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Pullback attractors for three-dimensional Navier-Stokes-Voigt equations with delays
Boundary Value Problems volume 2013, Article number: 191 (2013)
Our aim in this paper is to study the existence of pullback attractors for the 3D Navier-Stokes-Voigt equations with delays. The forcing term containing the delay is sub-linear and continuous with respect to u. Since the solution of the model is not unique, which is caused by the continuity assumption, we establish the existence of pullback attractors for our problem by using the theory of multi-valued dynamical system.
Let be an open, bounded and connected set. We consider the following problem for three-dimensional Navier-Stokes-Voigt (NSV) equations with delays in continuous and sub-linear operators:
Here is the velocity vector field, ν is a positive constant, α is a characterizing parameter of the elasticity of the fluid, p is the pressure, g is the external force term which contains memory effects during a fixed interval of time of length , is an adequate given delay function, is the initial velocity field at the initial time , φ is the initial datum on the interval .
Equation (1.1) with becomes the classical three-dimensional Navier-Stokes (NS) equation. In the past decades, many authors [1–6] investigated intensively the classical three-dimensional incompressible NS equation. For the sake of direct numerical simulations for NS equations, the NSV model of viscoelastic incompressible fluid has been proposed as a regularization of NS equations.
Equation (1.1) governs the motion of a Klein-Voigt viscoelastic incompressible fluid. Oskolkov  was the first to introduce the system which gives an approximate description of the Kelvin-Voigt fluid (see, e.g., [8, 9]). In 2010, Levant et al.  investigated numerically the statistical properties of the Navier-Stokes-Voigt model. Kalantarov and Titi  studied a global attractor of a semigroup generated by equation (1.1) for the autonomous case. Recently, Luengo et al.  obtained asymptotic compactness by using the energy method, and they further got the existence of pullback attractors for three-dimensional non-autonomous NSV equations.
Let us recall some related results in the literature. Yue and Zhong  studied the long time behavior of the three-dimensional NSV model of viscoelastic incompressible fluid for system (1.1) by using a useful decomposition method. The authors in [12, 14] deduced the existence of -pullback attractors for 3D non-autonomous NSV equations using the energy method. As we know from , delay terms appear naturally. In recent years, Caraballo and Real [16–18] developed a fruitful theory of existence, uniqueness, stability of solutions and global attractors for Navier-Stokes models including some hereditary characteristics such as constant, variable delay, distributed delay, etc. However, our present problem has no uniqueness of solutions. To overcome the difficulty, we may cite the results by Ball  and by Marín-Rubio and Real . Gal and Medjo  proved the existence of uniform global attractors for a Navier-Stokes-Voigt model with memory. As commented before, in comparison with three-dimensional Navier-Stokes equations, there is no regularizing effect. Our result of this paper is to establish the existence of pullback attractors for three-dimensional NSV equations in when the external forcing term and the function is continuous with respect to u. Another difficulty is to obtain that the multi-valued processes are asymptotically compact. In 2007, Kapustyan and Valero  presented a method suitable for verifying the asymptotic compactness. The authors  applied this method to 2D Navier-Stokes equations with delays in continuous and sub-linear operators. We shall apply the energy method to prove that our multi-valued processes are asymptotically compact by making some minor modifications caused by the term in (1.1).
This paper is organized as follows. In Section 2, we recall briefly some results on the abstract theory of pullback attractors. In Section 3, we introduce some abstract spaces necessary for the variational statement of the problem and give the proof of the global existence of solutions. In Section 4, we consider the asymptotic behavior of problem (1.1).
2 Basic theory of pullback attractors
By using the framework of evolution processes, thanks to [20, 23, 24], we now briefly recall some theories of pullback attractors and the related results. On the one hand, we have to overcome some difficulties caused by multi-valued processes. On the other hand, since our model is non-autonomous, we should use the related results for classical multi-valued processes in [23, 24], but which are not completely adapted to our model.
Let be a metric space, and let be the class of nonempty subsets of X. As usual, we denote by the Hausdorff semi-distance in X between and , i.e.,
where denotes the distance between two points x and y in X.
We now formulate an abstract result in order to establish the existence of pullback attractors for the multi-valued dynamical system associated with (1.1).
Definition 2.1 A multi-valued process U is a family of mappings for any pair of real numbers such that
If the above relation is not only an inclusion but also an equality, we say that the multi-valued process is strict. For example, the relation generalized by 3D non-autonomous NSV equation (see, e.g., ) is an equality, while the relation generalized by 3D NS equations (see, e.g., ) is strict.
Definition 2.2 Suppose that is a family of sets. A multi-valued process U is said to be -asymptotically compact if for any , any sequences with , , and , the sequence is relatively compact in X.
Lemma 2.1 If a multi-valued process U is -asymptotically compact, then the sets are nonempty compact subsets of X, where
Furthermore, attracts in a pullback sense to at time t, i.e.,
Indeed, it is the minimal closed set with this property.
Definition 2.3 The family of subsets is said to be pullback-absorbing with respect to a multi-valued process U if for every and all bounded subset B of X, there exists a time such that
Lemma 2.2 Let the family of sets be pullback-absorbing for the multi-valued process U, and let U be -asymptotically compact. Then, for any bounded sets B of X, it holds that
Definition 2.4 Suppose that U is a multi-valued process. A family is said to be a pullback attractor for a multi-valued process U if the set is compact for any and attracts at time t to any bounded set in a pullback sense, i.e.,
We can see obviously that a pullback attractor does not need to be unique. However, it can be considered unique in the sense of minimal, that is, the minimal closed family with such a property. In this sense, we obtain the following property and the existence of pullback attractors.
Lemma 2.3 
Assume that U is a multi-valued process, and U is -asymptotically compact and a family is pullback-absorbing for U. Then, for any and any bounded subset B of X, the set
is a nonempty compact subset contained in , which attracts to B in a pullback sense. In fact, defined above is the minimal closed set with this property.
Furthermore, for any bounded set B, the set is a pullback attractor. From Definitions 2.3 and 2.4, it is easy to see that .
If there exists a time such that is bounded, for bounded B, then
As we know, for the single-valued processes, the continuity of processes provides invariance; while in the multi-valued processes, the upper semi-continuity (defined below) provides negatively invariance of the omega limit sets and the attractor. The following is the definition of the upper semi-continuity of the multi-valued processes (see in ).
Definition 2.5 Let U be a multi-valued process on X. It is said to be upper semi-continuous if for all , the mapping is upper semi-continuous from X into , that is to say, given a converging sequence , for some sequence such that for all n, there exists a subsequence of converging in X to an element of .
Lemma 2.4 
Assume that a multi-valued process U and a family . If, in addition, is -asymptotically compact and upper semi-continuous, then the family is negatively invariant, i.e.,
where B is a bounded set B of X. The family is also negatively invariant and the family defined in Lemma 2.3 is also negatively invariant.
Lemma 2.5 
Given a universe , if a multi-valued process U is -asymptotically compact, then, for any and for any , the omega limit set is a nonempty compact set of X that attracts to at time t in a pullback sense. Indeed, it is the minimal closed set with this property. If, in addition, the multi-valued process U is upper semi-continuous, then is negatively invariant.
From the above lemmas, we obtain the following results which are rather similar to Theorem 3 in . We only sketch it here.
Lemma 2.6 Suppose that is a universe and is pullback -absorbing for a multi-valued process U, which is also -asymptotically compact. Then the results in Lemma 2.5 hold. Furthermore, the family , where , and the following results hold:
For each , the set defined above is compact.
attracts pullback to any .
Suppose U is upper semi-continuous, then is negatively invariant.
Assume , the minimal family of closed sets attracts pullback to elements of .
Assume , each is closed and the universe is inclusion-closed, then and it is the only family of which satisfies the above properties (1), (2) and (3).
If contains the families of fixed bounded sets, then defined in Lemma 2.3 is the minimal pullback attractor of bounded sets, and for each . In addition, if there exists some such that is bounded, then for all .
3 Introduction to some abstract spaces and the existence of solutions
We first recall some notations about the function spaces which will be used later to discuss the regularity of pullback attracting sets. Let us consider the following abstract space:
The symbols H, V denote the closures of in , , respectively. In other words, H= the closure of in with the norm and the inner product , where for ,
V= the closure of in with the norm associated to the inner product , where for ,
We shall use to denote the norm of . The value of a functional from on an element from V is denoted by brackets . It follows that , and the injections are dense and compact.
Define for all , where P is the ortho-projector from onto H. Considering the properties of the operator A, we have as
for every function u, v, , and the operator as
Obviously, is a continuous trilinear form such that
Moreover, b and B satisfy the following:
Now, we make some assumptions. The given delay function ρ satisfies , and there is a constant independent of t satisfying
Moreover, we assume that satisfies the following assumptions:
(H1) is measurable for all .
(H2) For all , is continuous.
(H3) There exist two functions . The above and for all , for , such that , , .
As to the initial datum, we assume
(H4) , and , where .
Next, we shall consider the solution of (1.1).
Definition 3.1 It is said that u is a weak solution to (1.1) if u belongs to for all such that coincides with in and satisfies equation (3.2) in for all .
If u is a solution of (3.2), then it is easy to get , and for all . From the property of the operator A, we have . On the other hand, reasoning as in , we have .
Now, we define a functional , where is a differentiable and nonnegative strictly increasing function given by . We have
From the above analysis, taking into account for all , we obtain that . Hence, it is clear that u is a weak solution to (1.1) if , for all , and satisfies the energy equality
in the distributional sense on .
Theorem 3.1 Suppose that (H1)-(H4) hold. Then there exists a global solution u to (3.2).
Proof We shall prove the result by the Faedo-Galerkin scheme and compactness method. For convenience and without loss of generality, we set the initial time . As to different value τ, we only proceed by translation.
Consider the Hilbert basis of H formed by the eigenfunctions of the above operator A, i.e., . In fact, these elements allow to define the operator , which is the orthogonal projection of H and V in with their respective norms.
Denote , where , , are unknown real functions satisfying the finite-dimensional problem
with . From , we can obtain the local well-posedness of this finite-dimensional delay problem. The following provides estimates which imply that the solutions are well defined in the whole .
By (3.4), we obtain
Multiplying (3.5) by , summing from to , and using the properties of the operator b, we easily get
Observing (H3) and using the Young inequality, we have
Considering the above inequality in (3.6) and observing that
we easily get
From the above inequality and the Gronwall inequality, one has that is bounded in , also in , for any . Observing (3.5), we know that the sequence is bounded in for all . The reason is the same as in .
By the compactness of the injection of V into H, using the above estimates and the Ascoli-Arzelà theorem, we deduce that there exist a subsequence (we relabel the same) and for any with in . Recalling (3.9) and the above analysis, we have
strongly in for all , and
By the properties of operator A and (3.10), we deduce that weakly in . Reasoning as in  on page 76, we deduce that weakly in for any .
On the other hand, observing that by (3.11), (H2) and the hypothesis on ρ in (3.1), for any time t, we conclude for any
and as is bounded in the space obtained from (3.9), by (H3), we deduce
where . Thus we can easily derive from (3.12) and (3.13)
From the above discussion, passing to the limit, we prove that u is a global solution of (3.2) in the sense of Definition 3.1. □
Remark 3.1 We can obtain the uniqueness if there are additional assumptions on the forcing term g. For example, if we suppose that g satisfies (H1), (H3), for an arbitrary and , then for the solutions u, v,
then we obtain the uniqueness of solutions.
4 Existence of pullback attractors
In this section, we discuss the existence of pullback attractors for the 3D Navier-Stokes-Voight equations with delays in continuous and sub-linear operators. At first, we propose the assumptions for g given in Section 3:
(H1′) For all , is measurable.
(H2′) For all , is continuous.
(H3′) There are two nonnegative functions with for some and such that for any ,
To construct a multi-valued process, we introduce symbols and , where as two phase spaces. Let denote the set of global solutions of (1.1) in and the initial datum .
By Theorem 3.1, we know there exists a solution to problem (3.2) although we have no discussion on the uniqueness of solutions to problem (3.2). We may define two strict processes, as
Considering the regularity of the problem, the asymptotic behavior of the two processes shall be the same, as we shall see in what follows.
In order to simplify the calculation form, we introduce a function . For any , we set
From (4.1), we can find that
and for any σ: , then
Lemma 4.1 Suppose that (H1′)-(H3′) hold, for any initial datum and any , it holds
Proof Let u be a solution of (3.2), so . Multiplying (1.1) by and using the energy equality and the Poincaré inequality, we have
where . Thus
where we have denoted
and integrating (4.5) from τ to t, we deduce
where we set
Observing the above estimates, we easily deduce
Applying the Poincaré inequality and the Gronwall inequality to (4.8), we deduce that (4.4) holds. This finishes the proof of this lemma. □
Next, we shall prove that the processes and defined above are pullback-absorbing. To obtain this, we propose the assumptions (H4′)
and the relation among constants , d defined above, and α in (1.1) satisfies(H5′)
where the function is given by (4.1).
Before proving that the two multi-valued processes possess pullback-absorbing sets, we introduce the definition of the two natural tempered universes which shall play the key role for our main purpose.
Definition 4.1 Suppose that is the collection of the sets of all functions such that
Let be the class of all families such that for some . In the same way, let denote the class of all families satisfying for some .
Let be any fixed bounded subset of . Observing that , which is inclusion-closed, by (H4′) and (H5′), we deduce that the family is contained in . With regard to , we use the same method and obtain a similar conclusion if is included in .
The following lemma provides that there exist pullback-absorbing sets for the two processes mentioned above.
Lemma 4.2 Suppose that (H1′)-(H6′) hold and the constants α, d, , satisfy and .
Then, for any and any family , there exits such that any initial datum and any for any satisfy that , where the positive continuous function is given by
Let be included which is given by
Then the set and is -pullback absorbing for the process . Therefore, is -pullback absorbing for the process .
Proof Since the proof is a consequence of the definition of , we only sketch it here. From Lemma 4.1, (H5′) and (H6′) , we have
We complete the proof of (1). Estimate (2) is a consequence of (1). □
For the two processes and , they possess pullback-absorbing sets. In order to apply Lemma 2.3 to obtain the existence of pullback attractors, it is necessary to prove that the two multi-valued processes are asymptotically compact. This will be done in the following lemma.
Lemma 4.3 Suppose that the assumptions in Lemma 4.2 hold. The two processes and are -asymptotically compact.
Proof For any , a sequence with and a sequence with with , we shall prove that the sequence is relatively compact in . By the properties concerning operator b mentioned in Section 3, we deduce that
It is easy to get the above estimate which is independent of n. The sequences of and possess their subsequence, relabeled the same in suitable spaces such that there exist and satisfying
According to the assumptions on a function g and analogously as in Theorem 3.1, we deduce that strongly in V a.e. .
By the Lebesgue theorem and the uniform estimate of in , we deduce that the function converges to the function strongly. Therefore, for any , we have and
The uniform estimate of in implies that the sequence is equicontinuous in for any . In addition, the sequence is bounded, which is independent of n in . Using the Ascoli-Arzelà theorem, we can obtain
From the uniform boundedness of the sequence in , for any , we can also obtain , weakly in V.
By the analogous argument, for any compact sequence and , we obtain that the sequence is convergent to weakly in V. To achieve our result in Lemma 4.3, we only need to prove
The proof is slightly different from Proposition 6 in  or in . We only sketch it here. We use a contradiction argument. Suppose that it is not true, then there would exist a value ε, a sequence (relabeled the same) , and with satisfying for all . We shall see in V. In order to achieve the last claim, because the sequence is weakly convergent to in V, we only need the convergence of the norms above. In other words, as .
From the weak convergence of in V, we get
Therefore we have to check that
From the energy equality, for any , we obtain
where or . For any , define the continuous functions and as
By (4.14), it is clear that J and are non-increasing functions. By the convergence (4.10), for any , we have that . Using the same analysis method as in , we can deduce that for , , which gives (4.13) as desired. □
We can apply the technical method for any family in in Lemma 4.3. Suppose that the assumptions in Lemma 4.3 hold. We can deduce that the processes and are -asymptotically compact and -asymptotically compact.
Lemma 4.4 Suppose that (H1′)-(H6′) hold. The two processes and are semi-continuous and that and have compact values in their respective topologies.
Proof In fact, the upper semi-continuity of the process can be obtained by similar arguments to those used for the Galerkin sequence in Theorem 3.1.
As to the process , applying the same energy-procedure in Lemma 4.3, we shall obtain that in any set of solutions possesses a converging subsequence in this process, whence the assertion in Lemma 4.4 follows. □
According to the results in Section 2, the following two theorems shall be obtained, which are our result in this paper. Observing Lemmas 4.3 and 4.4, and applying Lemma 2.6, we obtain the following theorem.
Theorem 4.1 Suppose that (H1′)-(H6′) hold. For any , then there exist global pullback attractors and for the process in the universe of fixed bounded sets and in , respectively. Moreover, they are unique in the sense of Lemma 2.5 and negatively and strictly invariant for U respectively, and the following holds:
The above theorem proves that there exist pullback attractors in the framework, while we shall prove that there exist pullback attractors in the framework in the following theorem.
Theorem 4.2 Suppose that the assumptions in Theorem 4.1 hold. For any , there exist global pullback attractors and for the process in the universes of fixed bounded sets and in . They are unique in the sense of Lemma 2.5 and negatively and strictly invariant for U, respectively, and we have . Moreover, the relationship between the attractors for and for is as follows:
where is the continuous mapping defined by .
By Theorem 3.1, we can conclude that maps into bounded sets in if , and also maps bounded sets from into bounded sets of .
Noting that is the minimal closed set, and using Lemma 2.5 and the above arguments, we deduce that also attracts bounded sets in in a pullback sense. Therefore the inclusion holds.
As to the opposite inclusion of the first identification in (4.15), for any bounded set B, it follows from the continuous injection and the attractor . Thus , whence the opposite inclusion of the first identification in (4.15) holds.
Analogously, it is obvious that the second relation in (4.15) holds. □
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This work was supported in part by the NNSF of China (No. 11271066, No. 11031003), by the grant of Shanghai Education Commission (No. 13ZZ048), by Chinese Universities Scientific Fund (No. CUSF-DH-D-2013068).
The authors declare that they have no competing interests.
Haiyan carried out the main part of this manuscript. Yuming participated in the discussion and corrected the main theorems. All authors read and approved the final manuscript.