 ) 2 , respectively.The norms in  and  are denoted, respectively, by | ⋅ | and ‖ ⋅ ‖.We denote by P :  2 (T 2 ) 2 →  the Helmholtz-Leray orthogonal projection operator and by  = −PΔ the Stokes operator, subject to periodic boundary conditions, with domain () =  2 (T 2 ) 2 ∩ .We note that in the space periodic case The operator  −1 is a self-adjoint positive definite compact operator from  into .By 0 < (2/) 2 =  1 ≤  2 ≤ ⋅ ⋅ ⋅ , we denote the eigenvalues of  in the 2 case.It is well known that, in two dimensions, the eigenvalues of operator  satisfy Weyl's type formula (see, e.g., [13,15]); namely, there exists a constant  0 > 0 such that By we denote the scalar product and the norm in , respectively.Let   be the dual space of .For every V ∈   , we denote by ⟨V, ⟩ the value of the functional V from   on a vector  ∈ .
The Poincaré inequalities read For every  1 ,  2 ∈ V, we define the bilinear operator In the following lemma, we list certain relevant inequalities and properties of  (see, e.g., [11]).
Lemma 1.The bilinear operator B defined in (9) satisfies the following.
can be extended as a continuous bilinear map  :  ×  →   .In particular,  satisfies the following inequalities: Moreover, for every  1 ,  2 ,  3 ∈ , we have and in particular We apply the operator P to both sides of ( 2) and obtain an equivalent system: The initial condition is posed at  = ,  ∈ R: In order to clarify the assumptions on the external force  0 , we introduce the following notation.Given a Banach space , we denote by  2  (R; ) the subspace of  2 loc (R; ) of translation bounded functions; that is, for Ψ() ∈  2  (R; ), we have We now give from [10] the definition and some properties of translation compact functions.
Definition 2. A function Ψ ∈  2 loc (R; ) is said to be translation compact in  2 loc (R; ) if the set of its translations {Ψ( + ℎ), ℎ ∈ R} is precompact in  2 loc (R; ) for the local convergence topology.
The set is called the hull of the function Ψ in the space  2 loc (R; ), where [⋅]  denotes the closure in the space .Note that if Ψ is translation compact in  2 loc (R; ), then its hull H(Ψ) is compact in  2 loc (R; ).The hull H() of (, ) in the space  2 loc (R; ) is The following proposition gives the existence and uniqueness of weak solutions of problems ( 17)-( 18) (see [10] for the proof).
The following result gives the existence and the structure of the uniform attractor of the process {  0 (, )} (see [10] for the proof).

The 2D Leray-𝛼 Model and Its Uniform Attractor
3.1.The 2D Leray- Model.We consider the following system with periodic boundary conditions: This system is an approximation of the 2D Navier-Stokes system discussed in the previous section.The unknown functions are the vector fields ) and the scalar function  = (, ).In (25),  is a fixed positive parameter which is called the subgrid length scale of the model.For  = 0, the function V =  and we obtain exactly the 2D Navier-Stokes system.We can rewrite system (25) in an equivalent form using the standard projector P in  and excluding the pressure as in the previous section, where all the necessary notations were defined.We obtain the system We supplement system (26) with the initial data It follows from the embedding theorem in R 2 that  2 (T 2 ) ⊂  ∞ (T 2 ).In particular, we have the energy inequality ∀ ∈  2 ∩ , where V =  +  2  and () is a constant that depends on .We obtain from inequality (28) that where We study weak solutions V(, ) of system (25) belonging to the space We now formulate the theorem on the existence and uniqueness of weak solutions of problems ( 26)-( 27).
Proof of Theorem 5.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 Using Poincaré inequality (7), we arrive at where  = ] 1 .Applying Lemma 6 with we get that is, This proves (32).Multiplying (26) by V, we have Recall that From (29), we have Replacing ( 43) and ( 44) in (42), we get Let us set () = ‖V()‖ 2 and obtain Using Gronwall's lemma, we obtain From the estimate in (33), we deduce from (47) that where This ends the proof of Theorem 5.
Remark 7. We note that the estimates in (32) and (33) are independent of .This fact plays the key role in the proof of the convergence of solutions of the 2D Leray-model to the solution of the 2D Navier-Stokes system as  → 0 + .

The Uniform Attractor
A  of the 2D Leray- Model.In this subsection, we prove the existence of the uniform attractor for the 2D Leray- model.We consider the process {U   0 (, )},  ≥ ,  ∈ R corresponding to problems (26)-(27).More precisely, the mapping U   0 (, ) :  →  is defined by for all V  ∈ ,  ≥ ,  ∈ R, where V is solution of (26)-(27).It follows from (32) that the process {U   0 (, )} has the uniform (w.r.t. ∈ R) absorbing set where (R;) and the set  0 is bounded in .Therefore, for any bounded (in ) set O, there exists a time (O) such that for all  > (O) and  ∈ R.
Proposition 8.The process {U   0 (, )} associated with ( 26)-( 27) is uniformly compact in  and has a uniformly absorbing set  1 (bounded in ) defined by where  0 is given by ( 51).Moreover, the process {U   0 (, )} has a uniform attractor A  which satisfies Proof.From (34) and (51), it is clear that  1 is bounded in  and hence is relatively compact in .From (34), it is also clear that  1 is uniform (with respect to  ∈ R) absorbing set for the process {U   0 (, )}.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 A  given in Proposition 8 satisfies the following: (ii) A  is the minimal set that satisfies (i).

The Structure of the Uniform Attractor of the 2D Leray-𝛼
Model.We consider the system We assume that  0 is translation compact in the space  2 loc (R; ).Let H( 0 ) be the hull of  0 in  2 loc (R; ).For all  ∈ H( 0 ), the problem has a unique solution V() and the estimates in (32)-(34) hold.
Proof.The uniform boundedness of the family of processes {U   (, )},  ∈ H( 0 ), follows from (32) and the fact that This estimate also implies that the set (R;) , is uniformly (with respect to  ∈ H( 0 ) absorbing.The set is also uniformly absorbing.By (34), the set  1 is bounded in  and therefore, by the compactness of the embedding  → ,  1 is precompact in .Hence the family {U   (, )},  ∈ H( 0 ), is uniformly compact.
Let us verify the ( × H( 0 ), )-continuity of the processes {U   (, )},  ∈ H( 0 ).We consider two symbols  1 and  2 and the corresponding solutions V 1 and V 2 of problem (56) with initial data V 1 and V 2 , respectively.Denote The function  satisfies the equation We take the inner product of (60) with ; we obtain Using the estimate in (10), we arrive at Also we have Using ( 62) and ( 63) in (61), we get Let us set () = |()| 2 and we obtain Using Gronwall's lemma, we obtain With the estimate in (33), we get The estimate in (67) proves that ∫   ‖V 2 ()‖ 2  is bounded, and (66) implies the ( × H( 0 ), )-continuity of the family of processes {U   (, )},  ∈ H( 0 ).This ends the proof of the proposition.
Theorem 10.If  0 is translation compact in   2 (R; ), then the process {U  0 (, )} corresponding to (55) with external force  0 (, ) has the uniform (with respect to  ∈ R) attractor A  that coincides with the uniform (with respect to  ∈ H( 0 )) attractor A  H( 0 ) of the family of processes {U   (, )},  ∈ H( 0 ). Moreover, where K   is the kernel of the process {U   (, )}.The kernel K   is nonempty for all  ∈ H( 0 ).
In the next section, we study the asymptotic behavior of the uniform attractor of the 2D Leray- model.

Convergence of the Uniform Attractors of the 2D Leray-𝛼 Model
In the previous sections, we have proven the existence and the structure of the uniform attractor: Our aim in this section is to prove the convergence of the uniform attractors A  to the uniform attractor A 0 as  approaches 0; that is, lim if The following proposition is the key.
(2)   ⇀  in H( 0 ) as  → ∞. ( Then V is a weak solution of the 2D Navier-Stokes system with external force ; that is, V ∈ K  .
For the proof of this proposition, we need an estimate for the derivative   V in which constants are independent of  similar to that proven for V in (32)-(33).Proposition 12. Let  0 ∈  2  (R; ) and let V  ∈ .Then any solution V() of ( 26 From inequalities (10) and (72), we get We deduce that where (78) This ends the proof of the proposition.
(a) A  of the process {U   0 (, )} generated by the solutions of the 2D Leray- model.(b) A 0 of the process {U  0 (, )} generated by the solutions of the 2D Navier-Stokes system.