On the Convergence of the Uniform Attractor for the 2D Leray-α Model
We consider a nonautonomous 2D Leray- model of fluid turbulence. We prove the existence of the uniform attractor . We also study the convergence of as goes to zero. More precisely, we prove that the uniform attractor converges to the uniform attractor of the 2D Navier-Stokes system as tends to zero.
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 , the author considers some special classes of nonautonomous dynamical systems and studies the existence and uniqueness of uniform attractors. In , 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 and is a parameter. Precise assumptions on the external force are given below. Formally, the above system is the 2D Navier-Stokes system when .
The 2D Leray- model has received much attention over the past years (see  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  as a large eddy simulation subgrid scale model of 3D turbulence. In , 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 , analogous results were proven for the 3D Navier-Stokes- model. In , 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 Leray- model has been studied analytically in  and computationally in . In , the authors investigated the rate of convergence of four alpha models (2D Navier-Stokes- model, 2D Leray- model, 2D modified Leray- model, and 2D simplified Bardina model) in the 2D case subject to periodic boundary conditions. In particular, they showed upper bounds in terms of for the difference between solutions of the 2D -models and solutions of the 2D Navier-Stokes system. They found that all the four -models have the same order of convergence and error estimates. We also note that the autonomous and nonautonomous 2D Navier-Stokes- models were considered in [6, 19]. In , they proved that the global attractors of the 2D Navier-Stokes- model converge to a subset of the global attractor of the 2D Navier-Stokes system when approaches 0. In , the authors studied the convergence of the uniform attractors of the 2D Navier-Stokes- model when tends to zero. They found that the uniform attractors of the 2D Navier-Stokes- model converge to the uniform attractor of the 2D Navier-Stokes system when approaches zero.
The purpose of this paper is to prove analogous results for the nonautonomous 2D Leray- model. More precisely, we prove that the uniform attractors for the 2D Leray- model converge to the uniform attractor of the 2D Navier-Stokes system when 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- 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- model. In Section 4, we prove the convergence of the uniform attractors of the 2D Leray- model to the uniform attractor of the 2D Navier-Stokes system as approaches zero.
2. The 2D Navier-Stokes System and Its Uniform Attractor
We consider the nonautonomous 2D Navier-Stokes system with periodic boundary conditions:In (2), is the unknown vector field in describing the motion of the fluid. The scalar function is the unknown pressure and is a given field of external force. Let be the set of trigonometric polynomials of two variables with periodic domain and spatial average zero; that is, for every , . We then setWe denote by and the closure of in and , respectively. The norms in and are denoted, respectively, by and .
We denote by the Helmholtz-Leray orthogonal projection operator and by the Stokes operator, subject to periodic boundary conditions, with domain . We note that in the space periodic caseThe operator is a self-adjoint positive definite compact operator from into . By , we denote the eigenvalues of in the 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 such thatBywe denote the scalar product and the norm in , respectively. Let be the dual space of . For every , we denote by the value of the functional from on a vector . The operator is an isomorphism from to . In particular for all .
The Poincaré inequalities readFor every , we define the bilinear operatorIn the following lemma, we list certain relevant inequalities and properties of (see, e.g., ).
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 , we haveand in particularWe apply the operator to both sides of (2) and obtain an equivalent system:The initial condition is posed at :
In order to clarify the assumptions on the external force , we introduce the following notation. Given a Banach space , we denote by the subspace of of translation bounded functions; that is, for , we haveWe now give from  the definition and some properties of translation compact functions.
Definition 2. A function is said to be translation compact in if the set of its translations is precompact in for the local convergence topology.
The setis called the hull of the function in the space , where denotes the closure in the space . Note that if is translation compact in , then its hull is compact in . The hull of in the space isThe following proposition gives the existence and uniqueness of weak solutions of problems (17)-(18) (see  for the proof).
Now, we are given a field external force that is translation compact function in . In particular, is translation bounded in .
We denote by the kernel of the process with the external force . Let us recall that is the family of all complete solutions , of (23) which are bounded in the norm of . The set is called the kernel section at .
The following result gives the existence and the structure of the uniform attractor of the process (see  for the proof).
Proposition 4. If is translation compact function in , then the process corresponding to (17) with external force has the uniform attractor that coincides with the uniform attractor of the family of processes andwhere is the kernel of the process . The kernel is nonempty for all .
3. The 2D Leray-α Model and Its Uniform Attractor
3.1. The 2D Leray-α 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 or and the scalar function . In (25), is a fixed positive parameter which is called the subgrid length scale of the model. For , the function and we obtain exactly the 2D Navier-Stokes system.
We can rewrite system (25) in an equivalent form using the standard projector in and excluding the pressure as in the previous section, where all the necessary notations were defined. We obtain the systemWe supplement system (26) with the initial dataIt follows from the embedding theorem in that . In particular, we have the energy inequality, where and is a constant that depends on . We obtain from inequality (28) thatwhere .
Consider an arbitrary function . Then, from (29), we conclude thatWe study weak solutions of system (25) belonging to the space . ThenWe now formulate the theorem on the existence and uniqueness of weak solutions of problems (26)-(27).
Lemma 6. Let a real function , be uniformly continuous and satisfy the inequalitywhere , for all , and . Suppose also thatThen .
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 . Let us prove the estimate in (32). We take the scalar product of (26) with and use relation (16); we obtainUsing Poincaré inequality (7), we arrive atwhere . Applying Lemma 6 withwe getthat is,This proves (32). Multiplying (26) by , we haveRecall thatFrom (29), we haveReplacing (43) and (44) in (42), we getLet us set and obtainUsing Gronwall’s lemma, we obtainFrom the estimate in (33), we deduce from (47) thatwhereThis 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 .
3.2. The Uniform Attractor of the 2D Leray-α Model
In this subsection, we prove the existence of the uniform attractor for the 2D Leray- model. We consider the process corresponding to problems (26)-(27). More precisely, the mapping is defined byfor all , where is solution of (26)-(27). It follows from (32) that the process has the uniform absorbing setwhere and the set is bounded in . Therefore, for any bounded (in ) set , there exists a time such thatfor all and .
Proposition 8. The process associated with (26)-(27) is uniformly compact in and has a uniformly absorbing set (bounded in ) defined bywhere is given by (51). Moreover, the process has a uniform attractor which satisfies
Proof. From (34) and (51), it is clear that is bounded in and hence is relatively compact in . From (34), it is also clear that is uniform (with respect to ) absorbing set for the process . The rest of the proof of the proposition follows the general theory on uniform global attractors . This ends the proof of the proposition.
From the general theory on uniform global attractors in , the global attractor given in Proposition 8 satisfies the following:(i)For any bounded (in ) set , as .(ii) is the minimal set that satisfies (i).
3.3. The Structure of the Uniform Attractor of the 2D Leray-α Model
We consider the systemWe assume that is translation compact in the space . Let be the hull of in . For all , the problemhas a unique solution and the estimates in (32)–(34) hold. For , system (56) generates a process that satisfies the same properties as the process . The family of processes , acting on corresponds to (56).
Proposition 9. The family of processes , corresponding to (56) is uniformly (with respect to ) bounded, uniformly compact, and -continuous.
Proof. The uniform boundedness of the family of processes , follows from (32) and the fact thatThis estimate also implies that the set , where , is uniformly (with respect to absorbing. The setis also uniformly absorbing. By (34), the set is bounded in and therefore, by the compactness of the embedding , is precompact in . Hence the family , is uniformly compact.
Let us verify the -continuity of the processes . We consider two symbols and and the corresponding solutions and of problem (56) with initial data and , respectively. DenoteThe function satisfies the equationWe take the inner product of (60) with ; we obtainUsing the estimate in (10), we arrive atAlso we haveUsing (62) and (63) in (61), we get Let us set and we obtainUsing Gronwall’s lemma, we obtainWith the estimate in (33), we getThe estimate in (67) proves that is bounded, and (66) implies the -continuity of the family of processes . This ends the proof of the proposition.
Theorem 10. If is translation compact in , then the process corresponding to (55) with external force has the uniform (with respect to ) attractor that coincides with the uniform (with respect to ) attractor of the family of processes .
Moreover,where is the kernel of the process . The kernel is nonempty for all .
In the next section, we study the asymptotic behavior of the uniform attractor of the 2D Leray- model.
4. 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:(a) of the process generated by the solutions of the 2D Leray- model.(b) of the process generated by the solutions of the 2D Navier-Stokes system. Our aim in this section is to prove the convergence of the uniform attractors to the uniform attractor as approaches ; that is, if .The following proposition is the key.
Proposition 11. Let , and a sequence of functions satisfy the following conditions:(1) as .(2) in as .(3) in as .Then is a weak solution of the 2D Navier-Stokes system with external force ; that is, .
Proof. Consider the operator , where . We note thatFrom inequalities (10) and (72), we getWe deduce thatwhere . Using the triangle inequality, it follows from (26) thatwhere . This proves (70).
For the proof of (71), we use inequalities (11) and (72) and we getWe then haveIt follows from (26) thatThis ends the proof of the proposition.
Proof of Proposition 11. We prove that is a weak solution of the 2D Navier-Stokes system on every interval . The function satisfies the equationFrom the estimates in (32)-(33) and (71), we haveSince each bounded sequence in a reflexive Banach space has a weakly convergent subsequence (see , Theorem 21.D, p. 255), we can choose a subsequence of such thatas . The convergence (82) uses the fact that the generalized derivatives are compatible with the weak limits (see , Proposition 23.19, p. 419). From (83), we obtainIn order to establish the equality, it is sufficient to prove that the sequence converges to in as . Notice thatIndeed, the function satisfies the equationSince is bounded in , then, passing to a subsequence, we may assume that converges to a function weakly in ; that is,Then the sequence weakly in andTherefore, in equality (86), we may pass to the limit in the space and obtain thatThen, (87) and (89) imply (85).
From (71), the sequences and are bounded in . Then the Aubin compactness theorem  implies that, passing to a subsequence, we may assume that and converge to strongly in . Therefore, we may assume that