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.
Keywords:3D Navier-Stokes-Voigt equations; pullback attractors; delay terms; multi-valued process
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.
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).
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.
Indeed, it is the minimal closed set with this property.
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.
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 .
Given a universe, if a multi-valued processUis-asymptotically compact, then, for anyand for any, the omega limit setis a nonempty compact set ofXthat attracts toat timetin a pullback sense. Indeed, it is the minimal closed set with this property. If, in addition, the multi-valued processUis upper semi-continuous, thenis 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 2.6Suppose thatis a universe andis pullback-absorbing for a multi-valued processU, which is also-asymptotically compact. Then the results in Lemma 2.5 hold. Furthermore, the family, where, and the following results hold:
(7) Ifcontains the families of fixed bounded sets, thendefined in Lemma 2.3 is the minimal pullback attractor of bounded sets, andfor each. In addition, if there exists somesuch thatis bounded, thenfor 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:
Moreover, b and B satisfy the following:
As to the initial datum, we assume
Next, we shall consider the solution of (1.1).
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 .
Theorem 3.1Suppose 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.
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
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
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 .
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:
Considering the regularity of the problem, the asymptotic behavior of the two processes shall be the same, as we shall see in what follows.
From (4.1), we can find that
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. □
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.
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.
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.
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 .
Therefore we have to check that
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.
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 4.1Suppose that (H1′)-(H6′) hold. For any, then there exist global pullback attractorsandfor the processin 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 forUrespectively, and the following holds:
Theorem 4.2Suppose that the assumptions in Theorem 4.1 hold. For any, there exist global pullback attractorsandfor the processin the universes of fixed bounded sets and in. They are unique in the sense of Lemma 2.5 and negatively and strictly invariant forU, respectively, and we have. Moreover, the relationship between the attractors forand foris as follows:
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. □
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.
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).
Kloeden, P, Valero, J: The weak connectedness of the attainability set of weak solutions of the three-dimensional Navier-Stokes equations. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci.. 463, 1491–1508 (2007). Publisher Full Text
Ladyzhenskaya, O: In memory of A. P. Olskolkov. J. Math. Sci.. 99, 799–801 (2000). Publisher Full Text
Oskolkov, A: Theory of nonstationary flows of Kelvin-Voigt fluids. J. Math. Sci.. 28, 751–785 (1985). Publisher Full Text
Levant, B, Ramos, F, Titi, ES: On the statistical properties of the 3D incompressible Navier-Stokes-Voigt model. Commun. Math. Sci.. 8, 277–293 (2010). Publisher Full Text
Kalantarov, V, Titi, E: Global attractors and determining modes for the 3D Navier-Stokes-Voigt equations. Chin. Ann. Math., Ser. B. 30, 697–714 (2009). Publisher Full Text
García-Luengo, J, Marín-Rubio, P, Real, J: Pullback attractors for three-dimensional non-autonomous Navier-Stokes-Voigt equations. Nonlinearity. 25, 905–930 (2012). Publisher Full Text
Manitius, AZ: Feedback controllers for a wind tunnel model involving a delay: analytical design and numerical simulation. IEEE Trans. Autom. Control. 29, 1058–1068 (1984). Publisher Full Text
Caraballo, T, Real, J: Navier-Stokes equations with delays. R. Soc. Lond. Proc., Ser. A, Math. Phys. Eng. Sci.. 457, 2441–2453 (2001). Publisher Full Text
Caraballo, T, Real, J: Asymptotic behavior for two-dimensional Navier-Stokes equations with delays. R. Soc. Lond. Proc., Ser. A, Math. Phys. Eng. Sci.. 459, 3181–3194 (2003). Publisher Full Text
Caraballo, T, Real, J: Attractors for 2D Navier-Stokes equations with delays. J. Differ. Equ.. 205, 271–297 (2004). Publisher Full Text
Gal, CG, Medjo, TT: A Navier-Stokes-Voigt model with memory. Math. Methods Appl. Sci. (2013). Publisher Full Text
Kapustyan, AV, Valero, J: Weak and strong attractors for the 3D Navier-Stokes system. J. Differ. Equ.. 240, 249–278 (2007). Publisher Full Text
Caraballo, T, Łukaszewicz, G, Real, J: Pullback attractors for asymptotically compact nonautonomous dynamical systems. Nonlinear Anal. TMA. 64, 484–498 (2006). Publisher Full Text
Marín-Rubio, P, Real, J: On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems. Nonlinear Anal. TMA. 71, 3956–3963 (2009). Publisher Full Text
García-Luengo, J, Marín-Rubio, P, Real, J: Pullback attractors in V for non-autonomous 2D-Navier-Stokes equations and their tempered behaviour. J. Differ. Equ.. 252, 4333–4356 (2012). Publisher Full Text